Painting by Russian artist oral account. Nikolai Bogdanov-Belsky

When I come to the Tretyakov Gallery with another group, then, of course, I know that mandatory list of paintings that you cannot pass by. I keep everything in my head. From start to finish, 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 not being flawed. But there are some that are completely and not required to be shown. And my choice here depends only on me. From my location to the group, from the mood, but also the availability of free time.

Well, the painting "Oral Account" by the artist Bogdan-Belsky is exclusively for the soul. And I can't get past it. Yes, 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, don't force them.

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

Here we stand, and for a long time we examine with interest 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 see. Well, as if I started to give comments at a time when the ear wants to enjoy the melody that has captured us.

Nevertheless, some explanations still need to be made. Even necessary. What do we see? And we see eleven village boys immersed in the thought process in search of an answer to a mathematical equation written on the blackboard by their cunning teacher.

Thought! So much in this sound! Thought in commonwealth with difficulty created man. Auguste Rodin gave us the best evidence of this in his Thinker. But when I look at this famous sculpture, and I saw its original in the Rodin Museum in Paris, then it gives rise to some strange feeling in me. And, oddly enough, it is a feeling of fear, and even horror. Some kind of bestial power emanates from the mental tension of this creature, placed in the courtyard of the museum. And I involuntarily see wonderful discoveries that this creature sitting on a rock is preparing for us in its tormenting mental effort. For example, the discovery of the atomic bomb, which threatens 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 that can wipe out all life on earth.

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

The picture was painted in 1895. A few years earlier, in 1887, the infamous circular was adopted.

This circular, approved by Emperor Alexander III and given the ironic name “on the cook’s children” in society, instructed the educational authorities to admit only well-to-do children to the gymnasium and progymnasium, that is, “only such children who are in the care of persons representing sufficient guarantee of the right over them home supervision and in providing them with the convenience necessary for their studies. My God, what a wonderful clerical syllable.

And further in the circular it was explained that “with the steadfast observance of this rule, gymnasiums and pro-gymnasiums will be freed from the admission of children of coachmen, lackeys, cooks, laundresses, small shopkeepers and similar people to them.

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

Though some people might get lucky. Because they were all lucky with the teacher. He was famous. Moreover, he was a teacher from God. His name was Sergey Alexandrovich Rachinsky. Today, he is almost unknown. And he so deserved all his life to remain in our memory. Take a closer look at him. Here he sits surrounded by his bastard 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 also by his entire mental make-up, by vocation. And he loved children.

Having gained learning, he returned to his native village of Tatevo. And he built this school that we see in the picture. Yes, and with a hostel for village children. Because, let's tell the truth, he did not accept everyone at 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 the chosen ones. He wanted to work with selected material. And he got the desired result. Therefore, do not be surprised that such a difficult task 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 forget his first teacher. No, he 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, along with other subjects, painting and drawing. 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 to the Trinity-Sergius Lavra, to 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, on Myasnitskaya Street. And what teachers he had! Polenov, Makovsky, Pryanishnikov. And then Repin. One of the paintings of the young artist "The Future Monk" was bought by Empress Maria Feodorovna herself.

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

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

The teacher wrote a mathematical equation on the blackboard. You can easily see it. And rewrite. And try to decide. 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 spent at least five minutes on it. Try it too. And I don't even bother. Because I didn't have such a teacher at school. Yes, I think that even if I had, I would not have succeeded. Well, I'm not a mathematician. And to this day.

And I realized this already in the fifth grade. Even though I was still very small, but even then I realized that all these brackets and squiggles in no way, in no way, would be useful to me in life. They won't come out sideways. And in no way these numbers did not excite my soul. On the contrary, they were only indignant. 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 icons useless and even harmful. And they evoked nothing but a quiet and unspoken hatred in me. And when all sorts of cosines with tangents came, complete darkness ensued. It pissed me off that all this algebraic bullshit only kept me away from 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 sines are. But I don’t feel any pain or regret about it either. The absence of this knowledge did not affect everything in my already and not small life. It is still a mystery to me today how electrons run at incredible speed inside an iron wire for terrible distances, creating an electric current. Yes, and that's not all. In some small fraction of a second, they can suddenly stop and run together back. Well, let them run, I think. Whoever is interested, let him do it.

But that's not the point. And the question was that even in those small years of my life I did not understand why it was necessary to torment me with something that my soul completely rejected. And I was right in my painful doubts.

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 must lay down so that the country does not lag behind others in its development, following the lead of losers 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 and a diamond with grains of gold, not to mention nuggets, is also not found. Well, so I and others like me were this very dump breed, which was all that was needed to nurture mathematicians and even mathematical prodigies that the country needed. But how could I then know about it with all my attempts to solve the equations that the good 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 escape from this obvious truth.

So it was, so it is, and so it will always be. 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 of my students, and in each group there are approximately 12 of them, two to three students will know the language. The rest are crap. Or dump rock, if you like. For various reasons.

It is you in the picture that you see eleven enthusiastic boys with burning eyes. But this is a picture. But life is not like that at all. And any teacher will tell you that.

There are different reasons why not. To be clear, let me give you the following example. A mother comes to me and asks how long it will take me to teach her boy French. I don't know what to answer her. I mean, I know, of course. But I don’t know how to answer without offending the assertive mother. And she should answer the following:

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 plant at least three tutor professors 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 have them at all. So the genes, God or someone else unknown to me decided. Here, for example, a girl wants to learn ballroom dancing, but God did not give her a sense of rhythm, no plasticity, or, just oh horror, an appropriate 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 moving thing. Today it is not, but tomorrow it appeared. That is what happened to me myself. My first teacher of French, dear Rosa Naumovna, seemed to be very surprised when she learned that it was her subject that would become the work of my whole life.

*****
But back to the teacher Rachinsky. I confess that I am immeasurably more interested in his portrait than in the personality of the artist. He was a well-born nobleman and not at all a poor man. He had his own estate. And to all this he had a learned head. After all, it was he who first translated The Origin of Species by Charles Darwin 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, which was 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 moved 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. He, 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, as it seemed to me, put forward ideas in this work, well, more than original ones. And Tolstoy himself was fascinated by the following thought of the great educator 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 the plants grown on another, one tree to bear the fruit of another. He mixes and confuses climates, elements, seasons. He disfigures his dog, his horse, his slave. He turns everything upside down, distorts everything, loves the ugly, the monstrous. He does not want to see anything the way nature created it, not excluding man: and he needs to train a man, like a horse for an arena, he needs to remake in his own way, as he uprooted a tree in his garden.

And in his declining years, Tolstoy tried to put into practice 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 souls 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, with his own money, a school and a hostel for gifted village children. And then he completely became the ideologist of the parochial school in the countries.

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

“If you please remember 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 in his estate, in the most remote wilderness of the Belsky district of the Smolensk province, and lives there without a break here for more than 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 became a true benefactor of the area, having founded and leads, with the help of 4 priests, 5 public schools, which now represent a model for the whole earth. This is a wonderful person. Everything 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 Nicholas II himself writes in the name of Sergei Rachinsky:

“The schools you founded and run, being among the parochial ones, have become a nursery of educated figures in the same spirit, a school of labor, sobriety and good morals, and a living model for all such institutions. The care that is close to my heart for public education, which you worthily serve, prompts me to express my sincere gratitude to you. I stay with you, benevolent Nikolay"

In conclusion, having plucked up courage, I want to add a few words of my own to the statements of the two persons mentioned above. These words will be about the teacher.

In the world there are a lot of professions. All living things on Earth are busy trying to prolong their existence. And above all, in order to find something to eat. Both herbivores and carnivores. Both the big ones and the smallest ones. All! And the man too. But a person has a lot of such opportunities. The choice of activities is overwhelming. That is, the occupations that a person indulges in in order to earn his bread, his 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 a routine, daily repetition of the same thing. The same mental and physical actions. Even in the so-called creative professions. I won't 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 necessary for retirement. And there's nothing you can do about it. Such is our human universe. It is arranged in a life who as can.

But, I repeat, there are few professions in which the whole life and the whole work of life is based solely on spiritual need. One of them is the teacher. Capitalized. I know what I'm talking about. Since I myself have been in this topic for many years. A teacher is both an earthly cross, and a calling, 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 profession written in the work book in the column - a teacher.

And you need to prove your right to be a teacher every day, from the very second when you crossed the threshold of the class. And sometimes it's not so easy. Do not think that beyond this threshold only happy moments of your life are waiting for you. And you should also not count on the fact that the small people will meet you all in anticipation of the knowledge that you are ready to put into their heads and souls. That the entire class space is inhabited entirely by angelic, incorporeal cherubs. These cherubs know how to bite like that sometimes. And how much it hurts too. This nonsense needs to be put out of your head. On the contrary, one must remember that in this bright room with huge windows, ruthless animals are waiting for you, who still have a difficult path to becoming human. And it is the teacher who must lead them along this path.

I distinctly remember one such "cherub" when I first came to class during my internship. I was warned. There is one boy there. Not very simple. And God help you deal with it.

How much time has passed, but I still remember it. If only because he had some strange last name. Noak. 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 when I appeared. Everyone got up. Except him. I realized that this Noak wanted to immediately declare to me and everyone else in this manner who is their boss here.

Sit down, children, I said. Everyone sat down and waited with interest to continue. Noack's leg remained in the same position. I approached him, still not knowing what to do or what to say.

Are you going to sit like this the whole lesson? Very uncomfortable posture! - I said, feeling a wave of hatred rise in me for this insolent, intent on disrupting 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 he even spat in the direction of the window. And then, not realizing what I was doing, I grabbed him by 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 unusual silence in the classroom. As if it were completely empty. Everyone looked at me dumbfounded. "Vo 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 the long journey of my teaching.

Ways of happy peak joyful moments and cruel disappointments. At the same time, I remember another teacher. Teacher Melnikov from the film "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 a reasonable, good eternal, and henbane grows - a thistle,” he once said in his hearts. And he wanted to leave school. At all! And he didn't leave. Because if you are a real teacher, then this is for you forever. Because you understand that you will not find yourself in any other business. Do not express yourself to the fullest. Got it - be patient. It is a great duty and a great honor to be a teacher. And this is exactly how Sergei Alexandrovich Rachinsky understood this, who, of his own free will, placed himself at the black blackboard for his entire life term.

P.S. If you still tried to solve this equation on the board, then the correct answer will be 2.

Lesson Objectives:

• development of the ability to observe;
• development of the ability to think;
• development of the ability to express thought;
• instilling an interest in mathematics;
• touching the art of N.P. Bogdanov-Belsky.

DURING THE CLASSES

Teaching is the work that educates and shapes a person.

Four pages from the life of a painting

Page one

The painting “Mental Account” was painted in 1895, that is, 110 years ago. This is a kind of anniversary of the picture, which is the creation of human hands. What is shown in the picture? Some boys have gathered around the blackboard and are looking at something. Two boys (these are the ones in front) turned away from the blackboard and remember something, or maybe they count. One boy whispers something into the ear of a man, presumably the teacher, while the other appears to be eavesdropping.

- And why are they in bast shoes?

“Why are there no girls here, only boys?”

Why are they standing with their backs to the teacher?

– What are they doing?

You have probably already understood that students and a teacher are depicted here. Of course, the costumes of the students are unusual: some of the guys are wearing bast shoes, and one of the characters in the picture (the one in the foreground), in addition, has a torn shirt. It is clear that this picture is not from our school life. Here is the inscription on the picture 1895 - the time of the old pre-revolutionary school. The peasants then lived in poverty, they themselves and their children wore bast shoes. The artist depicted peasant children here. Only at that time, few of them could study even in elementary school. Look at the picture: after all, only three of the students are wearing bast shoes, and the rest are in boots. Obviously, guys from rich families. Well, why girls are not depicted in the picture is also not difficult to understand: after all, at that time, girls, as a rule, were not accepted to school. Teaching was “not their business”, and not all of the boys studied.

Page two

This picture is called "Mental Account". See how the boy in the foreground of the picture thinks intently. It is evident that the teacher gave a difficult task. But, probably, this student will soon finish his work, and there should be no mistake: he takes mental counting very seriously. But the student who whispers something in the teacher's ear, apparently, has already solved the problem, only his answer is not quite correct. Look: the teacher listens attentively to the student's answer, but there is no approval on his face, which means that the student did something wrong. Or maybe the teacher patiently waits for others to count correctly, just like the first one, and therefore is in no hurry to approve his answer?

- No, the first one will give the correct answer, the one in front: it is immediately clear that he is the best student in the class.

And what task did the teacher give them? Can't we solve it too?

- But try it.

I will write on the board as you used to write:

(10 10+11 11+12 12+13 13+14 14):365

As you can see, each of the numbers 10, 11, 12, 13 and 14 must be multiplied by itself, the results added up, and the resulting sum divided by 365.

– This is the task (you won’t solve such an example soon, and even in your mind). But still try to count verbally, in difficult places I will help you. Ten ten is 100, everyone knows that. Eleven times eleven is also easy to count: 11 10=110, and even 11 is 121 in total. 144. I also calculated that 13 13=169 and 14 14=196.

But while I was multiplying, I almost forgot what numbers I got. Then I remembered them, and after all, these numbers still need to be added, and then the sum should be divided by 365. No, you yourself will not be able to calculate this.

- I'll have to help a little.

- What numbers did you get?

- 100, 121, 144, 169 and 196 - this was counted by many.

- Now you probably want to add all five numbers at once, and then divide the results by 365?

We will do it differently.

- Well, let's add the first three numbers: 100, 121, 144. How much will it be?

How much should be divided?

– Also on 365!

- How much will it be if the sum of the first three numbers is divided by 365?

- One! - everyone will figure it out.

- Now add the other two numbers: 169 and 196. How much will it be?

– Also 365!

- Here is an example, and quite simple. It turns out that only two!

- Only to solve it, you need to know well that the sum can be divided not all at once, but in parts, each term separately, or in groups of two or three terms, and then add up the resulting results.

Page three

This picture is called "Mental Account". It was painted by the artist Nikolai Petrovich Bogdanov-Belsky, who lived from 1868 to 1945.

Bogdanov-Belsky knew his little heroes very well: he grew up in their environment, was once a shepherd boy. “... I am the illegitimate son of a poor woman, that's why Bogdanov, and Belsky became the name of the county,” the artist said about himself.

He was lucky to get into the school of the famous Russian teacher Professor S.A. Rachinsky, who noticed the artistic talent of the boy and helped him get an art education.

N.P. Bogdanov-Belsky graduated from the Moscow School of Painting, Sculpture and Architecture, studied with such famous artists as V.D. Polenov, V.E. Makovsky.

Many portraits and landscapes were painted by Bogdanov-Belsky, but he remained in the memory of people, first of all, as an artist who managed to poetically and faithfully tell about the smart rural children eagerly reaching for knowledge.

Which of us is not familiar with the paintings “At the Doors of the School”, “Beginners”, “Composition”, “Village Friends”, “At the Sick Teacher”, “Voice Test”, - these are the names of just some of them. Most often, the artist depicts children at school. Charming, trusting, concentrated, thoughtful, full of lively interest and always marked by a natural mind - Bogdanov-Belsky knew and loved peasant children like that, immortalized in his works like that.

Page Four

The artist depicted non-fictional students and teachers in this picture. From 1833 to 1902, the famous Russian teacher Sergei Aleksandrovich Rachinsky, a remarkable representative of Russian educated people of the century before last, lived. He was a doctor of natural sciences and a professor of botany at Moscow University. In 1868 S.A. Rachinsky decides to go to the people. “He is taking the exam” for the title of primary school teacher. At his own expense, he opens a school for peasant children in the village of Tatyevo, Smolensk province, and becomes a teacher there. So, his students counted so well orally that all the visitors to the school were surprised at this. As you can see, the artist depicted S.A. Rachinsky with his students at the lesson of oral problem solving. By the way, the artist N.P. Bogdanov-Belsky was a student of S.A. Rachinsky.

This picture is a hymn to the teacher and the student.

This picture is called "Mental Accounting at the Rachinsky School", and it was painted by the same boy who stands in the picture in the foreground.
He grew up, graduated from this parochial school of Rachinsky (by the way, a friend of K.P. Pobedonostsev, an ideologist of parochial schools) and became a famous artist.
Do you know what we are talking about?

P.S. By the way, did you solve the problem?

"Verbal counting. In the folk school of S. A. Rachinsky ”- a painting by the artist N. P. Bogdanov-Belsky painted in 1985.

On the canvas we see a lesson in oral counting in a village school of the 19th century. The teacher is a very real, historical person. This is a mathematician and botanist, professor of Moscow University Sergey Alexandrovich Rachinsky. Carried away by the ideas of populism in 1872, Rachinsky came from Moscow to his native village of Tatevo and created a school there with a hostel for village children. In addition, he developed his own method of teaching oral counting. By the way, the artist Bogdanov-Belsky himself was a student of Rachinsky. Pay attention to the problem written on the board.

Can you decide? Try it.

About the rural school of Rachinsky, which at the end of the 19th century instilled in village children the skills of oral counting and the basics of mathematical thinking. The illustration to the note, a reproduction of Bogdanov-Belsky's painting, shows the process of solving the fraction 102+112+122+132+142365 in the mind. Readers were asked to find the simplest and most rational method of finding the answer.

As an example, a calculation variant was given, in which it was proposed to simplify the numerator of the expression by grouping its terms in a different way:

102+112+122+132+142=102+122+142+112+132=4(52+62+72)+112+(11+2)2=4(25+36+49)+121+121 +44+4=4×110+242+48=440+290=730.

It should be noted that this solution was found "honestly" - in the mind and blindly, while walking with a dog in a grove near Moscow.

More than twenty readers responded to the invitation to send their solutions. Of these, slightly less than half propose to represent the numerator in the form

102+(10+1)2+(10+2)2+(10+3)2+(10+4)2=5×102+20+40+60+80+1+4+9+16.

This is M. Graf-Lyubarsky (Pushkino); A. Glutsky (Krasnokamensk, Moscow Region); A. Simonov (Berdsk); V. Orlov (Lipetsk); Kudrin (Rechitsa, Republic of Belarus); V. Zolotukhin (Serpukhov, Moscow region); Y. Letfullova, 10th grade student (Ulyanovsk); O. Chizhova (Kronstadt).

The terms were even more rationally represented as (12−2)2+(12−1)2+122+(12+1)2+(12+2)2, when the products of ±2 by 1, 2 and 12 cancel each other out, Zlokazov; M. Likhomanova, Yekaterinburg; G. Schneider, Moscow; I. Gornostaev; I. Andreev-Egorov, Severobaykalsk; V. Zolotukhin, Serpukhov, Moscow region

Reader V. Idiatullin offers his own way of converting sums:

102+112+122=100+200+112−102+122−102=300+1×21+2×22=321+44=365;

132+142=200+132−102+142−102=200+3×23+4×24=269+94=365.

D. Kopylov (St. Petersburg) recalls one of the most famous mathematical discoveries of S. A. Rachinsky: there are five consecutive natural numbers, the sum of the squares of the first three of which is equal to the sum of the squares of the last two. These numbers are on the blackboard. And if the students of Rachinsky knew by heart the squares of the first fifteen to twenty numbers, the task was reduced to adding three-digit numbers. For example: 132+142=169+196=169+(200−4). Hundreds, tens and ones are added separately, and it remains only to calculate: 69−4=65.

Yu. Novikov, Z. Grigoryan (Kuznetsk, Penza region), V. Maslov (Znamensk, Astrakhan region), N. Lakhova (St. Petersburg), S. Cherkasov (Tetkino village, Kursk region) solved the problem in a similar way .) and L. Zhevakin (Moscow), who also proposed a fraction calculated in a similar way:

102+112+122+132+142+152+192+22365=3.

A. Shamshurin (Borovichi, Novgorod region) used a recursive formula like A2i=(Ai−1+1)2 to calculate the squares of numbers, which greatly simplifies calculations, for example: 132=(12+1)2=144+24+1 .

Reader V. Parshin (Moscow) tried to apply the rule of rapid raising to the second power from the book by E. Ignatiev “In the realm of ingenuity”, found an error in it, derived his own equation and applied it to solve the problem. In general, a2=(a−n)(a+n)+n2, where n is any number less than a. Then
112=10×12+12,
122=10×14+22,
132=10×16+32
and so on, then the terms are grouped rationally so that the numerator eventually becomes 700 + 30.

Engineer A. Trofimov (Ibresi village, Chuvashia) made a very interesting analysis of the numerical sequence in the numerator and converted it into an arithmetic progression of the form

X1+x2+...+xn, where xi=ai+1−ai.

For this progression, the statement

Xn=2n+1, i.e. a2n+1=a2n+2n+1,

Where does equality come from?

A2n+k=a2n+2nk+n2

It allows you to mentally count the squares of two or three digit numbers and can be used to solve the Rachinsky problem.

And finally, the correct answer turned out to be possible to obtain by estimates, and not by exact calculations. A. Polushkin (Lipetsk) notes that, although the sequence of squares of numbers is not linear, one can take the square of the average number - 12 five times, rounding it up: 144 × 5≈150 × 5 = 750. A 750:365≈2. Since it is clear that mental counting must operate with integers, this answer is certainly correct. It was received in 15 seconds! But it can still be checked additionally by making an estimate “from below” and “from above”:

102×5=500.500:365>1
142×5=196×5<200×5=1000,1000:365<3.

More than 1, but less than 3, hence - 2. V. Yudas (Moscow) made exactly the same estimate.

G. Poloznev (Berdsk, Novosibirsk region), the author of the note “A Prediction That Came True,” rightly noted that the numerator must certainly be a multiple of the denominator, that is, equal to 365, 730, 1095, etc. An estimate of the magnitude of partial sums unambiguously indicates the second number.

It is difficult to say which of the proposed methods of calculation is the simplest: everyone chooses his own based on the characteristics of his own mathematical thinking.

For more details, see: http://www.nkj.ru/archive/articles/6347/ (Science and Life, Oral Counting)

This painting also depicts Rachinsky and the author.

Working in a rural school, Sergei Aleksandrovich Rachinsky brought to the people: Bogdanov I. L. - an infectious disease specialist, doctor of medical sciences, corresponding member of the USSR Academy of Medical Sciences;
Vasiliev Alexander Petrovich (September 6, 1868 - September 5, 1918) - archpriest, confessor of the royal family, pastor-teetotaler, patriot-monarchist;
Sinev Nikolai Mikhailovich (December 10, 1906 - September 4, 1991) - Doctor of Technical Sciences (1956), Professor (1966), Honored. worker of science and technology of the RSFSR. In 1941 - deputy. ch. tank building designer, 1948-61 - early. Design Bureau at the Kirov Plant. In 1961-91 - deputy. prev. state to-that of the USSR on the use of atomic energy, laureate of the Stalin and State. prizes (1943, 1951, 1953, 1967); and many others.

S.A. Rachinsky (1833-1902), a representative of an ancient noble family, was born and died in the village of Tatevo, Belsky district, and meanwhile was a corresponding member of the Imperial St. Petersburg Academy of Sciences, who devoted his life to creating a Russian rural school. Last May marked the 180th anniversary of the birth of this outstanding Russian man, a true ascetic (there is an initiative to canonize him as a saint of the Russian Orthodox Church), a tireless worker, a rural teacher forgotten by us and an amazing thinker, whose L.N. Tolstoy learned to build a rural school, P.I. Tchaikovsky received recordings of folk songs, and V.V. Rozanov was spiritually instructed in matters of writing.

By the way, the author of the above-mentioned painting, Nikolai Bogdanov (Belsky is a pseudonym prefix, since the painter was born in the village of Shitiki, Belsky district, Smolensk province) came from the poor and was just a student of Sergei Alexandrovich, who created about three dozen rural schools and, at his own expense, helped his brightest students to realize themselves professionally, who became not only rural teachers (about forty people!) Or professional artists (three pupils, including Bogdanov), but also, say, a teacher of the king’s children, as a graduate of the St. Petersburg Archpriest Alexander Vasilyev of the Theological Academy, or a monk of the Trinity-Sergius Lavra, like Titus (Nikonov).

Rachinsky built not only schools, but also hospitals in Russian villages, the peasants of the Belsky district called him nothing more than "father of their own." Through the efforts of Rachinsky, sobriety societies were recreated in Russia, uniting tens of thousands of people throughout the empire by the beginning of the 1900s. Now this problem has become even more urgent, drug addiction has now grown to it. It is gratifying that the teetotal path of the educator is again taken up, that sobriety societies named after Rachinsky are reappearing in Russia, and this is not some AlAnon (an American society of anonymous alcoholics, reminiscent of a sect and, unfortunately, leaked to us in the early 1990s ). At the same time, we recall that before the October Revolution of 1917, Russia was one of the most non-drinking countries in Europe, second only to Norway.

Professor S.A. Rachinsky

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The writer V. Rozanov drew attention to the fact that the Tatev school of Rachinsky became the mother school, from which “more and more bees fly off to the side and in a new place do the deed and faith of the old. And this faith and deed consisted in the fact that Russian ascetic teachers looked at teaching as a holy mission, a great service to the noble goals of raising spirituality among the people.

* * *

“Did you manage to meet the heirs of Rachinsky’s ideas in modern life?” - I ask Irina Ushakova, and she talks about a man who shared the fate of the people's teacher Rachinsky: both his lifetime veneration and post-revolutionary scolding. In the 1990s, when she was just beginning to study the activities of Rachinsky, I. Ushakova often met with the Tatev school teacher Alexandra Arkadyevna Ivanova and wrote down her memoirs. Father A.A. Ivanova, Arkady Averyanovich Seryakov (1870-1929), was Rachinsky's favorite student. He is depicted in the painting by Bogdanov-Belsky "At the Sick Teacher" (1897) and, it seems, we see him at the table in the painting "Sunday Readings in a Rural School"; on the right, under the portrait of the sovereign, Rachinsky is depicted and, I think, Fr. Alexander Vasiliev.

N.P. Bogdanov-Belsky. Sunday readings at a rural school, 1895

In the 1920s, when the darkened people, along with the tempters, destroyed all the good things of the nobles along with the lord's estates, the Rachinsky family crypts were desecrated, the temple in Tatev was turned into a repair shop, the estate was plundered. All teachers, pupils of Rachinsky, were expelled from the school.

Remains of a house in the Rachinsky estate (photo 2011)

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In the book “S.A. Rachinsky and his school”, published in Jordanville in 1956 (our emigrants kept this memory, unlike us), tells about the attitude of the chief prosecutor of the Holy Synod K.P. Pobedonostsev, who on March 10, 1880 wrote to the heir to the crown prince, Grand Duke Alexander Alexandrovich (we read, as if, about our days): “The impressions of St. Petersburg are extremely difficult and bleak. To live at such a time and to see at every step people without direct activity, without a clear thought and firm decision, preoccupied with the small interests of their own self, immersed in the intrigues of their ambition, hungry for money and pleasure and idly chatting, is simply tearing the soul ... Kind impressions come only from within Russia, from somewhere in the countryside, from the wilderness. There is still a whole spring, from which it still breathes freshness: from there, and not from here, is our salvation.

There are people there with a Russian soul who do a good deed with faith and hope... Still, it is gratifying to see at least one such person... My friend Sergei Rachinsky, a truly kind and honest man. He was a professor of botany at Moscow University, but when he was tired of the strife and intrigues that arose there between the professors, he left the service and settled in his village, far from all railways ... He truly became a benefactor of the whole area, and God sent people to him - from the priests and landowners who work with him ... This is not chatter, but deed and true feeling.

On the same day, the heir to the crown prince answered Pobedonostsev: “... how you envy people who can live in the wilderness and bring true benefit and be far from all the abominations of city life, and especially St. Petersburg. I am sure that there are many such people in Russia, but we don’t hear about them, and they work quietly in the wilderness, without phrases and boasting ... "

N.P. Bogdanov-Belsky. At the school door, 1897

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N.P. Bogdanov-Belsky. Verbal counting. In the folk school S.A. Rachinsky, 1895

* * *

The "May Man" Sergei Rachinsky passed away on May 2, 1902 (according to the Art. Art.). Dozens of priests and teachers, rectors of theological seminaries, writers, scientists gathered for his burial. In the decade before the revolution, more than a dozen books were written about the life and work of Rachinsky, the experience of his school was used in England and Japan.

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

Now 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 in a one-room wooden school, while our children are taught so badly?!

Don't be quick to get angry. Take a look at the picture. Don't you think that the teacher looks too intelligent, somehow like a professor, and is dressed with obvious pretense? Why does the classroom have such a high ceiling and an expensive stove with white tiles? Did the village schools and the teachers in them really look like this?

Of course they didn't look like that. The picture is called "Mental counting in the folk school of S.A. Rachinsky." Sergei Rachinsky, a professor of botany at Moscow University, a man with certain government connections (for example, a friend of the chief prosecutor of the Synod Pobedonostsev), a landowner, abandoned all his affairs in the middle of his life, 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 that it taught for one year. In such a school they taught then 3-4 years (and in two-class schools - 4-5 years, in three-class 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 them all within the same lesson. It was quite a tricky thing: while the children of one year of study were doing some writing exercise, the children of the second year were answering at the blackboard, the children of the third year were reading a textbook, etc., and the teacher alternately paid attention to each group.

Rachinsky's pedagogical theory was very original, and its different parts somehow poorly converged 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 knew by heart a certain number of prayers would certainly grow up as a highly moral person, and the very sounds of the Church Slavonic language would 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 they needed to quickly count in their minds. Rachinsky was not very interested in teaching mathematical theory, but he did very well in mental arithmetic at his school. The students firmly and quickly answered 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 shown in the painting was the most complex mathematical operation studied at his school.

And finally, Rachinsky was a supporter of a very practical teaching of the Russian language - the 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’s clear that a peasant could come in handy in everyday life: simple letters, petitions, etc. Even at Rachinsky’s school some manual labor was taught, the children sang in chorus, And that's where education ends.

Rachinsky was a real enthusiast. School became his whole life. Rachinsky's children lived in a hostel and were organized into a commune: they performed all the housekeeping work for themselves and the school. Rachinsky, who had no family, spent all the time with the children from early morning until late in the evening, and since he was a very kind, noble and sincerely attached person to children, his influence on the students was enormous. By the way, Rachinsky gave the first child who solved the problem a gingerbread (in the literal sense of the word, he did not have a whip).

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 primary folk 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 elementary school teachers and priests, so he prepared children mainly for theological and teacher's seminaries. There were also significant exceptions - first of all, this is 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 influence in the capital's intellectual circles. The most important was the acquaintance with the ultra-influential Pobedonostsev. Under a certain influence of the ideas of Rachinsky, the spiritual department decided that there would be no sense in the Zemstvo school - the liberals would not teach children well - and in the mid-1890s began to develop their own independent network of parochial schools.

In some ways, the parish schools were similar to the Rachinsky school - they had a lot of Church Slavonic and prayers, and the rest of the subjects were reduced accordingly. But, alas, the dignity of the Tatev school was not transferred to them. Priests showed little interest in school affairs, ran schools under pressure, did not teach in these schools themselves, and hired the most third-rate teachers, and paid them noticeably less than in zemstvo schools. The peasants took a dislike to the parochial school, because they realized that they almost didn’t teach anything useful there, and prayers were of little interest to them. By the way, it was the teachers of the church school, recruited from pariahs of the clergy, who turned out to be one of the most revolutionized professional groups of that time, and it was through them that socialist propaganda actively penetrated into the village.

Now we see that this is a common thing - any author's pedagogy, designed for the deep involvement and enthusiasm of the teacher, immediately dies with mass reproduction, falling into the hands of uninterested and sluggish people. But for the time it was a big bummer. Church-parish schools, which by 1900 accounted for about a third of primary public schools, turned out to be disliked by everyone. When, beginning in 1907, the state began to allocate large amounts of money to primary education, there was no question of subsidizing church schools through the Duma; almost all the funds went to the Zemstvo.

The more common zemstvo school was quite different from the Rachinsky school. For starters, the Zemstvo considered the Law of God completely useless. It was impossible to refuse his teaching, for political reasons, so the zemstvos pushed him into a corner as best they could. The law of God was taught by an underpaid and neglected parish priest, with corresponding results.

Mathematics at 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. Up to raising to a degree, training did not reach, so the students of an ordinary elementary school simply would not understand the task depicted in the picture.

The zemstvo school tried to turn the teaching of the Russian language into world science, through the so-called explanatory reading. The method consisted in the fact that while dictating the educational text in the Russian language, the teacher also additionally explained to the students what the text itself says. In such a palliative way, the lessons of the Russian language 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. In terms of economics, the mass public school was much poorer, the mathematics course in it was shorter and simpler, and 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, how do students solve the problem on the board? Only direct, head-on: 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 materials 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.

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

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

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

Nearby stands 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 picture 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 the craving for knowledge on the part of the simple Russian people.

Warm colors bring 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 came together to make the only right decision. This is very 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 painting to his teacher, the great genius of mathematics, whom he knew and respected well. Now the picture is in Moscow in the Tretyakov Gallery, if you are there, be sure to take a look at the pen of the great master.

description-kartin.com

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

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

On the wave 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 mental counting, instilling in village children his skills and the 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 reigned in the classroom.

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

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

Solution of the Rachinsky problem

First way to solve

There are several ways to solve this expression. If you learned the 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: (100+121+144+169+196) divided by 365, which eventually becomes the quotient of 730 and 365, which is: intermediate answers.

The second way to solve

If you didn’t learn the squares of numbers up to 20 in school, then a simple method based on the use of a reference number may come in handy. This method allows you to simply and quickly multiply any two numbers less than 20. The method is very simple, you need to add the unit of the second to the first number, multiply this amount by 10, and then add the product of units. 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 task can be solved in the same way as shown in the first method.

The third solution

Another way involves using a simplification of the numerator of a fraction, based on the use of the formulas of 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, simply divide this number by 2 and multiply by 10, which equals 720. Then we divide this expression by 365 and get: 2.

The fourth solution

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

Rachinsky sequences for mental counting

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

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

10 2 +11 2 +12 2 = 13 2 +14 2 (the 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 Rachinsky sequence, it is enough just to write an equation of the following form (note that always in such a sequence the number of summed squares on the right is one less than on the left):

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

This equation reduces to a quadratic equation and is easily solved. In this case, "n" is 3, which corresponds to the first Rachinsky sequence described above (3 2 +4 2 = 5 2).

Thus, the solution of the famous Rachinsky example can be mentally generated even faster than described in this article, simply by knowing the second Rachinsky 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 tasks for mental counting" by Sergey Rachinsky.

Evgeny Buyanov