Painting oral counting at the Rachinsky school. Nikolay Bogdanov-Belsky

When I come to the Tretyakov Gallery with another group, then, of course, I know that obligatory list of paintings that you cannot pass by. I keep everything in my head. From beginning to end, lined up in one line, these paintings should tell the story of the development of our painting. With all that is not a small part of our national heritage and spiritual culture. These are all pictures, so to speak, of the first order, which cannot be avoided without the history being flawed. But there are some that seem to be completely unnecessary for the show. And my choice here depends only on me. From my location to the group, from my mood, and also the availability of free time.

Well, the painting "Verbal Account" by the artist Bogdan - Belsky is exclusively for the soul. And I just can't get past her. And how to get through, because I know in advance that the attention of our foreign friends in this particular picture will manifest itself to such an extent that it will be simply impossible not to stop. Well, do not drag them away by force.

Why? This artist is not one of the most famous Russian painters. His name is known for the most part by experts - art critics. But this picture will make, nevertheless, stop anyone. And it will attract the attention of a foreigner no less.

Here we are, and for a long time we look with interest at everything in it, even the smallest details. And I understand that I don't need to explain much here. Moreover, I feel that with my words I can even interfere with the perception of what I saw. Well, it’s as if I’m starting to give comments at a time when the ear wants to enjoy the melody that has captured us.

And nevertheless, it is still necessary to make some explanations. Even necessary. What do we see? And we see eleven village boys immersed in the thought process in search of the answer to the mathematical equation written on the blackboard by their cunning teacher.

Thought! How much of this sound! Thought in collaboration with difficulty created man. The best evidence of this was shown to us by Auguste Rodin with his Thinker. But when I look at this famous sculpture, and I saw its original in the Rodin Museum in Paris, then in me it gives rise to some strange feeling. And, oddly enough, this is a feeling of fear, and even horror. Some kind of bestial power emanates from the mental strain of this creature, placed in the courtyard of the museum. And I involuntarily see the wonderful discoveries that this creature sitting on a rock is preparing for us in its painful mental effort. For example, the discovery of the atomic bomb threatening to destroy humanity itself along with this Thinker. And we already know for certain that this bestial man will come to the invention of a terrible bomb capable of erasing all life on earth.

But the boys of the artist Bogdan - Belsky do not scare me at all. Against. I look at them and feel a warm sympathy for them is born in my soul. I want to smile. And I feel the joy that rushes to my heart from the contemplation of the touching scene. The mental search expressed in the faces of these boys delights and excites me. And it also makes you think about something else.

The painting was painted in 1895. And a few years earlier, in 1887, the infamous circular had been passed.

This circular, approved by Emperor Alexander III and received in society an ironic name "about cook's children", instructed the educational authorities to admit only wealthy children to the gymnasium and gymnasium, that is, "only those children who are in the care of persons who present sufficient assurance about the correct home supervision and providing them with the convenience they need for their studies. " My God, what a wonderful clerical style.

And further in the circular it was explained that “with the unswerving observance of this rule, grammar schools and progymnasiums will be freed from the children of coachmen, lackeys, cooks, laundresses, small shopkeepers and the like from admission to them.

Like this! Now look at these young, quick-witted Newtons in sandals and tell me how many chances they have to become "reasonable and great."

Although maybe someone will be lucky. Because they were all lucky with a teacher. He was famous. Moreover, he was a teacher from God. His name was Sergei Alexandrovich Rachinsky. Today they hardly know him. And he deserved it with all his life to remain in our memory. Take a closer look at him. Here he is, surrounded by his bast students.

He was a botanist, mathematician, and also a professor at Moscow University. But most importantly, he was a teacher not only by profession, but throughout his entire spiritual makeup, by vocation. And he loved children.

Having gained scholarship, he returned to his native village of Tatevo. And he built this school that we see in the picture. And even with a dormitory for village children. Because, let’s tell the truth, he didn’t take everyone to school. He himself selected, unlike Leo Tolstoy, whom he accepted into his school all the surrounding children.

Rachinsky created his own method for oral counting, which, of course, not everyone could learn. Only a select few. He wanted to work with selected material. And he achieved the desired result. Therefore, do not be surprised that such a difficult problem is solved by children in bast shoes and shirts for graduation.

And the artist Bogdanov - Belsky himself went through this school. And how could he have forgotten his first teacher. No, I couldn't. And this picture is a tribute to the memory of a beloved teacher. And Rachinsky taught at this school not only mathematics, but also painting and drawing along with other subjects. And he was the first to notice the boy's attraction to painting. And he sent him to continue studying this subject not just anywhere, but in the Trinity-Sergius Lavra, in the icon-painting workshop. And then - more. The young man continued to comprehend the art of painting at the no less famous Moscow School of Painting, Sculpture and Architecture, which is on Myasnitskaya Street. And what kind of teachers he had! Polenov, Makovsky, Pryanishnikov. And then also Repin. One of the paintings of the young artist "The Future Monk" was bought by the Empress Maria Feodorovna herself.

That is, Sergei Alexandrovich gave him a ticket to life. And after that, how could an already accomplished artist thank his teacher? But only this very picture. This is the biggest thing he could do. And he did the right thing. Thanks to him, we, too, have today a visible image of this wonderful person, the teacher of Rachinsky.

The boy was lucky, of course. Just incredibly lucky. Well, who was he? The bastard son of a farm laborer! And what future could he have if he did not get into the school of the famous teacher.

The teacher wrote a math equation on the blackboard. You can easily see it. And rewrite. And try to solve. Once there was a math teacher in my group. He carefully rewrote the equation on a piece of paper in a notebook and began to solve. And I decided. And I spent at least five minutes on it. Try it yourself. But I don’t even undertake it. Because I didn't have such a teacher at school. Yes, I think that even if I had, nothing would have worked for me. Well, I'm not a mathematician. And to this day.

And I realized this already in the fifth grade. Even though I was still quite small, but even then I realized that all these brackets and squiggles would not be useful to me in any way in my life. They will not come out in any way. And these tsiferki did not excite my soul in any way. On the contrary, they only outraged. And I do not have a soul for them to this day.

At that time, I still unconsciously found my attempts to solve all these numbers with all sorts of badges useless and even harmful. And they aroused nothing but quiet and unspoken hatred in me. And when all sorts of cosines with tangents came, then there was complete darkness. It pissed me off that all this algebraic bullshit only pulled me away from the more useful and exciting things in the world. For example, from geography, astronomy, drawing and literature.

Yes, since then I have not learned what cotangents and sinuses are. But I do not feel any suffering or regret about this either. The lack of this knowledge, well, did not affect everything in my no longer small life. It is still a mystery to me today how electrons run at incredible speed inside an iron wire over terrible distances, creating an electric current. And that's not all. In some small fraction of a second, they suddenly can stop and run back together. Well, let them run, I think. Who cares, so let him do it.

But that is not the question. And the question was that, even in those small years of mine, I did not understand why it was necessary to torment me with what my soul completely rejected. And I was right in these painful doubts of mine.

Later, when I became a teacher myself, I found the answer to everything. And the explanation is that there is such a bar, such a level of knowledge that a public school should lay down so that the country does not lag behind in its development from others, following the lead of poor students like me.

To find a diamond or a grain of gold, you need to process tons of waste rock. It is called dump, unnecessary, empty. But without this unnecessary breed, a diamond with grains of gold, not to mention nuggets, cannot be found either. Well, I and others like me were this very dump breed, which was only needed to raise mathematicians and even mathematical geeks needed for the country. But how could I then know about this with all my attempts to solve the equations that the kind teacher wrote to us on the blackboard. That is, with my torments and inferiority complexes, I contributed to the birth of real mathematicians. And there is no way to get away from this obvious truth.

So it was, so it is, and it will always be so. And I know this for certain today. Because I am not only a translator, but also a French teacher. I teach and I know for sure that out of my students, and there are about 12 of them in each group, two to three students will know the language. The rest sucks. Or a dump, if you like. For various reasons.

You in the picture see eleven keen boys with glowing eyes. But this is a picture. But in life, it is not at all like that. And any teacher will tell you this.

The reasons are different, why not. To be clear, I will give the following example. Mom comes to me and asks how long it will take me to teach her boy French. I don't know how to answer her. That is, I know, of course. But I don’t know how to answer without offending the assertive mother. And she needs to answer the following:

A language in 16 hours is only on TV. I do not know the degree of interest and motivation of your boy. There is no motivation - and put at least three professors-tutors with your dear child, nothing will come of it. And then there is such an important thing as abilities. And some have these abilities, while others do not. So genes, God or someone else unknown to me decided. For example, a girl wants to learn ballroom dancing, but God did not give her either a sense of rhythm, or plasticity, or, just about horror, the corresponding figure (well, she became fat or lanky). And so you want. What are you going to do here if nature itself has risen across. And so it is in every case. And in language learning too.

But, really, in this place I want to put a big comma to myself. Not so simple. Motivation is a mobile thing. Today she is not, and tomorrow she appeared. That is what happened to me myself. My first teacher of French, dear Rosa Naumovna, seemed to be very surprised to learn that it was her subject that would become the work of my whole life.

*****
But back to the teacher Rachinsky. I confess that his portrait interests me immeasurably more than the personality of the artist. He was a well-born nobleman and not a poor man at all. He had his own estate. And to all this he had a learned head. After all, it was he who first translated Charles Darwin's The Origin of Species into Russian. Although here is a strange fact that struck me. He was a deeply religious person. And at the same time he translated the famous materialistic theory, absolutely disgusting to his soul

He lived in Moscow on Malaya Dmitrovka, and was familiar with many famous people. For example, with Leo Tolstoy. And it was Tolstoy who pushed him to the cause of public education. Even in his youth, Tolstoy was fond of the ideas of Jean Jacques Rousseau, the Great Enlightener was his idol. That, for example, wrote a wonderful pedagogical work "Emil or about education". I not only read it, but wrote a term paper on it at the institute. To tell the truth, Rousseau, it seemed to me, put forward ideas in this work well, well, than the original ones. And Tolstoy himself was carried away by the following thought of the great enlightener and philosopher:

“Everything comes out good from the hands of the Creator, everything degenerates in the hands of man. He forces one soil to nourish plants grown on another, one tree to bear the fruit of another. He mixes and confuses climates, elements, seasons. He mutilates his dog, his horse, his slave. He inverts everything, distorts everything, loves ugliness, monstrous. He does not want to see anything the way nature created - not excluding man: he also needs to train a man like a horse for an arena;

And in his declining years, Tolstoy tried to implement the above wonderful idea. He wrote textbooks and manuals. Wrote the famous "ABC" He also wrote children's stories. Who does not know the famous Filippok or the story about the bone.
*****

As for Rachinsky, here, as they say, two kindred spirits met. So much so that inspired by the ideas of Tolstoy, Rachinsky left Moscow and returned to his ancestral village of Tatevo. And he built, following the example of the famous writer, a school and a dormitory for gifted village children with his own money. And then he completely became the ideologist of the parish school in the countries.

This is his activity in the field of public education was noticed at the very top. Read what Pobedonostsev writes about him to Emperor Alexander III:

“You will deign to recall how several years ago I reported to you about Sergei Rachinsky, a respectable man who, having left his professorship at Moscow University, went to live on his estate, in the most remote forest wilderness of the Belsky district of the Smolensk province, and lives there without a break here. for over 14 years, working from morning to night for the benefit of the people. He breathed a completely new life into a whole generation of peasants ... He truly became a benefactor of the area, founding and leading, with the help of 4 priests, 5 public schools, which now represent a model for the whole earth. This is a wonderful person. All that he has, and all the means of his estate, he gives to the penny for this business, limiting his needs to the last degree "

And here is what Nikolai II himself writes to Sergei Rachinsky:

“The schools that you founded and run, including parish schools, have become nurseries of educated leaders in the same spirit, a school of labor, sobriety and good morals, and a living model for all such institutions. The concern close to my heart for public education, which you serve worthily, prompts Me to express my sincere gratitude to you. My benevolent Nikolai is staying with you "

In conclusion, plucking up the courage, I want to add a few words from myself to the statements of the two above-mentioned persons. These words will be about the teacher.

In the world there are a lot of professions. All life on Earth is busy to prolong its existence. And above all, in order to find something for themselves to eat. Both herbivores and carnivores. Both the largest and the smallest. Everything! And man too. But a person has a great many opportunities. The choice of activities is enormous. That is, the occupations that a person indulges in in order to earn his own bread, for a living.

But of all these occupations, there is an insignificant percentage of those professions that can give complete satisfaction to the soul. The vast majority of all other things come down to the routine, daily repetition of the same thing. The same actions of a mental and physical nature. Even in the so-called creative professions. I will not even name them. Without the slightest chance for spiritual growth. Stamp the same nut all your life. Or ride on the same rails, literally and figuratively, until the end of your work experience required for retirement. And there's nothing you can do about it. This is our human creation. One gets in life as best he can.

But, I repeat, there are few professions in which the whole life and the whole work of life is based solely on a spiritual need. One of them is the Teacher. With a capital letter. I know what I'm talking about. Since I myself have been in this topic for many years. The teacher is the earthly cross, and a vocation, and torment, and joy all together. Without all this, there is no teacher. And there are enough of them, even among those who have a teacher in their work record book.

And you have to prove your right to be a teacher every day, from the very second when you crossed the threshold of the class. And this is sometimes so difficult. Do not think that beyond this threshold only the happy moments of your life await you. And it is also not necessary to count on the fact that all small people will meet you in anticipation of knowledge that you are ready to put into their heads and souls. That the entire classroom space is inhabited entirely by angelic, disembodied cherubs. These cherubim know how to bite at times. And how much it hurts. This whim needs to be thrown out of the head. On the contrary, we must remember that in this light room with huge windows, ruthless animals await you, which still have a difficult path to become human. And it is the teacher who must guide them along this path.

I distinctly remember one such "cherub" when I first appeared in class during my internship. I was warned. There is one boy there. It is not very simple. And God will help you to cope with it.

How long has passed, but I still remember it. If only because he had some strange last name. Noack. That is, I knew that the PLA is the People's Liberation Army of China. But here ... I went in and instantly figured out this asshole. This sixth grader, who was sitting at the last desk, put one of his feet on the table at my appearance. They all stood up. Except him. I realized that this Noak wanted to immediately in such a manner tell me and everyone else about who is their boss here.

Sit down, children, ”I said. Everyone sat down and waited with interest for the continuation. Noak's leg remained in the same position. I approached him, not yet knowing what to do and what to say.

Why are you going to sit for the whole lesson? A very uncomfortable position! - I said, feeling how a wave of hatred rises in me for this impudent person who intends to disrupt my first lesson in my life.

He did not answer, turned away and made a forward movement with his lower lip as a sign of complete contempt for me, and even spat in the direction of the window. And then, no longer realizing what I was doing, I grabbed the collar and kicked him out of the classroom into the corridor with a kick in the ass. Well, he was still young and hot. There was an extraordinary silence in the classroom. As if it were completely empty. Everyone looked at me in a daze. "In gives" - someone whispered loudly. A desperate thought flashed through my head: “That's it, I have nothing else to do at school! End!" And I was very wrong. This was only the beginning of my pre-long path as a teacher.

Paths of happy peak joyful moments and cruel disappointments. At the same time, I remember another teacher. Teacher Melnikov from the movie "We'll Live Until Monday." There was a day and an hour when a deep depression befell him. And it was from what! “You sow here reasonable, good eternal, and henbane grows - a thistle,” he once said in his hearts. And he wanted to leave school. At all! And he did not leave. Because if you are a real teacher, then this is already for you forever. Because you understand that you will not find yourself in any other business. You cannot express yourself to the fullest. Took it - be patient. It is a great duty and great honor to be a teacher. And this is exactly how Sergei Aleksandrovich Rachinsky understood it, having voluntarily set himself up for his entire life term at the black chalkboard.

P.S. If you did try to solve this equation on the board, then the correct answer would be 2.

known to many. The painting depicts a village school at the end of the 19th century during an arithmetic lesson while solving a fraction in the head.

The teacher is a real person, Sergei Alexandrovich Rachinsky (1833-1902), botanist and mathematician, professor at Moscow University. In the wake of populism in 1872, Rachinsky returned to his native village of Tatevo, where he created a school with a hostel for peasant children, developed a unique method of teaching oral counting, instilling in village children his skills and foundations of mathematical thinking. Bogdanov-Belsky, himself a former student of Rachinsky, dedicated his work to an episode from the life of a school with a creative atmosphere that prevailed in the classroom.

However, for all the fame of the picture, few of those who saw it delved into the content of the "difficult task" that is depicted on it. It consists in quickly finding the result of the calculation by verbal counting:

The talented teacher cultivated in his school oral counting, based on the virtuoso use of the properties of numbers.

The numbers 10, 11, 12, 13, and 14 have an interesting twist:

10 2 + 11 2 + 12 2 = 13 2 + 14 2 .

Indeed, since

100 + 121 + 144 = 169 + 196 = 365,

Wikipedia offers the following method for calculating the value of the numerator:

10 2 + (10 + 1) 2 + (10 + 2) 2 + (10 + 3) 2 + (10 + 4) 2 =

10 2 + (10 2 + 2 10 1 + 1 2) + (10 2 + 2 10 2 + 2 2) + (10 2 + 2 10 3 + 3 2) + (10 2 + 2 · 10 · 4 + 4 2) =

5 100 + 2 10 (1 + 2 + 3 + 4) + 1 2 + 2 2 + 3 2 + 4 2 =

500 + 200 + 30 = 730 = 2365.

As for me, it is too tricky. It's easier to do otherwise:

10 2 + 11 2 + 12 2 + 13 2 + 14 2 =

= (12 - 2) 2 + (12 - 1) 2 + 12 2 + (12 + 1) 2 + (12 + 2) 2 =

5 12 2 + 2 4 + 2 1 = 5 144 + 10 = 730,

The above reasoning is quite possible to carry out orally - 12 2 , of course, you need to remember, the doubled products of the squares of binomials to the left and right of 12 2 mutually annihilate and they can not be counted, but 5 144 = 500 + 200 + 20, - not difficult.

We will use this technique and orally find the amount:

48 2 + 49 2 + 50 2 + 51 2 + 52 2 = 5 50 2 + 10 = 5 2500 + 10 = 12510.

Let's complicate:

84 2 + 87 2 + 90 2 + 93 2 + 96 2 = 5 8100 + 2 9 + 2 36 = 40500 + 18 + 72 = 40590.

Rachinsky's row

Algebra gives us a means to pose the question of this interesting feature of the series of numbers

10, 11, 12, 13, 14

more broadly: is it the only row of five consecutive numbers, the sum of the squares of the first three of which is equal to the sum of the squares of the last two?

Denoting the first of the sought numbers by x, we have the equation

x 2 + (x + 1) 2 + (x + 2) 2 = (x + 3) 2 + (x + 4) 2.

It is more convenient, however, to denote by x not the first, but the second of the sought numbers. Then the equation will have a simpler form

(x - 1) 2 + x 2 + (x + 1) 2 = (x + 2) 2 + (x + 3) 2.

Expanding the brackets and making simplifications, we get:

x 2 - 10x - 11 = 0,

where

x 1 = 11, x 2 = -1.

Therefore, there exist two series of numbers with the required property: the Raczynski series

10, 11, 12, 13, 14

and a number

2, -1, 0, 1, 2.

Indeed,

(-2) 2 +(-1) 2 + 0 2 = 1 2 + 2 2 .

Two!!!

I would like to end with bright and touching memories of the author of the author's blog V. Iskra in the article About squares of two-digit numbers and not only about them ...

Once, in the year about 1962, our "mathematician", Lyubov Iosifovna Drabkina, gave this problem to us, 7th grade students.

At that time I was very fond of the newly appeared KVN. I was rooting for the team of the Moscow Region city of Fryazino. "Fryazintsy" were distinguished by their special ability to apply logical "express analysis" to solve any problem, "pull out" the most tricky question.

I couldn't do it quickly in my head. However, using the "Fryazin" method, I figured out that the answer should be expressed as an integer. Otherwise, it is no longer "verbal counting"! This number could not be one - even if there were the same 5 hundred in the numerator, the answer would be clearly larger. On the other hand, he clearly fell short of the number "3".

- Two!!! - I blurted out, a second ahead of my friend, Lenya Strukov, the best mathematician of our school.

- Yes, indeed two, - Lenya confirmed.

- What did you think? - asked Lyubov Iosifovna.

- I didn’t count. Intuition - I replied to the laughter of the whole class.

- If you didn’t count - the answer doesn’t count - “skalamburila” Lyubov Iosifovna. Lyonya, didn’t you count it either?

- No, why, Lenya answered gravely. I had to add 121, 144, 169 and 196. I added the first and third, second and fourth numbers in pairs. This is more comfortable. It turned out 290 + 340. The total amount, including the first hundred, is 730. Divide by 365 - we get 2.

- Well done! But for the future, remember - in a series of two-digit numbers - the first five of its representatives - have an amazing property. The sum of the squares of the first three numbers in the row (10, 11 and 12) is equal to the sum of the squares of the next two (13 and 14). And this amount is equal to 365. It's easy to remember! So many days a year. If the year is not a leap year. Knowing this property, the answer can be obtained in a second. Without any intuition ...

* * *

... Years have passed. Our city has acquired its own "Wonder of the World" - mosaic paintings in underground passages. There were many transitions, and even more pictures. The topics were very different - the defense of Rostov, space ... In the central passage, under the crossroads of Engels (now Bolshaya Sadovaya) - Voroshilovsky, they made a whole panorama about the main stages of the life of a Soviet person - a maternity hospital - a kindergarten - a school, a graduation party ...

In one of the "school" pictures one could see a familiar scene - the solution of a problem ... Let's call it "Rachinsky's problem" ...

... Years passed, people passed ... Merry and sad, young and not so. Someone recalled their school, while others "wiggled their brains" ...

The master tilers and artists, led by Yuri Nikitovich Labintsev, did a wonderful job!

Now the “Rostov miracle” is “temporarily unavailable”. Trade came to the fore, both literally and figuratively. Nevertheless, let's hope that in this common phrase - the main thing is the word "temporarily" ...

Sources: Ya.I. Perelman. Entertaining algebra (Moscow, "Nauka", 1967), Wikipedia,

The famous Russian artist Nikolai Petrovich Bogdanov-Belsky wrote a unique and incredibly life story in 1895. The work is called "Oral account", and in the full version "Oral account. In the folk school of S. A. Rachinsky. "

Nikolay Bogdanov-Belsky. Verbal counting. In the folk school of S. A. Rachinsky

The painting, painted in oil on canvas, depicts a 19th century rural school during an arithmetic lesson. Students solve an interesting and difficult example. They are deep in thought and looking for the right solution. Someone thinks at the blackboard, someone stands aside and tries to compare knowledge that will help in solving the problem. Children are completely absorbed in looking for an answer to the question posed, they want to prove to themselves and the world that they can do it.

Nearby is a teacher, whose prototype is Rachinsky himself - a famous botanist and mathematician. No wonder the picture was given such a name, it is in honor of a professor at Moscow University. The canvas depicts 11 children and only one boy quietly whispers in the teacher's ear, perhaps the correct answer.

The painting depicts a simple Russian class, the children are dressed in peasant clothes: bast shoes, pants and shirts. All this very harmoniously and succinctly fits into the plot, unobtrusively bringing to the world a craving for knowledge on the part of the common Russian people.

The warm color scheme carries the kindness and simplicity of the Russian people, there is no envy and falsehood, there is no evil and hatred, children from different families with different incomes have come together to make the only right decision. This is sorely lacking in our modern life, where people are used to living in a completely different way, regardless of the opinions of others.

Nikolai Petrovich dedicated the picture to his teacher, the great genius of mathematics, whom he knew and respected well. Now the painting is in Moscow in the Tretyakov Gallery, if you are there, be sure to take a look at the pen of the great master.

opisanie-kartin.com

Nikolay Petrovich Bogdanov-Belsky (December 8, 1868, village Shitiki, Belsky district, Smolensk province, Russia - February 19, 1945, Berlin, Germany) - Russian itinerant artist, academician of painting, chairman of the Kuindzhi Society.

The painting depicts a village school at the end of the 19th century during an arithmetic lesson while solving a fraction in the head. The teacher is a real person Sergey Alexandrovich Rachinsky (1833-1902), botanist and mathematician, professor at Moscow University.

In the wake of populism in 1872, Rachinsky returned to his native village of Tatevo, where he created a school with a hostel for peasant children, developed a unique method of teaching oral counting, instilling in village children his skills and foundations of mathematical thinking. Bogdanov-Belsky, himself a former student of Rachinsky, dedicated his work to an episode from the life of a school with a creative atmosphere that prevailed in the classroom.

An example is written on the chalkboard for the students to solve:

The task depicted in the picture could not be offered to the students of a standard elementary school: the study of the concept of a degree was not provided for in the curriculum of one-class and two-class elementary public schools. However, Rachinsky did not follow the model training course; he was confident in the excellent mathematical abilities of most of the peasant children and considered possible a significant complication of the mathematics curriculum.

Solution of the Rachinsky problem

The first solution

There are several ways to solve this expression. If you learned squares of numbers up to 20 or up to 25 at school, then most likely it will not cause you much difficulty. This expression is equal to: (100 + 121 + 144 + 169 + 196) divided by 365, which ultimately converts to the quotient 730 and 365, which equals: 2. To solve an example in this way, you may need to use the skills of mindfulness and the ability to keep in mind several intermediate answers.

Second solution

If you did not learn the meaning of squares of numbers up to 20 at school, then you may find it useful to use a simple method based on the use of a reference number. This method allows you to simply and quickly multiply any two numbers less than 20. The method is very simple, you need to add one to the first number of the second, multiply this sum by 10, and then add the product of ones. For example: 11 * 11 = (11 + 1) * 10 + 1 * 1 = 121. The rest of the squares are also:

12*12=(12+2)*10+2*2=140+4=144

13*13=160+9=169

14*14=180+16=196

Then, having found all the squares, the problem can be solved in the same way as shown in the first method.

Third way of solving

Another method involves using a simplification of the numerator of the fraction, based on the use of the formulas for the square of the sum and the square of the difference. If we try to express the squares in the numerator of the fraction through the number 12, we get the following expression. (12 - 2) 2 + (12 - 1) 2 + 12 2 + (12 + 1) 2 + (12 + 2) 2. If you know well the formulas for the square of the sum and the square of the difference, then you will understand how this expression can be easily reduced to the form: 5 * 12 2 + 2 * 2 2 + 2 * 1 2, which equals 5 * 144 + 10 = 730. To multiply 144 by 5, you just need to divide this number by 2 and multiply by 10, which equals 720. Then we divide this expression by 365 and get: 2.

Fourth solution

Also, this problem can be solved in 1 second if you know the Raczynski sequences.

Raczynski sequences for mental arithmetic

To solve the famous Rachinsky problem, you can also use additional knowledge about the laws of the sum of squares. We are talking about exactly those sums that are called Rachinsky sequences. So mathematically, you can prove that the following sums of squares are equal:

3 2 +4 2 = 5 2 (both sums are 25)

10 2 +11 2 +12 2 = 13 2 +14 2 (sum is 365)

21 2 +22 2 +23 2 +24 2 = 25 2 +26 2 +27 2 (which is 2030)

36 2 +37 2 +38 2 +39 2 +40 2 = 41 2 +42 2 +43 2 +44 2 (which equals 7230)

To find any other Raczynski sequence, you just need to compose an equation of the following form (note that in such a sequence, the number of squares to be summed is always one less on the right than on the left):

n 2 + (n+1) 2 = (n+2) 2

This equation reduces to a quadratic equation and is easy to solve. In this case, "n" equals 3, which corresponds to the first Rachinsky sequence described above (3 2 + 42 = 5 2).

Thus, the solution to the famous example of Raczynski can be done in the head even faster than described in this article, simply by knowing the second Raczynski sequence, namely:

10 2 +11 2 +12 2 +13 2 +14 2 = 365 + 365

As a result, the equation from the picture of Bogdan-Belsky takes the form (365 + 365) / 365, which undoubtedly equals two.

Also, the Rachinsky sequence can be useful for solving other problems from the collection "1001 Problems for Mental Counting" by Sergei Rachinsky.

Evgeny Buyanov

Many have seen the painting "Oral Counting in the People's School". The end of the 19th century, a folk school, a blackboard, an intelligent teacher, poorly dressed children, 9-10 years old, are enthusiastically trying to solve a problem written on the blackboard in their minds. The first person who decides to communicate the answer to the teacher in his ear, in a whisper, so that others do not lose interest.

Now let's look at the problem: (10 squared + 11 squared + 12 squared + 13 squared + 14 squared) / 365 = ???

Heck! Heck! Heck! Our children at the age of 9 will not solve such a problem, at least in their minds! Why were grimy and barefoot village children taught so well from one room in a wooden school, while our children are taught so poorly ?!

Do not rush to be indignant. Take a closer look at the picture. Don't you think that the teacher looks too intelligent, somehow professorial, and is dressed with an obvious pretense? Why is there such a high ceiling and an expensive stove with white tiles in the classroom? Was this what the village schools and teachers looked like?

Of course, they didn't look like that. The picture is called "Oral counting in the folk school of S.A. Rachinsky". Sergei Rachinsky is a professor of botany at Moscow University, a person with certain government connections (for example, a friend of the chief prosecutor of the Synod Pobedonostsev), a landowner - in the middle of his life he dropped everything, went to his estate (Tatevo in the Smolensk province) and started there (of course, for own account) experimental folk school.

The school was one-class, which did not mean at all that it was taught for one year. At that time, they taught in such a school for 3-4 years (and in two-grade schools - 4-5 years, in three-grade schools - 6 years). The word one-class meant that children of three years of study make up a single class, and one teacher deals with all of them within one lesson. It was quite a tricky thing: while the children of one year of schooling were doing some written exercise, the children of the second year answered at the blackboard, the children of the third year read the textbook, etc., and the teacher paid attention to each group in turn.

Rachinsky's pedagogical theory was very original, and its different parts somehow did not agree well with each other. Firstly, Rachinsky considered the teaching of the Church Slavonic language and the Law of God to be the basis of education for the people, and not so much explanatory as consisting in memorizing prayers. Rachinsky firmly believed that a child who knows a certain number of prayers by heart will certainly grow up to be a highly moral person, and the very sounds of the Church Slavonic language will already have a moral-improving effect. For practice in the language, Rachinsky recommended that children be hired to read the Psalter over the dead (sic!).

Secondly, Rachinsky believed that it was useful for the peasants and needed to be quick in their minds. Rachinsky was not very interested in teaching mathematical theory, but he was very good at oral counting in his school. The students answered firmly and quickly how much change per ruble should be given to someone who buys 6 3/4 pounds of carrots at 8 1/2 kopecks per pound. The squaring depicted in the painting was the most difficult mathematical operation studied in his school.

And finally, Rachinsky was a supporter of very practical teaching of the Russian language - students were not required to have any special spelling skills or good handwriting, they were not taught theoretical grammar at all. The main thing was to learn to read and write fluently, albeit in a clumsy handwriting and not very competently, but it is clear that what could be useful to a peasant in everyday life: simple letters, petitions, etc. Even at the Rachinsky school, some manual labor was taught, the children sang in chorus, and this was where the whole education ended.

Rachinsky was a real enthusiast. School became his whole life. Rachinsky's children lived in a dormitory and were organized into a commune: they performed all the housekeeping work for themselves and the school. Rachinsky, who did not have a family, spent all the time with the children from early morning until late at night, and since he was a very kind, noble and sincerely attached person to children, his influence on students was enormous. By the way, Rachinsky gave out a gingerbread to the first child who solved the problem (in the literal sense of the word, he did not have a stick).

The school classes themselves took 5-6 months a year, and the rest of the time Rachinsky worked individually with older children, preparing them for admission to various educational institutions of the next level; the elementary public school was not directly connected with other educational institutions and after it it was impossible to continue education without additional training. Rachinsky wanted to see the most advanced of his students as primary school teachers and priests, so he prepared children mainly for theological and teaching seminaries. There were also significant exceptions - first of all, it was the author of the painting himself, Nikolai Bogdanov-Belsky, whom Rachinsky helped to get into the Moscow School of Painting, Sculpture and Architecture. But, oddly enough, Rachinsky did not want to lead peasant children along the main path of an educated person - gymnasium / university / public service.

Rachinsky wrote popular pedagogical articles and continued to enjoy a certain amount of influence in the capital's intellectual circles. The most important was the acquaintance with the ultra-hydraulic Pobedonostsev. Under a certain influence of Rachinsky's ideas, the clerical department decided that there would be no use from the zemstvo school - the liberals would not teach children good things - and in the mid-1890s it began to develop its own independent network of parish schools.

In some ways, the parish schools were similar to the Rachinsky school - they had a lot of Church Slavonic language and prayers, and the rest of the subjects were accordingly reduced. But, alas, the dignity of the Tatev school was not passed on to them. The priests were not very interested in school affairs, they managed schools out of hand, they themselves did not teach in these schools, and they hired the most third-rate teachers, and paid them noticeably less than in zemstvo schools. The peasants disliked the parish school, because they realized that they hardly teach anything useful there, and they were not very interested in prayers. By the way, it was the teachers of the church school, recruited from the pariahs of the clergy, who turned out to be one of the most revolutionary professional groups of that time, and it was through them that socialist propaganda actively penetrated the countryside.

Now we see that this is a common thing - any author's pedagogy, calculated on the deep involvement and enthusiasm of the teacher, immediately dies during mass reproduction, falling into the hands of disinterested and sluggish people. But for that time, it was a big bummer. Parish schools, which by 1900 accounted for about a third of the elementary public schools, turned out to be disgraceful to everyone. When, starting in 1907, the state began to allocate big money to primary education, there was no question of passing subsidies to church schools through the Duma, almost all the funds went to the Zemstvo people.

The more widespread zemstvo school was quite different from the Rachinsky school. For a start, the Zemstvo people considered the Law of God completely useless. It was impossible to refuse to teach him, for political reasons, so the zemstvos pushed him into a corner as best they could. The law of God was taught by a parish priest who was paid little and ignored, with appropriate results.

Mathematics in the zemstvo school was taught worse than at Rachinsky, and to a lesser extent. The course ended with operations with simple fractions and non-metric units. The teaching did not reach the level of elevation, so the students of an ordinary elementary school simply would not understand the problem depicted in the picture.

The zemstvo school tried to transform the teaching of the Russian language into world studies, through the so-called explanatory reading. The technique consisted in the fact that dictating the educational text in the Russian language, the teacher also additionally explained to the students what the text itself says. In this palliative way, Russian language lessons also turned into geography, natural history, history - that is, into all those developing subjects that could not find a place in the short course of a one-class school.

So, our picture depicts not a typical, but a unique school. This is a monument to Sergei Rachinsky, a unique personality and teacher, the last representative of that cohort of conservatives and patriots, to which the well-known expression "patriotism is the last refuge of a scoundrel" could not yet be attributed. The mass public school was economically much poorer, the mathematics course in it was shorter and simpler, and the teaching was weaker. And, of course, the students of an ordinary elementary school could not only solve, but also understand the problem reproduced in the picture.

By the way, what method do schoolchildren use to solve the problem on the blackboard? Only straight, in the forehead: multiply 10 by 10, remember the result, multiply 11 by 11, add both results, and so on. Rachinsky believed that the peasant did not have writing utensils at hand, so he taught only oral methods of counting, omitting all arithmetic and algebraic transformations that required calculations on paper.

For some reason, only boys are depicted in the picture, while all the materials show that children of both sexes studied with Rachinsky. What this means is not clear.

In one of the halls of the Tretyakov Gallery, you can see the famous painting by the artist N.P. Bogdanov-Belsky "Oral account". It depicts a lesson in a rural school. Classes are conducted by an old teacher. Village boys in poor peasant shirts and bast shoes crowded around. They are concentratedly and enthusiastically solving the problem proposed by the teacher ... A plot familiar to many from childhood, but not many know that this is not an artist's fiction and that behind all the characters in the picture there are real people painted by him from nature - people whom he knew and loved, and the main character is an elderly teacher, a person who played a key role in the artist's biography. His fate is amazing and extraordinary - after all, this person is a wonderful Russian educator, teacher of peasant children Sergei Alexandrovich Rachinsky (1833-1902)

N.P. Bogdanov-Belsky "Oral counting in the Rachinsky public school" 1895.

Future teacher S.A. Rachinsky.

Sergei Aleksandrovich Rachinsky was born in the estate of Tatevo, Belsky district, Smolensk province, into a noble family. His father, Alexander Antonovich Rachinsky, in the past, a participant in the December movement, was exiled for this to his family estate Tatevo. Here, on May 2, 1833, the future teacher was born. His mother was the sister of the poet E.A. Baratynsky and the Rachinsky family closely communicated with many representatives of Russian culture. In the family, parents paid great attention to the comprehensive education of their children. All this was very useful for Rachinsky in the future. Having received an excellent education at the Faculty of Natural Sciences of Moscow University, he travels a lot, meets interesting people, studies philosophy, literature, music and much more. After some time, he wrote several scientific papers and received a doctorate and the department of professor of botany at Moscow University. But his interests were not limited to the scientific framework. The future rural teacher was engaged in literary creativity, wrote poetry and prose, played the piano perfectly, was a collector of folklore - folk songs and handicrafts. Khomyakov, Tyutchev, Aksakov, Turgenev, Rubinstein, Tchaikovsky and Tolstoy often visited his apartment in Moscow. Sergei Alexandrovich was the author of the libretto for two operas by P.I. Tchaikovsky, who listened to his advice and recommendations and dedicated his first string quartet to Rachinsky. With L.N. Tolstoy Rachinsky was connected by friendly and family relations, since the niece of Sergei Alexandrovich, the daughter of his brother, the rector of the Petrovskaya (now Timiryazevskaya) Academy, Konstantin Alexandrovich Rachinsky - Maria was the wife of Sergei Lvovich, the son of Tolstoy. An interesting correspondence between Tolstoy and Rachinsky, full of discussions and disputes about public education.

In 1867, due to the prevailing circumstances, Rachinsky left the professorship at Moscow University, and with it all the bustle of metropolitan life, returned to his native Tatevo, opened a school there and devoted himself to teaching and educating peasant children. A few years later, the Smolensk village of Tatevo becomes known throughout Russia. Enlightenment and service to the common people will henceforth become the work of their entire life.

Professor of Botany at Moscow University Sergei Rachinsky.

Rachinsky develops an innovative, unusual for that time, system of teaching children. The combination of theoretical and practical training becomes the basis of this system. In the classroom, children were taught various crafts necessary for the peasants. The boys studied carpentry and bookbinding. We worked in the school garden and in the apiary. Natural history lessons were taught in the garden, in the field, and in the meadow. The pride of the school is the church choir and icon painting workshop. At his own expense, Rachinsky built a boarding school for children who came from far away and did not have housing.

N.P. Bogdanov-Belsky "Sunday Reading of the Gospel in the Rachinsky Folk School" 1895. In the picture, S.A. Rachinsky.

Children received a diversified education. In arithmetic lessons, they not only learned to add and subtract, but also mastered the elements of algebra and geometry, and in an accessible and fun form for children, often in the form of a game, along the way making amazing discoveries. It is precisely this discovery of the theory of numbers that is depicted on the blackboard in the painting "Oral Counting". Sergei Aleksandrovich gave the children to solve interesting problems and they had to be solved orally, in their minds. He said: "You can't run in the field for a pencil and paper, you have to be able to count in your head."

S. A. Rachinsky. Figure N.P. Bogdanov-Belsky.

Poor peasant shepherd Kolya Bogdanov from the village of Shitiki, Belsk district, was one of the first to go to the Rachinsky school. In this boy, Rachinsky discerned the talent of a painter and helped him develop, fully taking upon himself his future art education. In the future, all the work of the itinerant artist Nikolai Petrovich Bogdanov-Belsky (1868-1945) will be devoted to peasant life, school and beloved teacher.

In the painting "On the Threshold of School" the artist captured the moment of his first acquaintance with Rachinsky's school.

NP Bogdanov-Belsky "On the threshold of school" 1897.

But what is the fate of the Rachinsky folk school in our time? Is the memory of Rachinsky preserved in Tatev, once famous throughout Russia? These questions worried me in June 2000, when I first went there.

And finally, it is in front of me, stretched among green forests and fields, the village of Tatevo, Belsky district, the former Smolensk province, and nowadays referred to the Tver region. It was here that the famous Rachinsky school was created, which so influenced the development of public education in pre-revolutionary Russia.

Upon entering the estate, I saw the remains of a regular park with linden alleys and century-old oaks. A picturesque lake in the clear waters of which the park is reflected. The lake of artificial origin, fed by springs, was dug during the time of S.A. Rachinsky's grandfather, St. Petersburg Chief of Police Anton Mikhailovich Rachinsky.

Lake on the territory of the estate.

And so I come to a dilapidated manor house with columns. From the majestic building erected at the end of the 18th century, now only the skeleton remains. The restoration of the Trinity Church has begun. Near the church is the grave of Sergei Aleksandrovich Rachinsky - a modest stone slab with the Gospel words inscribed on it at his request: "A man will not live about bread alone, but about every verb that comes from the mouth of God." There, among the family tombstones, his parents, brothers and sisters are buried.

A manor house in Tatev today.

In the fifties, the landlord's house began to gradually collapse. In the future, the destruction continued, reaching its full apogee in the seventies of the last century.

Manor house in Tatev at the time of Rachinsky.

Church in Tatev.

The building of the wooden school has not survived. But the school has survived in another two-story brick house, the construction of which was conceived by Rachinsky, but carried out shortly after his death in 1902. This building, designed by a German architect, is considered unique. Due to a design error, it turned out to be asymmetrical - it lacks one wing. Only two more buildings were built according to the same project.

The building of the Rachinsky school today.

It was nice to know that the school is alive, active and in many ways surpasses the capital's schools. When I arrived at this school, there were no computers and other modern innovations, but there was a festive, creative atmosphere, teachers and children showed a lot of imagination, freshness, invention and originality. I was pleasantly surprised by the openness, cordiality, and hospitality with which I was greeted by the students and teachers, headed by the school director. Here the memory of their founder is anxiously kept. The school museum cherishes the relics associated with the history of the creation of this school. Even the external design of the school and classrooms was bright and unusual, so unlike the standard, official design that I had to see in our schools. These are windows and walls originally decorated and painted by the students themselves, and a code of honor invented by them on the wall, and their own school anthem and much more.

Memorial plaque on the wall of the school.

Within the walls of the Tatev school. These stained glass windows were made by the students of the school themselves.

At the Tatev school.

At the Tatev school.

At the Tatev school these days.

Museum of N.P. Bogdanov-Belsky in the former house of the manager.

N.P. Bogdanov-Belsky. Self-portrait.

All the characters in the painting "Oral Count" are painted from life and in them the inhabitants of the village of Tatevo recognize their grandfathers and great-grandfathers. I want to tell you a little about how the life of some of the boys depicted in the picture turned out. Local old-timers who knew some of them personally told me about this.

S.A. Rachinsky with his students on the doorstep of a school in Tatev. June 1891.

NP Bogdanov-Belsky "Oral counting in the Rachinsky public school" 1895.

Many people think that the artist depicted himself in the boy in the foreground of the painting - in fact, this is not so, this boy is Vanya Rostunov. Ivan Evstafievich Rostunov was born in 1882 in the village of Demidovo in a family of illiterate peasants. Only in the thirteenth year he entered the Rachinsky public school. Later he worked on a collective farm as an accountant, saddler, postman. In the absence of a mail bag, before the war he delivered letters in a cap. Rostunov had seven children. All of them studied at the Tatev secondary school. One of them is a veterinarian, another is an agronomist, the third is a military man, one daughter is a livestock technician, and another daughter was a teacher and director of the Tatev school. One son died during the Great Patriotic War, and another, upon returning from the war, soon died from the consequences of the wounds he received there. Until recently, Rostunov's granddaughter worked as a teacher at the Tatev school.

The boy standing on the far left in boots and a purple shirt - Dmitry Danilovich Volkov (1879-1966) became a doctor. During the Civil War, he worked as a surgeon in a military hospital. During the Great Patriotic War, he was a surgeon in a partisan unit. In peacetime, he treated the residents of Tatev. Dmitry Danilovich had four children. One of his daughters was a partisan in the same detachment as her father and died heroically at the hands of the Germans. Another son was a participant in the war. The other two children are a pilot and a teacher. The grandson of Dmitry Danilovich was the director of the state farm.

The fourth from the left, the boy depicted in the picture, is Andrei Petrovich Zhukov, he became a teacher, worked as a teacher in one of the schools created by Rachinsky and located a few kilometers from Tatev.

Andrey Olkhovnikov (second from right in the picture) also became a prominent teacher.

The boy on the far right is Vasily Ovchinnikov, a participant in the first Russian revolution.

The boy who dreamed and threw his hand behind his head is Grigory Molodonkov from Tatev.

Sergey Kupriyanov from the village of Gorelki whispers in the teacher's ear. He was the most capable of mathematics.

The tall guy who is pondering at the blackboard is Ivan Zeltin from the village of Pripechye.

The permanent exhibition of the Tatev Museum tells about these and other residents of Tatev. There is a section dedicated to the pedigree of each Tatev family. Merits and achievements of grandfathers, great-grandfathers, fathers and mothers. Achievements of a new generation of students of the Tatev school are presented.

Peering into the open faces of today's Tatev schoolchildren, so similar to the faces of their great-grandfathers from the painting by N.P. Bogdanov-Belsky, I thought that maybe the source of spirituality on which the Russian pedagogue ascetic, my ancestor Sergei Alexandrovich Rachinsky, so strongly hoped, might not have completely stalled.