Mental counting: how to learn to count in your mind. Mental counting in mathematics lessons Mental counting in art

Why count in the mind, if you can solve any arithmetic problem on a calculator. modern medicine and psychology prove that mental counting is an exercise for gray cells. Performing such gymnastics is necessary for the development of memory and mathematical abilities.

There are many tricks to simplify mental calculations. Everyone who has seen the famous painting by Bogdanov-Belsky "Mental Account" is always surprised - how do peasant children solve such a difficult task as dividing the sum of five numbers that must first be squared?

It turns out that these children are students of the famous teacher-mathematician Sergei Alexandrovich Rachitsky (he is also depicted in the picture). These are not geeks - students primary school village school of the 19th century. But they all already know how to simplify arithmetic calculations and have learned the multiplication table! Therefore, it is quite possible for these kids to solve such a problem!

Secrets of mental counting

There are methods of oral counting - simple algorithms that it is desirable to bring to automatism. After mastering simple techniques, you can move on to mastering more complex ones.

We add the numbers 7,8,9

To simplify the calculations, the numbers 7,8,9 must first be rounded up to 10, and then subtract the increase. For example, to add 9 to a two-digit number, you must first add 10 and then subtract 1, and so on.

Examples :

Add two digit numbers quickly

If the last digit of a two-digit number is greater than five, round it up. We perform the addition, subtract the “additive” from the resulting amount.

Examples :

54+39=54+40-1=93

26+38=26+40-2=64

If the last digit of a two-digit number is less than five, then add up by digits: first add tens, then ones.

Example :

57+32=57+30+2=89

If the terms are reversed, then you can first round the number 57 to 60, and then subtract 3 from the total:

32+57=32+60-3=89

Adding three-digit numbers in your mind

Quick counting and addition of three-digit numbers - is it possible? Yes. To do this, you need to parse three-digit numbers into hundreds, tens, units and add them one by one.

Example :

249+533=(200+500)+(40+30)+(9+3)=782

Subtraction features: reduction to round numbers

Subtracted are rounded up to 10, up to 100. If you need to subtract a two-digit number, you need to round it up to 100, subtract, and then add an amendment to the remainder. This is true if the correction is small.

Examples :

576-88=576-100+12=488

Mind subtracting three-digit numbers

If at one time the composition of numbers from 1 to 10 was well mastered, then subtraction can be done in parts and in the indicated order: hundreds, tens, units.

Example :

843-596=843-500-90-6=343-90-6=253-6=247

Multiply and Divide

Instantly multiply and divide in your mind? It is possible, but one cannot do without knowledge of the multiplication table. is the golden key to quick mental counting! It applies to both multiplication and division. Recall that in the elementary grades of a village school in the pre-revolutionary Smolensk province (the painting "Mental Counting"), children knew the continuation of the multiplication table - from 11 to 19!

Although in my opinion it is enough to know the table from 1 to 10 in order to be able to multiply larger numbers. For example:

15*16=15*10+(10*6+5*6)=150+60+30=240

Multiply and divide by 4, 6, 8, 9

Having mastered the multiplication table for 2 and 3 to automatism, making the rest of the calculations will be as easy as shelling pears.

For multiplication and division of two- and three-digit numbers, we use simple tricks:

multiplying by 4 is twice multiplying by 2;

to multiply by 6 means to multiply by 2 and then by 3;

multiplying by 8 is three times multiplying by 2;

multiplying by 9 is twice multiplying by 3.

For example :

37*4=(37*2)*2=74*2=148;

412*6=(412*2) 3=824 3=2472

Similarly:

divided by 4 is twice divided by 2;

divide by 6 is first divide by 2 and then by 3;

divided by 8 is three times divided by 2;

Divide by 9 is twice divided by 3.

For example :

412:4=(412:2):2=206:2=103

312:6=(312:2):3=156:3=52

How to multiply and divide by 5

The number 5 is half of 10 (10:2). Therefore, we first multiply by 10, then we divide the result in half.

Example :

326*5=(326*10):2=3260:2=1630

Yet easier rule division by 5. First, multiply by 2, and then divide by 10.

326:5=(326 2):10=652:10=65.2.

Multiply by 9

To multiply a number by 9, it is not necessary to multiply it twice by 3. It is enough to multiply it by 10 and subtract the multiplied number from the resulting number. Compare which is faster:

37*9=(37*3)*3=111*3=333

37*9=37*10 - 37=370-37=333

Also, particular patterns have long been noticed that greatly simplify the multiplication of two-digit numbers by 11 or by 101. So, when multiplied by 11, a two-digit number seems to move apart. The numbers that make it up remain at the edges, and their sum is in the center. For example: 24*11=264. When multiplying by 101, it is enough to attribute the same to a two-digit number. 24*101= 2424. The simplicity and logic of such examples is admirable. Such tasks are very rare - these are entertaining examples, the so-called little tricks.

Counting on fingers

Today you can still meet many defenders of "finger gymnastics" and the method of mental counting on the fingers. We are convinced that learning to add and subtract by bending and unbending fingers is very visual and convenient. The range of such calculations is very limited. As soon as the calculations go beyond one operation, difficulties arise: it is necessary to master the next technique. Yes, and bending your fingers in the era of iPhones is somehow undignified.

For example, in defense of the "finger" technique, the technique of multiplying by 9 is given. The trick of the technique is as follows:

• To multiply any number within the first ten by 9, you need to turn your palms towards you.
• Counting from left to right, bend the finger corresponding to the number being multiplied. For example, to multiply 5 by 9, you need to bend the little finger on your left hand.
• The remaining number of fingers on the left will correspond to tens, on the right - units. In our example - 4 fingers on the left and 5 on the right. Answer: 45.

Yes, indeed, the solution is quick and visual! But this is from the field of tricks. The rule only works when multiplying by 9. Isn't it easier to learn the multiplication table to multiply 5 by 9? This trick will be forgotten, and a well-learned multiplication table will remain forever.

There are also many more similar tricks using fingers for some single mathematical operations, but this is relevant while you use it and is immediately forgotten when you stop using it. Therefore, it is better to learn standard algorithms that will remain for life.

Oral account on the machine

First, you need to know the composition of the number and the multiplication table well.

Secondly, you need to remember the methods of simplifying calculations. As it turned out, there are not so many such mathematical algorithms.

Thirdly, in order for the technique to turn into a convenient skill, it is necessary to constantly conduct brief “brainstorming sessions” - to practice oral calculations using one or another algorithm.

Workouts should be short: mentally solve 3-4 examples using the same technique, then move on to the next one. We must strive to use every free minute - and useful, and not boring. Thanks to simple training, all calculations over time will be done at lightning speed and without errors. This is very useful in life and will help out in difficult situations.

This article was written by me several years ago for a tutoring site. When posting, the site administrator misrepresented not only my last name, but also the purpose of my article. I intended it for schoolchildren, and the administrator of that site redirected it to .... beginner tutors, with the title "What calculations does a math tutor do in her head?" At the same time, the ceiling of the mental calculation indicated by him in his article on this topic is reduced only to calculating in the mind the multiplication of a two-digit number by a single-digit one. He writes: "Let's say it's 29x7. The "soundtrack" from the tutor could be the following:" 29 is twenty and 9. Twenty by 7 will be .... (student answers 14), and 9 by 7 will be .... (student answers 63) One hundred and forty and sixty-three will be ... "" Not only is there an error in this text (Twenty by seven will be 140, not 14) - you need to check, read what is written (!!!), not only is thirty much more convenient multiply by seven and subtract seven, so this technique in the article of that tutor is the only one (????) in the matter of mental counting.
What happens? Are fast mental counting skills superfluous for schoolchildren and only tutors can use them? But no! In my classes, I always welcome when a student strives to count in his mind. Yes, this is usually not taught in school. But as experience shows, every schoolchild can use the skills of quick oral counting if desired. And this in itself is useful, because it allows you to "feel" the numbers and understand how much can be obtained by multiplication, and how much cannot. It is only important to learn to think a little differently than they teach in school. And after all, these techniques can be useful to a student throughout the entire school curriculum, and at exams, where, as you know, it is not allowed to use a calculator.
For example, you need to subtract 9487 from 11531. How do they teach at school? It is necessary to write a column, while constantly occupying, counting the difference. Meanwhile, if you borrow several times, you can easily make a mistake where you borrowed and where you didn’t. And you can calculate it in your mind in a completely different way, without even thinking in a column. It can be seen that in the minuend, the numbers are mostly small, and in the subtrahend, mostly large. Then we consider in this way: How much more is 11531 than 11000? - By 531. How much is 9487 less than 10000? - At 513. Between 11,000 and 10,000 is one thousand.

11531 – 9487 = 11000 + 531 – (10000 – 513) = 11000 – 10000 + 531 + 513 = 2044
This technique is most conveniently remembered with the help of a picture:

Now let's look at a more complicated example - multiplication. How much will 64 * 15 be? What is 15? 15 is 1.5 * 10. How is a number multiplied by 1.5, i.e. for one and a half? To do this, you need to add half of itself to this number. If the example does not include 1.5, but 15, or 150, then a certain number of zeros must also be added to the right. Thus, 64 plus half of this number, that is, we attribute 32 and zero.
That is, 64 + 32 = 96; 96 * 10 = 960.

64 * 15 = 64 * 1,5 * 10 = (64 + 32) * 10 = 960

Now multiply 84 by 25. A similar example, but in this case you can calculate different ways. You can think of 25 as 2.5 * 10. In other words, take 84 twice and add 42 to the result, and then multiply by 10.

84 * 25 = (84 + 84 + 42) * 10 = 2100
And assign zero. And it is possible in another way. 84 * 0.25 * 100. That is, we break 25 into 0.25 and 100. Why do we need this? The fact is that 0.25 is ¼ (one fourth). In other words, we divide 84 by 4, we get 21, and we assign two zeros. It turns out the same 2100:

84 * 25 = 84 * 0,25 * 100 = 84: 4 * 100 = 2100
It may seem that such techniques can hardly be needed at school, which in school curriculum there are only examples of type 29x7. Meanwhile, some textbooks are full of examples that involve the use of fast counting methods, it is only important to be able to recognize these methods. It is important to note in this connection that in the textbooks of the 6th grade there are often tasks "Calculate in the most rational way", and in the textbooks of the next grades such tasks are usually absent. This does not mean that such methods should be forgotten in high school. Here is an example from a real class with an 8th grade student. He met in one task
375 * 48. It would seem that you can multiply three-digit numbers by two-digit numbers only in a column. But the result of multiplying these two numbers is easier to get mentally. What is 375?
- That's 125 * 3. The number 125 is 0.125 * 1000 (one-eighth times a thousand). Therefore, we turn 375 into 0.375 (three eighths) * 1000. We get

48 * 375 = 48 * 0,375 * 1000 = 48 * 3: 8 * 1000 = 48: 8 * 3 * 1000 = 18000
Knowing this technique, all actions are obtained automatically in the mind and the student can be sure that he did not make a mistake anywhere. Whereas when counting in a column, where in fact it is necessary to perform several actions, the probability of error is much greater.
For a quick mental calculation, it’s good to know by heart not only the multiplication table, but also the table of squares, at least up to thirty. Practice shows that this is relatively easy, and there are schoolchildren with such knowledge. In addition, this knowledge sometimes allows not only to square, but also to count in the mind examples like 39 * 26, using the decomposition technique into "known" factors. It is easy to see that 39 is 13 * 3,
and 26 is 13 * 2. Knowing by heart that 13 * 13 = 169, only 169 * 6 remains. 170 * 6 will be 170 * 3 * 2 = 1020 and minus 6, it turns out 1014.

39 * 26 = 3 * 13 * 2 * 13 = 169 * 6 = 170 * 6 – 6 = 1014

By the way, about the table of squares. Yes, the table of squares is published on the flyleaf of textbooks, it is published in collections for preparing for exams, it is allowed to be used in the exam. It turns out that it is not necessary to know the table of squares by heart. However, before the revolution, when there were no calculators and computers, schoolchildren, at least in the Rachinsky school (the artist N.P. Bogdanov-Belsky has a painting "Mental Counting", reminiscent of this), were able to square numbers up to 100 in mind. Not in a column, but in the mind. How did they do it? It would seem that the process is rather time-consuming, even if, for example, the abbreviated multiplication formulas are used. Indeed, let's take, for example, the number 96 and square it using the formula for the square of the sum (90 + 6) 2 . Three terms are obtained, which are sometimes inconvenient to add up. It is even less convenient if we take the formula for the square of the difference (100 - 4) 2 . However, there is a simpler trick, but for now it’s worth making a digression and talking about abbreviated multiplication formulas. Curiously, in the school curriculum, these formulas are used in various sections of mathematics - from algebraic fractions to trigonometric transformations, but not for quick multiplication of numbers. Only with a direct study of the topic are several examples given with the help of these formulas, and such tasks are found at entrance exams to lyceums. Why? Yes, because it is not very convenient to make calculations in the mind using these formulas, and the methods are not universal. Of course, in some cases, these formulas can be used for a quick calculation. This is especially true for the difference of squares formula. Indeed, if you need to multiply 37 by 43, 26 by 32, 35 by 25, etc. (if the difference between the numbers is even), then the difference of squares formula can achieve a quick result, although this again requires knowing the table of squares (37 * 43 \u003d (40 - 3) * (40 + 3) \u003d 1600 - 9 \u003d 1591; 26 * 32 \u003d (29 - 3) * (29 + 3) \u003d 841 - 9 \u003d 832;
35 * 25 = (30 + 5) * (30 - 5) = 900 - 25 = 875). Another way of squaring is more convenient than using abbreviated multiplication formulas. For example, let's take the same number 96 squared.
First, let's look at the rule for quickly squaring numbers ending in 5. For example, 25 squared, 35 squared, 45 squared, 95 squared. The rule is. To do this, multiply the number of tens of the squared number (for example, 9 in 95) by a number that is one more (that is, 10 in the case of 95) and assign 25. It turns out 9025. Let's calculate in this way, for example 85 2:

85 2 = 8 * 9 * 100 + 25 = 7225
(we multiply by 100 because the product 8 * 9 gives us the first two digits of the final result).
I won’t comment on why it turns out this way within the framework of this article, I will only note that this rule also applies to three-digit numbers, which has become common, for example, at the OGE, and in the opposite direction - in the form of extracting an arithmetic square root of a five-digit number ending in...25. In all likelihood, the compilers of the assignments began to take into account that the table of squares published everywhere includes squaring only two-digit numbers, and it is necessary to check schoolchildren with something that goes beyond this table. In fairness, it must be said that in schools, some teachers introduce students to this technique. Although it is usually not said that with its help you can easily get the result of squaring any number from the table. How it's done? Among the numbers that are squared, there is a so-called. "base" numbers. These are, firstly, 10, 20, 30, 40, .... 90 and, secondly, 15, 25, 35 ... 95. These are the numbers that are very easy to square. Now we take the number 96 and square it. To do this, add 95 and 96 to 9025. Add 200 and subtract (5 + 4 are numbers that complement 95 and 96 to 100). We write the result - 9216. Why is that?

96 * 96 = (95 + 1) * 96 = 95 * 96 + 1 * 96 = 95 * (95 + 1) + 1 * 96 = 95 * 95 + 95 + 96 = 9216.
In a similar way, with appropriate training, you can square any number from the table of squares, up to showing tricks of fast counting or phenomenal memory in front of classmates. For those who are still so afraid big numbers, the principle of action can be explained in simple example. 4 squared. This will be 16. Now let's square 5. This will be 25. Knowing 4 squared, the result of the next squared number is obtained by adding the sum of the squared numbers to the previous one. For example, 5 squared is 4 squared + 5 + 4 (i.e. 16 + 9).
A student who has become proficient in applying these methods of rapid mental counting may well come up with his own methods, carefully peering into the numbers and find their own patterns in them. As experience shows, this desire teaches him not to make mistakes in counting, and the search for his own methods instills in him an interest in the subject, allows him to be creative in his study and find something of his own in it. Some schoolchildren tend to show off such skills in front of their classmates, or even completely pro-demon-stri-ro-vat "trick" by counting large numbers in their minds. This is to be welcomed, although not in all schools teachers believe that students can calculate something in their minds, and not on a calculator. In my memory, there is a completely anecdotal case from the series “you can’t think of it on purpose”, when a student in the 5th grade wrote: 22 + 33 = 55. It would seem that what is wrong here? But the teacher crossed it out for him, offering to rewrite the same thing ... in a column. Instead of teaching children to count in their heads, sometimes there are "incredulous" teachers who believe that if the column is not written, it means that the student counted with a calculator.
In individual lessons with a math tutor, it can be useful to pay attention to the study of fast mental counting techniques.

© Alexander Mirov, math tutor, Moscow

This CME is now dedicated to Science What we call mathematics with love. It will help to bring up Such accuracy of thought, To know everything in our life, Measure and calculate. Find the essential. Sum (minus, plus, equality, term, divisor). Geometry (figure, point, properties, theorem, equation). 2. Verification of definitions. Having given a definition to a particular concept, you must be sure that it is true. Correctness can be checked by interchanging the condition and conclusion in the definition. If the sentence remains true when changing places, then the definition is given by us correctly. Check the correctness of the definitions: A square is a quadrilateral. Addition is a mathematical operation. 3. Name a group of numbers in one word: a) 2, 4, 7, 9, 6; 6) 13,18,25,33,48,57. 1. 1. Find the essential. Triangle (plane, vertex, center, side, perpendicular). Difference (subtraction, plus, minus, sum, term). 2. Verification of definitions. The circle is geometric figure. An even number is a natural number. 3. Name a group of numbers in one word: a) 2, 4, 8.12, 44, 56; b) 1, 13.77.83.95. The first letter is in the word "marmot", but it is not in the word "lesson". And then think about it and a short word Among the smart guys you will find from anyone. Take two letters from your mother without embarrassment, But in general you will get the result from addition. The preposition is at my beginning, At the end is a country house. And we all decided the whole Both at the blackboard and at the table. At the beginning of the word - an oral account, Then the consonant sound comes. Hard hair of animals then, But in general we will find the result. Game "Compositor" Mother centipede bought boots for three daughters. How many pairs of boots did Mom have to buy? To find his bride, the prince forced his soldiers to bypass 12 settlements. Each of them had 40 girls. How many girls tried on the shoe in total? How to write the number 100 in five units? The hare had 4 sons and a sweet daughter. One day he brought home a bag of 60 apples. How many apples did each of the rabbits get if the hare divided them equally between them? The brave tailor killed 7 flies with one blow. How many flies did he kill in total if he hit 11? The children went for a walk with their dogs. One grandfather tells them: “Look, guys, don’t lose your heads and don’t break your legs.” One boy said, "We only have 36 legs and 13 heads, so we won't get lost." How many dogs and how many boys? A) One egg is boiled for 10 minutes. How long will it take to cook 2 eggs? B) The hare had 4 sons and a sweet daughter. Once he brought home a bag of 60 apples. How many apples did each of the rabbits get if the hare divided them equally between them. A) When a cat stands on 2 legs, it weighs 5 kg. How much will it weigh if it stands on 4 legs. B) There were 36 jackdaws sitting on three trees. When 6 jackdaws flew from the first tree to the second, and 4 jackdaws flew from the second to the third, then there were equal numbers of jackdaws on all three trees. How many jackdaws originally sat on each tree?

The mental counting process

The process of mental counting can be considered as a counting technology that combines human ideas and skills about numbers, mathematical algorithms of arithmetic.

There are three types mental arithmetic technologies that use different physical abilities person:

• audio motor counting technology;
• visual counting technology.

characteristic feature audiomotor mental counting is to accompany each action and each number with a verbal phrase like "twice two - four." The traditional counting system is precisely the audio-motor technology. The disadvantages of the audio-motor method of conducting calculations are:

• the absence in the memorized phrase of relationships with neighboring results,
• the impossibility to separate tens and units of the product in phrases about the multiplication table without repeating the entire phrase;
• the inability to reverse the phrase from the answer to factors, which is important for performing division with a remainder;
• slow playback speed of a verbal phrase.

Supercomputers, demonstrating high speeds of thinking, use their visual abilities and excellent visual memory. People who are proficient in speed calculations do not use words in the process of solving. arithmetic example in the mind. They show reality visual technology of mental counting, devoid of the main drawback - the slow speed of performing elementary operations with numbers.

Mental arithmetic in elementary school

The development of mental counting skills occupies a special place in elementary school and is one of the main tasks of teaching mathematics at this stage. It is in the first years of training that the main methods of oral calculations are laid, which activate the mental activity of students, develop memory, speech, the ability to perceive what is said by ear in children, increase attention and speed of reaction.

Simulators for mental counting

: Invalid or missing image

This section is missing information. Digital turntable design. The fixed base of the turntable is a plane with drawings of numbers arranged in a T-matrix format of three rows and three columns. A rotating plane (propeller) is superimposed on the base, on which arrows are drawn that suggest answers. The axis of rotation of the propeller coincides with the center of the fixed T-matrix. The only movement available is the rotation of the propeller about an axis. Addition. The principle of operation of a digital turntable is as follows. We write the sum of single-digit numbers A+B= in two digits of tens D and units E. All examples with the same value of the term +B will be called addition sheet. The number of units E of the addition example is shown by an arrow from A to E. This arrow is called sum unit indicator. The arrows on the addition sheet form broken lines lightning. Unit rule. Addition A + B is performed by moving along the arrow-pointer shown on the addition sheet (+B), from the number A to the number E of the sum units. Example 2+1. You will need an addition sheet (+1). Set the chip-label to the number 2 on the T-matrix. We move the chip along the lightning arrow coming out of point 2. The end of the pointer shows the sum 3. Example 7+7. We take the addition sheet (+7). Set the chip-label to the number 7 on the T-matrix. We move the chip along the “step up” arrow on the 7th lightning coming out of point A=7. The end of the pointer shows the units digit E=4. Apply tens rule. If there is an inversion on the unit indicator of the sum A->E, that is, A>E, then the tens digit of the sum D=1 . Let's carry out the following experiment with examples of multiplication by 3 (the third multiplication sheet 3xB=). Imagine that we are in the center of a large telephone T-matrix. Let's show with the left hand the direction from the center to the multiplier B. Let's set aside the right hand, making a right angle with the left hand. Then the right hand will show the number of units E example of 3xB multiplication. So, rule of units when multiplying by 3 formulated in two words: "units on the right"(from radial ray multiplier B). The rotation rule for rays (numbers) on the T-matrix can be considered as mnemonic rule, convenient for memorizing all examples of the 3rd multiplication sheet. If the teacher asks to calculate 3x7, the student will remember the picture of the T-matrix with the necessary rays and will read on it the numbers of the answer, naming the numbers words. However, when geometric calculations words are not needed in the mind, since words appear in the mind of the calculator after the picture, where the answer numbers are already indicated. Simultaneously with the picture that appears in the memory of a person, the number of the result has already been received and realized. It should be noted that the image elements in visual arithmetic are standardized, they can be considered as language of visual images, the sequence of which (corresponding to the algorithm) is equivalent to performing calculations. The pictures that come to mind can be dynamic like in the movies, or static, if both the initial data and the result numbers are shown on the same geometric diagram. One-step algorithms are preferable to multi-step ones. To remember the right picture to get the digits of the answer of an elementary example, a time interval of 0.1-0.3 seconds is required. Note that when solving elementary examples in a geometric way, there is no increase in the load on the psyche. In fact, the geometric account of a trained calculator is automatically a high-speed account. Computer at your fingertips. The indication of radial rays when multiplied by 3 can be performed with the palm of your hand right hand. Set aside the thumb of the right hand, tightly squeezing the rest of the fingers. Let's put the right palm on the center of the T-matrix, pointing the thumb at the multiplier B. Then the remaining fingers of the right hand will show the unit digit E of the product 3xB =). So, multiplication by 3 is implemented on the telephone matrix right hand rule". For example, 3x2=6 . Similarly: the rule of units for multiplying by 7 is left hand rule . The unit rule for multiplying by 9 is finger twine . Other geometric rules for units of multiplication can be shown in diagrams that have radial rays of the T-matrix. In this case, the multiplication of even numbers is performed on an even cross of digits of the T-matrix. A successful simulator are mechanical study guides- digital turntables using a digital telephone matrix. To show the value of the tens of the product AxB, you can use step models multiplication sheets, the appearance and features of which we remember in the same way as the terrain. The height of the hand above the base (floor) shows the value of tens. If the D number is greater than 5, then the bottom of the floor will correspond to D=5, and the upper level of the hand will correspond to 9. Phenomenal Counters The phenomenon of special abilities in mental counting has been around for a long time. As you know, many scientists possessed them, in particular, Andre Ampère and Karl Gauss. However, the ability to quickly count was also inherent in many people whose profession was far from mathematics and science in general. Until the second half of the 20th century, performances by specialists in oral counting were popular on the stage. Sometimes they organized demonstration competitions among themselves, which were also held within the walls of respected educational institutions, including, for example, Lomonosov Moscow State University. Among the well-known Russian "super counters": Among foreign: Although some experts assured that the matter is innate abilities, others convincingly argued the opposite: “the point is not only and not so much in some exceptional, “phenomenal” abilities, but in the knowledge of some mathematical laws, allowing you to quickly make calculations "and willingly disclosed these laws. The truth, as usual, turned out to be on a certain “golden mean” of a combination of natural abilities and their competent, industrious awakening, cultivation and use. Those who, following Trofim Lysenko, rely solely on will and assertiveness, with all the already well-known methods and methods of mental calculation, usually, with all their efforts, do not rise above very, very average achievements. Moreover, persistent attempts to “hard load” the brain with such activities as mental counting, blind chess, etc. can easily lead to overstrain and a noticeable drop in mental performance, memory and well-being (and in the most severe cases, to schizophrenia). On the other hand, gifted people, when using their talents indiscriminately in such an area as mental arithmetic, quickly “burn out” and cease to be able to show bright achievements for a long time and steadily. Oral counting competition Since 2004, the World Mental Computing Championship has been held every two years ( English), which gathers the best living phenomenal counters of the planet. Competitions are held to solve such problems as adding ten 10-digit numbers, multiplying two 8-digit numbers, calculating a given date according to the calendar from 1600 to 2100, the square root of a 6-digit number. The winner in the category "Best Universal Phenomenal Counter" is also determined based on the results of solving six unknown "problems with a surprise". Trachtenberg method Among those practicing mental arithmetic, the book "Quick Counting Systems" by the Zurich professor of mathematics Jacob Trachtenberg is popular. The history of its creation is unusual. In 1941, the Germans threw the future author into a concentration camp. To maintain clarity of mind and survive in these conditions, the scientist began to develop a system of accelerated counting. In four years, he managed to create a coherent system for adults and children, which he later outlined in a book. After the war, the scientist created and headed. Mental arithmetic in art In Russia, the picture of the Russian artist Nikolai Bogdanov-Belsky “Mental Account. In the folk school of S. A. Rachinsky ”, written in 1895. The task given on the board, which the students are thinking about, requires fairly high mental counting skills and ingenuity. Here is her condition: Unable to parse expression (executable file texvc not found; See math/README for setup help.): \frac(10^2 + 11^2 +12^2 + 13^2 + 14^2)(365) The phenomenon of rapid counting of an autistic patient is revealed in the film "Rain Man" by Barry Levinson and in the film "Pi" by Darren Aronofsky. Some methods of oral counting To multiply a number by a single-digit factor (for example, 34×9) orally, you must perform actions, starting with the most significant digit, sequentially adding the results (30×9=270, 4×9=36, 270+36=306) . For effective mental counting, it is useful to know the multiplication table up to 19 * 9. In this case, the multiplication 147*8 is mentally done like this: 147×8=140×8+7×8= 1120 + 56= 1176 . However, without knowing the multiplication table up to 19×9, in practice it is more convenient to calculate all such examples by reducing the multiplier to the base number: 147×8=(150−3)×8=150×8−3×8=1200−24=1176 , and 150×8=(150×2)×4=300×4=1200. If one of the multiplied is decomposed into single-valued factors, it is convenient to perform the action by successively multiplying by these factors, for example, 225×6=225×2×3=450×3=1350 . Also, 225×6=(200+25)×6=200×6+25×6=1200+150=1350 may be simpler. Several ways of oral counting: Multiply by 10. Assign zero to the right: 48 × 10 \u003d 480. Multiplication by 9. In order to multiply a number by 9, you need to add 0 to the multiplicand and subtract the multiplicand from the resulting number, for example 45 × 9=450−45=405. Multiplying by 5 is more convenient like this: first multiply by 10, and then divide by 2 Multiplication by 11 of a two-digit number. Move the numbers N and A apart, enter the sum (N + A) in the middle. e.g. 43×11 = = = 473. When multiplying by 1.5, the multiplied must be divided in half and added to the multiplied, for example 48 × 1.5 = 48/2 + 48 = 72. Can be applied when multiplying by 15 48 × 1.5 × 10 \u003d 720. The squaring of a number of the form (ending in five) is performed according to the scheme: we multiply N by N + 1, write it down in hundreds, and attribute 25 to the right. Formula: × = [ (N×(N+1)) ; 2; five ]. Proof (10N+5) × (10N+5) = (N×(N+1)) x 100 + 25 and assign 25 on the right). see also Write a review on the article "Oral Counting" Notes G. V. Dyudyaeva, N. V. Dolbilova // Teacher - student: problems, searches, finds: Collection of scientific papers. Release 8 [ Patent of the Russian Federation No. 2406160, 2009. Curds V. B. Digital turntables for addition, subtraction, multiplication and integer division using a telephone T-matrix] // A. Leonovich, Science and life, N4 1979 // Victor Pekelis, Technique for youth, N7 1974 Miracle counter // Divo-90. Miracles. Records. Achievements. - Moscow: "Divo", 1991. - S. 54. - 207 p. - 100,000 copies. Miracle counter // Divo 93. Miracles. Records. Achievements. - Moscow: "Divo", 1993. - S. 29. - 191 p. - 100,000 copies. - ISBN 5-87012-008-X.. Miracle counter // Book of records "Lefty". - Moscow: Publishing House "All Russia", 2004. - S. 123. - 336 p. - 4000 copies. // Divo-90. Miracles. Records. Achievements. - Moscow: "Divo", 1991. - S. 54. - 207 p. - 100,000 copies. Man-computer // Divo 93. Miracles. Records. Achievements. - Moscow: "Divo", 1993. - S. 29. - 191 p. - 100,000 copies. - ISBN 5-87012-008-X.. Man-computer // Divo. Miracles. Records. Achievements. - Moscow: "Divo", 1998. - S. 30. - 224 p. - 15,000 copies. - ISBN 5-87012-014-4.. Man-computer // Divo. Miracles. Records. Achievements. - Moscow: "Divo", 2001. - S. 29. - 287 p. - 10,000 copies. - ISBN 5-87012-017-9.. Man-computer // Book of Records "Lefty". - Moscow: Publishing House "All Russia", 2004. - S. 123. - 336 p. - 4000 copies. Man-calendar // Miracle 93. Miracles. Records. Achievements. - Moscow: "Divo", 1993. - S. 29. - 191 p. - 100,000 copies. - ISBN 5-87012-008-X.. Man-calendar // Divo. Miracles. Records. Achievements. - Moscow: "Divo", 1998. - S. 30-31. - 224 p. - 15,000 copies. - ISBN 5-87012-014-4.. Calendar in the head // Divo. Miracles. Records. Achievements. - Moscow: "Divo", 2001. - S. 29-30. - 287 p. - 10,000 copies. - ISBN 5-87012-017-9.. Calendar in the head // Divo. Miracles. Records. Achievements. - Moscow: "Divo", 2005. - S. 28-29. - 208 p. - ISBN 5-87012-023-3.. Man-calendar // Book of Records "Lefty". - Moscow: Publishing House "All Russia", 2004. - S. 123. - 336 p. - 4000 copies. (English) (English) “I think that Com. Goldstein D. N. is a calculator of the highest brand ... His work is based solely on memory and innate abilities. I am very pleased that my business has found in him a fairly well-deserved successor.” R. S. Arrago, Moscow, 5. 11. 1929 (est.) Alexander Khavronin. (Russian), Radio Liberty(December 8, 2006). Retrieved 29 September 2012. J. Trachtenberg "Quick Counting Systems" Perelman Ya. I. Quick account. Thirty simple methods of mental counting. (English) . Literature Bantova M. A. The system of formation of computational skills. //Begin. school - 1993.-№ 11.-p. 38-43. Beloshistaya A.V. Reception of the formation of oral computing skills within 100 // Primary School. - 2001.- № 7 Berman G. N. Account Receptions, ed. 6th, Moscow: Fizmatgiz, 1959. Borotbenko E I. Control of skills of oral calculations. //Begin. school - 1972. - No. 7. - p. 32-34. Vozdvizhensky A. Mental Computing. Rules and simplified examples of actions with numbers. - 1908. Volkova S., Moro M. I. Addition and subtraction multi-digit numbers. //Begin. school - 1998.-№ 8.-p.46-50 Voskresensky M.P. Abbreviated calculation methods: Integers. - M.: Tipo-lit. W. Richter, 1905.- 39 p. Wroblewski. How to learn to count easily and quickly. - M.-1932.-132s. Goldstein D.N. Simplified Computing Course. M.: State. educational-ped. ed., 1931. Goldstein D.N. Technique of fast calculations. M.: Uchpedgiz, 1948. Gonchar D. R. Oral Counting and Memory: Riddles, Developmental Techniques, Games // In Sat. Oral counting and memory. Donetsk: Stalker, 1997. ISBN 966-596-057-7. Demidova T. E., Tonkikh A. P. Rational computing techniques in primary course Mathematics // Elementary School. - 2002. - No. 2. - S. 94-103. Cutler E. McShane R. Trachtenberg fast counting system. - M.: Uchpedgiz. - 1967. -150s. Lipatnikova I. G. The role of oral exercises in mathematics lessons // Primary school. - 1998. - No. 2. Martel F. Quick counting tricks. - Pb. −1913. −34s. Martynov I.I. Mental arithmetic is for a schoolboy what scales are for a musician. // Primary School. - 2003. - No. 10. - S. 59-61. Melentiev P.V."Fast and verbal calculations." Moscow: Gostekhizdat, 1930. Perelman Ya. I. Quick account. L .: Soyuzpechat, 1945. Pekelis V.D. Your opportunities, man!. - 4th, revised. and additional - Moscow: Knowledge, 1984. - 272 p. - 200,000 copies. Robert Toque"2 + 2 = 4" (1957) (English edition: The Magic of Numbers (1960)). Sorokin A. S. Counting technique. M.: "Knowledge", 1976. Sukhorukova A. F. More emphasis on verbal calculations. //Begin. school - 1975.-No. 10.-p. 59-62. Curd V. B. Visual arithmetic and fast counting technology. M.: Book 1: Fundamentals. "Librocom", 2011. - 208 p. ISBN 978-5-397-01928-6. Faddeycheva T. I. Teaching Oral Computing // Elementary School. - 2003. - No. 10. Faermark D.S."The task came from the picture." M.: "Science". Links V. Pekelis.// Technique-youth, No. 7, 1974 S. Trankovsky.// Science and Life, No. 7, 2006. S. A. Rachinsky. Perelman Ya. I. L., 1941, 12 p. (in Math.Ru library) . from the painting “Oral Account. In the folk school of S. A. Rachinsky. An excerpt describing mental counting - Go to Italy, my friend, they will wait for you there. Just don't be too long! I'll be waiting for you too... – the Queen said with a gentle smile. Axel fell with a long kiss to her graceful hand, and when he raised his eyes, there was so much love and anxiety in them that the poor queen, unable to stand it, exclaimed: "Oh, don't worry, my friend! I am so well protected here that even if I wanted to, nothing could happen to me! Ride with God and come back soon... Axel looked at her beautiful face so dear to him for a long time, as if absorbing every line and trying to keep this moment in his heart forever, and then he bowed low to her and quickly walked along the path to the exit, without turning around and not stopping, as if afraid that if he turns around, he simply won’t have enough strength to leave ... And she saw him off with her huge blue eyes suddenly moistened, in which the deepest sadness lurked ... She was a queen and had no right to love him. But she was also just a woman whose heart entirely belonged to this purest, brave man forever ... without asking anyone for permission ... "Oh, how sad, isn't it?" Stella whispered softly. How I wish I could help them! – Do they need any help? I was surprised. Stella just nodded her curly head, without saying a word, and again began to show a new episode ... I was very surprised by her deep participation in this charming story, which so far seemed to me just a very sweet story of someone's love. But since I already knew quite well the responsiveness and kindness of Stella's big heart, somewhere in the depths of my soul I was almost sure that everything would certainly not be as simple as it seems at first, and I could only wait ... We saw the same park, but I had no idea how much time had passed there since we saw them in the last "episode". That evening, the whole park literally shone and shimmered with thousands of colored lights, which, merging with the shimmering night sky, formed a magnificent continuous sparkling fireworks. In terms of the splendor of the preparation, it was probably some kind of grandiose party, during which all the guests, at the bizarre desire of the queen, were dressed exclusively in white clothes and, somewhat reminiscent of the ancient priests, “organized” walked through the wonderfully lit, sparkling park, heading towards the beautiful stone gazebo, called by everyone - the Temple of Love. Temple of Love, vintage engraving And suddenly, behind the same temple, a fire broke out... Blinding sparks soared to the very tops of the trees, staining the dark night clouds with a bloody light. The admiring guests gasped in unison, approving the beauty of what was happening... But none of them knew that, according to the Queen's plan, this raging fire expressed all the power of her love... And only one person who was present at that evening understood the true meaning of this symbol. holiday... Excited, Axel leaned against the tree and closed his eyes. He still couldn't believe that all this stunning beauty was meant for him. Are you satisfied, my friend? whispered a soft voice behind him. - I'm delighted ... - Axel answered and turned around: it was, of course, she. Only for a moment they looked at each other with rapture, then the queen gently squeezed Axel's hand and disappeared into the night... Why has he always been so miserable in all his "lives"? - Stella was still sad for our “poor boy”. To be honest, I haven't seen any "unhappiness" yet, so I looked at her sad face in surprise. But for some reason, the little girl stubbornly refused to explain anything further ... The picture has changed dramatically. A luxurious, very large green carriage was rushing along the dark night road with might and main. Axel sat in the coachman's seat and, rather skillfully driving this huge carriage, looked back and looked around with obvious anxiety from time to time. It seemed that he was in a wild hurry somewhere or was running away from someone ... Inside the carriage sat the already familiar king and queen, and another pretty girl about eight years old, as well as two ladies still unknown to us. Everyone looked gloomy and agitated, and even the baby was quiet, as if she felt the general mood of adults. The king was dressed surprisingly modestly - in a simple gray frock coat, with the same gray round hat on his head, and the queen hid her face under a veil, and it was clear that she was clearly afraid of something. Again, this whole scene was very much like an escape ... Just in case, I looked again in Stella's direction, hoping for an explanation, but no explanation followed - the little girl watched what was happening very intently, and in her huge puppet eyes there was a deep sadness that was not childish at all. – Well, why?.. Why didn’t they listen to him?!.. It was so simple!.. – she was suddenly indignant. The carriage rushed all this time at almost insane speed. The passengers looked tired and somehow lost... Finally, they drove into some large unlit courtyard, with a black shadow of a stone building in the middle, and the carriage stopped abruptly. The place looked like an inn or a large farm. Axel jumped down to the ground and, approaching the window, was about to say something, when suddenly an imperious voice was heard from inside the carriage. male voice: “Here we will say goodbye, Count. It is unworthy of me to endanger you further. Axel, of course, who did not dare to object to the king, only had time, in parting, to briefly touch the queen's hand ... The carriage jerked ... and literally a second later disappeared into the darkness. And he was left standing alone in the middle of the dark road, wishing with all his heart to rush after them... Axel "inside" felt that he could not, had no right to leave everything to the mercy of fate! He just knew that without him, something would definitely go awry, and everything that he had organized so carefully and for so long would completely fail due to some ridiculous accident ... The carriage had not been seen for a long time, and poor Axel was still standing and looking after them, clenching his fists with all his might in hopelessness. Angry masculine tears rolled sparingly down his deathly-pale face... – This is the end already... I know, this is the end already... – he said quietly. “Will something happen to them?” Why are they running away? – not understanding what is happening, I asked. - Oh, yes! .. They will be caught very bad people and put in jail... even a boy. “Where do you see the boy here?” I was surprised. So he's just dressed up as a girl! Don't you understand? I shook my head. So far, I still didn’t understand almost anything here at all - neither about the royal escape, nor about the “bad people”, but I decided to just look further, without asking anything else. “These bad people offended the king and queen and wanted to capture them. So they tried to run. Axel arranged everything for them... But when he was ordered to leave them, the carriage went slower, because the king was tired. He even got out of the carriage to "breathe the air"... and that's where they recognized him. Well, they got it, of course... Pogrom at Versailles Arrest of the royal family Fear of what is happening... Seeing Marie Antoinette to the Temple Stella sighed... and again threw us into another "new episode" of this not so happy, but still beautiful story... This time everything looked sinister and even frightening. We found ourselves in some kind of dark, unpleasant room, as if it were a real evil prison. In a tiny, dirty, damp and stinking room, on a wooden couch with a straw mattress, sat exhausted by suffering, dressed in black, a thin gray-haired woman in whom it was completely impossible to recognize that fabulously beautiful, always smiling miracle queen, whom young Axel most of all loved in the world... Marie Antoinette at the Temple He was in the same room, completely shocked by what he saw and, not noticing anything around, stood on his knees, pressing his lips to her still beautiful, white hand, unable to utter a word ... He came to her completely desperate, having tried everything in the world and having lost the last hope of saving her ... and yet, again, he offered his almost impossible help ... He was obsessed with the only desire: to save her, no matter what ... He simply could not let her to die ... Because without her, his life, already unnecessary to him, would end ... They looked silently at each other, trying to hide the naughty tears that flowed down their cheeks in narrow paths ... Unable to take their eyes off each other, because they knew that if he failed to help her, this look could be the last for them .. . The bald jailer looked at the heartbroken guest and, not intending to turn away, watched with interest the sad scene of someone else's sorrow unfolding before him ... The vision disappeared and another one appeared, no better than the previous one - a terrible, screaming, armed with pikes, knives and guns, a brutal crowd mercilessly destroyed the magnificent palace ... Versailles... Then Axel reappeared. Only this time he was standing at the window in some very beautiful, richly furnished room. And next to him stood the same "friend of his childhood" Margarita, whom we saw with him at the very beginning. Only this time, all her arrogant coldness evaporated somewhere, and her beautiful face literally breathed with participation and pain. Axel was deathly pale and, pressing his forehead against the window glass, watched with horror something happening on the street ... He heard the crowd rustling outside the window, and in a terrifying trance he loudly repeated the same words: – My soul, I never saved you... Forgive me, my poor... Help her, give her the strength to endure this, Lord!... – Axel, please! .. You must pull yourself together for her sake. Well, please be reasonable! - with participation persuaded his old girlfriend. – Prudence? What kind of prudence are you talking about, Margarita, when the whole world has gone crazy?! .. - Axel shouted. - What is it for? For what?.. What did she do to them?!. Margarita unfolded some small piece of paper and, apparently not knowing how to calm him down, said: - Calm down, dear Axel, now listen better: “I love you, my friend... Don't worry about me. I miss only your letters. Perhaps we are not destined to meet again ... Farewell, the most beloved and most loving of people ... ". This was the last letter from the Queen, which Axel read thousands of times, but for some reason it sounded even more painful from someone else's lips... - What is it? What is going on there? - I could not stand it. – This beautiful queen is dying… She is being executed now. Stella replied sadly. Why can't we see? I asked again. “Oh, you don’t want to watch this, trust me. The little girl shook her head. - It's a pity, she's so unhappy... How unfair it is. “I would still like to see…” I asked. “Well, look…” Stella nodded sadly. On a huge square, chock-full of “strung-up” people, a scaffold ominously towered in the middle ... A deathly-pale, very thin and exhausted woman dressed in white proudly climbed it along the small, crooked steps. Her short-cropped blond hair was almost completely hidden by a modest white cap, and in her tired eyes, reddened from tears or insomnia, deep hopeless sadness was reflected ... Swaying slightly, because, due to her hands tightly tied behind her back, it was difficult for her to keep her balance, the woman somehow climbed onto the platform, still, from last strength trying to stand straight and proud. She stood and looked into the crowd, not lowering her eyes and not showing how truly she was terribly scared ... And there was no one around whose friendly look could warm the last minutes of her life ... No one who warmth could help her endure this terrifying moment when her life had to leave her in such a cruel way ... Before that, the raging, excited crowd suddenly suddenly fell silent, as if it ran into an insurmountable obstacle ... The women standing in the front rows silently cried. A slender figure on the scaffold approached the chopping block and stumbled a little and fell painfully to her knees. For a few short seconds, she raised her exhausted, but already pacified by the nearness of death face to the sky ... took a deep breath ... and proudly looked at the executioner, laid her tired head on the chopping block. The crying became louder, the women closed their children's eyes. The executioner approached the guillotine.... - God! Not!!! Axel screamed heartbreakingly. At that very moment, in the gray sky, the sun suddenly peeked out from behind the clouds, as if illuminating the last path of the unfortunate victim ... It gently touched her pale, terribly emaciated cheek, as if affectionately saying the last earthly "forgive me." There was a bright flash on the scaffold - a heavy knife fell, scattering bright scarlet splashes ... The crowd gasped. The blond head fell into the basket, it was all over... The beautiful queen went to where there was no more pain, no more bullying... There was only peace... There was deadly silence all around. There was nothing else to see... So the gentle and kind queen died, who until the very last minute managed to stand with her head held high, which was then so simply and ruthlessly torn down by the heavy knife of the bloody guillotine... Pale, frozen like a dead man, Axel looked through the window with unseeing eyes and it seemed that life was flowing out of him drop by drop, painfully slowly ... Carrying his soul far, far away, so that there, in light and silence, forever merge with the one whom he loved so deeply and selflessly... “My poor... My soul... How could I not die with you?.. Everything is over for me now...” Axel whispered with dead lips, still standing at the window. But everything will be “finished” for him much later, after some twenty long years, and this end will, again, be no less terrible than that of his unforgettable queen ... - Do you want to look further? Stella asked softly. I just nodded, unable to say a word. We already saw another, raging, brutalized crowd of people, and in front of it stood the same Axel, only this time the action took place many years later. He was still as handsome, only now almost completely gray-haired, in some kind of magnificent, very highly significant, military uniform He still looked fit and lean. And here, the same brilliant, smartest person stood in front of some half-drunk, brutalized people and, hopelessly trying to outshout them, tried to explain something to them... But, unfortunately, none of those present wanted to listen to him... swearing, fueling her anger, she began to press. He tried to fight them off, but they threw him to the ground, they began to brutally trample on his feet, tear off his clothes ... And some big man suddenly jumped on his chest, breaking his ribs, and without hesitation, easily killed him with a kick to the temple. Axel's naked, mutilated body was dumped on the side of the road, and there was no one who at that moment would have wanted to take pity on him, already dead ... There was only a rather laughing, drunken, excited crowd around ... who just needed to splash out on someone - something of his accumulated animal anger ... Axel's pure, suffering soul, finally freed, flew away to unite with the one that was his bright and only love, and had been waiting for him for so many long years... So, again, very cruelly, an almost unfamiliar man ended his life with Stella, but who became so close, a man named Axel, and ... the same little boy who, having lived only some short five years, managed to accomplish an amazing and only feat in his life, which any adult living on earth could be honestly proud of ... - What a horror! .. - I whispered in shock. - Why is it so? “I don’t know…” Stella whispered softly. “For some reason, people were very angry then, even worse than animals ... I looked a lot to understand, but I didn’t understand ...” the little girl shook her head. “They didn’t listen to reason, they just killed. And for some reason everything beautiful was also destroyed ... - And what about Axel's children or his wife? I asked, recovering from the shock. “He never had a wife - he always loved only his queen,” said little Stella with tears in her eyes. And then, suddenly, a flash seemed to flash in my head - I realized who Stella and I had just seen and for whom we were so worried from the bottom of our hearts! ... It was the French queen, Marie Antoinette, whose tragic life we ​​recently (and very briefly!) took place in a history lesson, and our history teacher strongly approved of the execution of which, considering such a terrible end to be very “correct and instructive” ... apparently because he taught us “Communism” mainly in history .. . Despite the sadness of what happened, my soul rejoiced! I just could not believe in the unexpected happiness that fell on me! .. After all, I had been waiting for this for so long! I almost squealed from the puppyish delight that seized me! .. Of course, I was so happy not because I did not believe in what was constantly happening to me. On the contrary, I always knew that everything that happened to me was real. But apparently to me, like to anyone ordinary person, and especially for a child, nevertheless, sometimes some, at least the simplest, confirmation was needed that I was not going crazy yet, and that now I can prove to myself that everything that happens to me is not just my sick fantasy or fiction, but real fact described or seen by other people. Therefore, such a discovery was a real holiday for me! .. I already knew in advance that as soon as I returned home, I would immediately rush to city ​​library to collect everything I can find about the unfortunate Marie Antoinette and will not rest until I find at least something, at least some fact that matches our visions ... I found, unfortunately, only two tiny books, which described not so many facts, but this was quite enough, because they fully confirmed the accuracy of what I had seen from Stella. Here is what I was able to find then: the queen's favorite person was a Swedish count named Axel Fersen, who selflessly loved her all his life and never married after her death; their parting before the departure of the count to Italy took place in the garden of the Petit Trianon - Marie Antoinette's favorite place - the description of which exactly coincided with what we saw; a ball in honor of the arrival of the Swedish king Gustav, held on June 21, where all the guests for some reason were dressed in white; an escape attempt in a green carriage organized by Axel (all the other six escape attempts were also organized by Axel, but none of them, for one reason or another, failed. True, two of them failed at the request of Marie Antoinette herself, since the queen did not wanted to run away alone, leaving her children behind); the beheading of the queen took place in complete silence, instead of the expected "happy rampage" of the crowd; a few seconds before the executioner's blow, the sun suddenly came out... The Queen's last letter to Count Fersen is reproduced almost exactly in the book "Memoirs of Count Fersen", and it almost exactly repeated what we heard, with the exception of just a few words. Already these small details were enough for me to rush into battle with tenfold strength! .. But that was only later ... And then, in order not to seem ridiculous or heartless, I tried my best to pull myself together and hide my delight about my wonderful “ insights." And to dispel Stellino's sad mood, she asked: - Do you really like the queen? - Oh yeah! She is kind and so beautiful ... And our poor "boy", he suffered so much here too ... I felt very sorry for this sensitive, sweet little girl, who, even in her death, was so worried about these people, completely alien and almost unfamiliar to her, as many do not worry about their own relatives ... – Perhaps there is some share of wisdom in suffering, without which we would not understand how precious our life is? I said uncertainly. - Here! Grandma says this too! - the girl was delighted. “But if people want only good, then why should they suffer? - Maybe because without pain and trials, even the most the best people would not truly understand the same goodness? I joked. But for some reason, Stella did not take it at all as a joke, but said very seriously: - Yes, I think you're right ... Do you want to see what happened to Harold's son next? she said more cheerfully. “Oh no, no more! I pleaded. Stella laughed happily. – Do not be afraid, this time there will be no trouble, because he is still alive! How is it alive? I was surprised. Immediately, a new vision appeared again and, continuing to surprise me unspeakably, it already turned out to be our century (!), And even our time ... A gray-haired, very pleasant man was sitting at the desk and thinking about something intently. The whole room was literally crammed with books; they were everywhere - on the table, on the floor, on the shelves, and even on the windowsill. A huge fluffy cat sat on a small sofa and, not paying any attention to the owner, concentratedly washed his face with a large, very soft paw. The whole atmosphere created an impression of "scholarship" and comfort.

In system subjects mathematics belongs special role. It equips students with the necessary knowledge, skills and abilities that are used in the study of other school subjects, especially in the study of geometry, algebra, physics and computer science. When studying this subject, students require a lot of strong-willed and mental efforts, developed imagination, concentration of attention, mathematics develops the personality of the student. In addition, the study of mathematics significantly contributes to the development logical thinking and broaden the horizons of students.

Mathematics is one of the most important sciences on earth and it is with it that a person meets every day in his life. That is why the teacher needs to develop in children an interest in this science, subject. In my opinion, it is possible to develop a cognitive interest in mathematics by using various kinds oral counting, and involving students in the preparation and conduct of this stage of the lesson and the lesson as a whole.

Oral counting in mathematics lessons can be represented by various forms of work with the class, students (mathematical, arithmetic and graphic dictations, mathematical lotto, rebuses, crossword puzzles, tests, conversations, surveys, warm-ups, "circular" examples and much more). It includes algebraic and geometric material, solving simple problems and tasks with ingenuity, discusses the properties of actions on numbers and quantities, and other issues, with the help of mental counting, you can create a problem situation, etc.

Oral counting is not a random stage of the lesson, it is in a methodical connection with the main topic and is of a problematic nature.

To achieve the correctness and fluency of oral calculations in each lesson of mathematics, 5-10 minutes are allotted for exercises in oral calculations.
Oral counting activates the mental activity of students. When they are performed, memory, speech, attention, the ability to perceive what is said by ear, speed of reaction are activated, develop.

This stage is an integral part in the structure of the mathematics lesson. It helps the teacher, firstly, to switch the student from one activity to another, secondly, to prepare students for studying a new topic, thirdly, tasks for repeating and summarizing the material covered can be included in the oral account, fourthly, it increases the intelligence of the students.

Goals At this stage of the lesson, you can define the following:

1) achievement of the set goals of the lesson;
2) development of computational skills;
3) development of mathematical culture, speech;
4) the ability to generalize and systematize, to transfer the acquired knowledge to new tasks.

Since oral exercises or mental counting is a stage of the lesson, it has its own tasks:

1. Reproduction and correction of certain knowledge, skills and abilities of students necessary for their independent activity in the lesson or conscious perception of the teacher's explanation.
2. The teacher's control over the state of knowledge of students.
3. Psychological preparation of students for the perception of new material.
4. Increasing cognitive interest.

When conducting oral counting, each teacher adheres to the following requirements:

• Exercises for mental counting are not chosen randomly, but purposefully.
• Tasks should be varied, the proposed tasks should not be easy, but they should not be “cumbersome”.
• Texts of exercises, drawings and notes, if required, should be prepared in advance.
• All students should be involved in oral counting.
• When conducting an oral account, evaluation criteria (encouragement) should be thought out.

An oral account can be built in the following form:

• Tasks for the development and improvement of attention. Such as: find a pattern and solve an example, continue the series.
• Tasks for development of perception, spatial imagination. For example, draw an ornament, a pattern; count how many lines.
• Tasks for the development of observation (find a pattern, what is superfluous?)
• Oral exercises using didactic games.

Oral calculation skills are formed in the process of students performing various exercises. Consider their main types:

1) Finding the values ​​of mathematical expressions.

A mathematical expression is proposed in one form or another, it is required to find its value. These exercises have many variations. You can offer numerical mathematical expressions and alphabetic ones (an expression with a variable), while the letters are assigned numerical values ​​and the numerical value of the resulting expression is found.

2) Comparison of mathematical expressions.

These exercises have a number of variations. Two expressions can be given, but it is necessary to establish whether their values ​​are equal, and if not equal, then which of them is greater or less.
Exercises may be offered in which the relation sign and one of the expressions are already given, and the other expression must be composed or supplemented: 8 (10 + 2) \u003d 8 10 + ...
The expressions of such exercises can include various numerical material: one-digit, two-digit, three-digit numbers and quantities. Expressions can have different actions.

The main role of such exercises is to promote the assimilation of theoretical knowledge about arithmetic operations, their properties, equalities, inequalities, etc. They also help develop computational skills.

3) Solution of equations.

These are, first of all, the simplest equations (x + 2 \u003d 10) and more complex ones (15 x - 9 \u003d 51)

The equation can be presented in different forms:

• From what number must 18 be subtracted to get 40?
• solution of the equation x 8 = 72;
• find the unknown number: 77 + x = 77 + 25
• Nikolai thought of a number, multiplied it by 5 and got 125. What number did Nikolai think?

The purpose of such exercises is to develop the ability to solve an equation, to help students learn the connections between the components and the results of arithmetic operations.

4) Problem solving.

Both simple and compound tasks are offered for oral work.

These exercises are included in order to develop the ability to solve problems, they help the assimilation of theoretical knowledge and the development of computational skills.
A variety of exercises excites interest in children, activates their mental activity.

Forms of perception of oral counting

1) Fluent auditory (read by a teacher, student, audio recording) - when perceiving a task by ear, a large load falls on memory, so students quickly get tired. However, such exercises are very useful: they develop auditory memory.

2) Visual (tables, posters, cards, notes on the blackboard, computer) - writing down the task makes calculations easier (no need to memorize numbers). Sometimes without a record it is difficult and even impossible to complete the task. For example, you need to perform an action with values ​​expressed in units of two names, fill in a table, or perform actions when comparing expressions.

3) Combined.

• feedback (showing answers using cards, mutual checking, guessing keywords, checking using the Microsoft Power Point computer program).
• assignments by options (ensure independence).
• exercises in the form of a game (“Dialogue”, “Math duel”, “Magic squares”, “Labyrinth of factors”, “Quiz”, “Magic number”, “Individual lotto”, “Best counter”, “Code exercises”, “Chip ”, “Who is faster”, “Flower, sun”, “Numerical mill”, “Numerical fireworks”, “Mathematical phenomenon”, “Silence”, “Mathematical relay race”). The ways and forms of using the listed games in mathematics lessons are considered in the work of V.P. Kovalenko “ Didactic games in math class."

Organization of classes on oral counting

When preparing for the lesson, the teacher must clearly define (based on the objectives of the lesson) the scope and content of oral assignments. If the purpose of the lesson is to present a new topic, then at the beginning of the lesson, you can make oral calculations on the material covered, you can also organize the work so that there is a smooth transition to a new topic. After presenting a new topic, it is appropriate to offer students oral tasks to develop skills and abilities on this topic. If the purpose of the lesson is repetition, then both the teacher and the students should prepare for oral calculations in the classroom. Students, with the advice of a teacher, can do their own counting in each lesson.
Oral counting can be combined with checking homework, consolidating the studied material, offered during a survey, and also specially set aside 5-7 minutes in a lesson for oral counting. The material for this can be selected from the textbook of special collections, mathematical encyclopedias or books, you can invite students to come up with tasks themselves.
Oral exercises should correspond to the topic and purpose of the lesson and help to assimilate the material studied in this lesson or previously covered. Depending on this, the teacher determines the place of oral counting in the lesson. If oral exercises are intended to repeat the material, the formation of computational skills and prepare for the study of new material, then it is better to conduct them at the beginning of the lesson before learning new material. If oral exercises are aimed at consolidating what has been learned in this lesson, then it is necessary to conduct an oral count after studying new material.
When selecting exercises for a lesson, it should be borne in mind that the preparatory exercises and the first exercises for consolidation, as a rule, should be simpler and more straightforward. Here it is unnecessary to strive for a special variety in the formulations and methods of work. Exercises for developing knowledge and skills, and especially for applying them in various conditions, on the contrary, should be more uniform. The wording of tasks, if possible, should be designed so that they are easily perceived by ear. To do this, they must be clear and concise, formulated easily and definitely, and not allow for different interpretations.
In addition to the fact that oral counting in mathematics lessons contributes to the development and formation of strong computational skills and abilities, it also plays an important role in instilling and increasing children's cognitive interest in mathematics lessons, as one of the most important motives for educational and cognitive activity, the development of logical thinking, and development of personal qualities of the child. In my opinion, by arousing interest and instilling love for mathematics with the help of various types of oral exercises, the teacher will help students to actively work with educational material, awaken in them the desire to improve methods of computing and solving problems, replacing less rational ones with more perfect ones. And this is the most important condition for the conscious assimilation of the material.
If the student likes the subject, then he will always with interest, enthusiastically master more and more knowledge, and an increase in interest in mathematics lessons can be achieved as follows:

1) Enriching the content with material on the history of science, which is often found on the pages of the textbook.
2) Solving problems of increased difficulty and non-standard problems. Selection of tasks is carried out from workbooks, didactic materials.
3) Emphasizing strength and grace, rationality of methods of calculations, proofs, transformations and research.
4) A variety of lessons, their non-standard construction, the inclusion in the lessons of elements that give each lesson a unique character, solution problem situations, the use of technical teaching aids (interactive whiteboard, computer, etc.), visual aids, a variety of oral counting.
5) Activation of the cognitive activity of students in the classroom using forms of independent and creative work.
6) Using various forms of feedback: systematic surveys, short-term oral and written tests, various tests, mathematical dictations, tests, along with tests provided for by the plan.
7) Variety homework. For example, invite students to write a fairy tale about a geometric figure, a poem about fractions, degrees.
8) Establishment of internal and interdisciplinary connections, showing and explaining the application of mathematics in life and in production.

For example, when studying triangles, you can tell that triangles are used in the game of billiards, bowling; during the construction of iron structures (Shukhov tower on Shabolovka); railway bridges; high-voltage power lines; introduce the legends of the Bermuda Triangle, the triangle of Pascal, Penrose and much more.

Students like to take part in the preparation for the lesson, therefore, in addition to homework, if desired, you can give the task to independently prepare an oral account for the lesson in accordance with the topic, and conduct it yourself in the next lesson (to be a teacher). You can also give the task to students to prepare an essay, report, come up with a puzzle, rebus, game (see. Attachment 1 ).

The children are very responsible and diligently prepare and conduct oral work in the classroom. When completing this task, they put in a lot of effort, since you need to come up with such tasks so that the class is interested, so that the tasks correspond to the topic of the lesson.

Saturation of lessons with varied, entertaining and useful computational tasks with a high density of current theoretical material on the topics studied is possible only through the improvement of the system of oral exercises in the classroom. This will allow, first of all, to teach students to learn, to delve into the meaning of what is being studied at every step of learning so as to be able to independently solve emerging problems.
This gives them self-confidence and encourages them to improve their results, the children begin to work actively in the lesson and they begin to like this subject.
It is also important to note the following, that primary and secondary school students quickly count, calculate mentally, orally, but for some reason in the upper grades, verbal counting is done using a calculator or with great difficulty without a calculator. It seems to me that we need to strive to ensure that this does not happen. And this, of course, can be achieved with the use of oral counting as an important and necessary element of the lesson.
Oral counting, as a mandatory stage of the lesson, should be carried out in mathematics lessons both in elementary grades and in middle and high grades.

Bibliography:

1. Berimets V.I."The use of various types of oral exercises as a means of increasing cognitive interest in the lesson of mathematics."
2. V. P. Kovalenko"Didactic games in mathematics lessons".
3. Zaitseva O.P. The role of mental counting in the formation of computational skills and in the development of the child's personality // Elementary School, 2001 No. 1
4. N.K. Vinokurov: “Let's think together”, M. “Growth”.