The famous Russian artist Nikolai Petrovich Bogdanov-Belsky painted a unique and incredible life story in 1895. The work is called “Oral Reckoning”, and in full version"Verbal counting. At the public school of S. A. Rachinsky."
Nikolai Bogdanov-Belsky. Verbal counting. At the public school of S. A. Rachinsky
The painting is done in oil on canvas and depicts a 19th century rural school during an arithmetic lesson. Schoolchildren solve interesting and complex example. They are deep in thought and searching for the right solution. Someone thinks at the board, someone stands on the sidelines and tries to collate 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.
Standing nearby is a teacher, whose prototype is Rachinsky himself, a famous botanist and mathematician. It is not for nothing that the painting 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, children are dressed in peasant clothes: bast shoes, trousers and shirts. All this fits very harmoniously and laconically into the plot, unobtrusively bringing to the world a thirst for knowledge on the part of the ordinary Russian people.
The warm color scheme brings the kindness and simplicity of the Russian people, there is no envy and falsehood, no evil and hatred, children from different families with different incomes came together to make the only right decision. This is sorely lacking in our modern life, where people are used to living completely differently, 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 painting is in Moscow in 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, Shitiki village, Belsky district, Smolensk province, Russia - February 19, 1945, Berlin, Germany) - Russian Itinerant artist, academician of painting, chairman of the Kuindzhi Society.
The painting shows a village school late XIX century during an arithmetic lesson while solving fractions in your head. Teacher - a real man, 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 dormitory for peasant children, developed a unique method of teaching mental arithmetic, instilling in the village children his skills and the basics of mathematical thinking. Bogdanov-Belsky, himself a former student of Rachinsky, dedicated his work to an episode from the life of the school with the creative atmosphere that reigned in the lessons.
There is an example written on the chalkboard that students need to solve:
The problem depicted in the picture could not be presented to students in a standard primary school: the curriculum of one-class and two-class primary public schools did not provide for the study of the concept of degree. However, Rachinsky did not follow the standard training course; he was confident in the excellent mathematical abilities of most peasant children and considered it possible to significantly complicate the mathematics curriculum.
Solution of Rachinsky's problem
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 becomes the quotient of 730 and 365, which equals: 2. To solve the example this way, you may need to use mindfulness skills and the ability to keep a few things in mind intermediate answers.
Second solution
If you didn’t learn the meaning of squares of numbers up to 20 at school, then a simple method based on the use of a reference number may be useful to you. 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 amount by 10, and then add the product of the units. For example: 11*11=(11+1)*10+1*1=121. The remaining 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.
Third solution
Another method involves using a simplification of the numerator of a 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 a 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 the formulas for the square of the sum and the square of the difference well, then you will understand how this expression can easily be 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.
Fourth solution
Also, this problem can be solved in 1 second if you know the Rachinsky sequences.
Rachinsky sequences for mental arithmetic
To solve the famous Rachinsky problem, you can also use additional knowledge about the laws of the sum of squares. It's about specifically about those sums that are called Rachinsky sequences. So it can be mathematically proven 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 (sum equals 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, simply construct an equation of the following form (note that in such a sequence the number of summable squares on the right is always one less than on the left):
n 2 + (n+1) 2 = (n+2) 2
This equation reduces to quadratic equation and is easy to solve. IN in this case"n" equals 3, which corresponds to the first Raczynski sequence described above (3 2 +4 2 = 5 2).
Thus, the solution to the famous Rachinsky example can be done in your mind even faster than was 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 Bogdan-Belsky’s painting takes the form (365 + 365)/365, which undoubtedly equals two.
Also, Rachinsky’s sequence can be useful for solving other problems from the collection “1001 problems for mental calculation” by Sergei Rachinsky.
Evgeny Buyanov
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 start to finish, these paintings, lined up in one line, should tell the story of the development of our painting. With all that is no 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 story being flawed. But there are also some that are not required to be shown at all. And my choice here depends only on me. From my disposition towards the group, from my mood, and also from the availability of free time.
Well, the painting “Oral Account” by the artist Bogdan-Belsky is purely 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 will be attracted to this particular picture to such an extent that it will be simply impossible not to stop. Well, don’t drag them away by force.
Why? This artist is not one of the most famous Russian painters. His name is known mostly to specialists - art critics. But this picture will nevertheless make anyone stop. And it will attract the attention of a foreigner no less.
So we stand, and for a long time we look at everything in it with interest, even the most small parts. 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, it’s as if I started giving comments at a time when the ear wants to enjoy the melody that has captured us.
Nevertheless, some clarifications 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! There is so much in this sound! Thought in commonwealth created man with difficulty. 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 it gives rise to some kind of Strange feeling. And, oddly enough, there is a feeling of fear, and even horror. Some kind of animal power emanates from the mental tension of this creature, placed in the courtyard of the museum. And I involuntarily see each other wonderful discoveries, which this creature sitting on a rock prepares for us in its painful mental effort. For example, the discovery of an atomic bomb, which threatens to destroy humanity itself along with this Thinker. And we already know for certain that this beast-like 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 frighten me at all. Against. I look at them and feel a warm sympathy for them arise in my soul. I want to smile. And I feel the joy that flows to my heart from contemplating the touching scene. The mental search expressed in the faces of these boys fascinates and excites me. It also makes you think about something else.
The painting was painted in 1895. A few years earlier, in 1887, the infamous circular was adopted.
By this circular, approved by the Emperor Alexander III and which received the ironic name in society “about cook’s children”, the educational authorities were ordered to admit only wealthy children to gymnasiums and pro-gymnasiums, that is, “only those children who are in the care of persons who provide sufficient guarantee of proper home supervision over them and in providing them with the necessary For training sessions facilities". My God, what a wonderful clerical style.
And further in the circular it was explained that “with strict observance of this rule, gymnasiums and pro-gymnasiums will be freed from the enrollment of children of coachmen, footmen, cooks, laundresses, small shopkeepers and the like.
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.”
Although maybe someone will get lucky. Because they were all lucky to have a teacher. He was famous. Moreover, he was a teacher from God. His name was Sergei Alexandrovich Rachinsky. Today he is hardly known. And he deserved it with all his life to remain in our memory. Take a closer look at him. Here he sits 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 also by his entire spiritual makeup, 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. And even with a hostel for village children. Because, let’s be honest, he didn’t accept everyone into school. He himself selected, unlike Leo Tolstoy, who accepted all the surrounding children into his school.
Rachinsky created own methodology for mental calculation, which, of course, not everyone could master. Only the chosen ones. He wanted to work with selected material. And he achieved the desired result. Therefore, do not be surprised that such a complex problem is solved by children in bast shoes and graduation shirts.
And the artist Bogdanov-Belsky himself went through this school. And how could he forget his first teacher? No, I couldn't. And this picture is a tribute to the memory of my 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 master 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 also Repin. One of the paintings young artist“The Future Monk” was bought by Empress Maria Feodorovna herself.
That is, Sergei Alexandrovich gave him a start in life. And how could an already accomplished artist thank his teacher after this? But only this very picture. This is the most he could do. And he did the right thing. Thanks to him, we also have a visible image of this today. wonderful person, Rachinsky's teacher.
The boy was lucky, of course. Just incredibly lucky. Well, who was he? Illegitimate son farmhands! And what kind of future could he have had if he had not gone to the school of the famous teacher?
The teacher wrote a mathematical equation on the board. You can see it easily. And rewrite. And try to decide. Once there was a math teacher in my group. He carefully copied the equation onto a piece of paper in a notebook and began to solve. And I decided. And he spent at least five minutes on it. Try it too. But I don’t even dare. Because at school I didn’t have such a teacher. Yes, I think that even if I had, nothing would have worked out 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 very small, I already realized that all these brackets and squiggles would in no way, in any way, be useful to me in life. They won't come out in any way. And these numbers didn’t bother my soul at all. On the contrary, they only outraged. And my soul does not lie with 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 quiet and unspoken hatred in me. And when all sorts of cosines and tangents arrived, it came complete darkness. It infuriated me that all this algebraic bullshit only distracted me 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 suffering or regrets about this. The lack of this knowledge did not affect my entire life, which is no longer small. It is still a mystery to me today how electrons run at incredible speeds inside an iron wire over terrible distances, creating an electric current. And that's not all. In a small fraction of a second, they can suddenly stop and run back together. Well, let them run, I think. Whoever is interested in this, let him do it.
But that's not the question. And the question was that even in those small years I didn’t understand why it was necessary to torment me with something that 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 must be laid Public School so that the country does not lag behind others in its development, 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 waste, unnecessary, empty. But without this unnecessary rock, a diamond with grains of gold, not to mention nuggets, cannot be found either. Well, I and people like me were this very dump breed, which was only needed to raise the mathematicians and even mathematical prodigies the country needed. But how could I know about this then with all my attempts to solve the equations that kind teacher wrote to us on the board. That is, with my torment and inferiority complexes I contributed to the birth of real mathematicians. And there is no way to escape 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 there are approximately 12 of them in each group, two or three students will know the language. The rest suck. Or dump rock, if you like. For various reasons.
In the picture you see eleven enthusiastic boys with sparkling eyes. But this is a picture. But in life it’s not at all like that. And any teacher will tell you this.
There are various reasons why this is not the case. To be clear, I will give next 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 needs to answer the following:
Language in 16 hours - this is only on TV. I don't know your boy's level of interest and motivation. There is no motivation - and even if you put at least three professor-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 have them at all. 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 a sense of rhythm, or plasticity, or, oh, horror, an appropriate figure (well, she became fat or lanky). And I want it that way. What will you do here if nature itself stands in the way? And so it is in every case. And in language learning too.
But, really, at this point I want to put a big comma on myself. Not so simple. Motivation is a moving thing. Today it is not there, but tomorrow it appears. That is, what happened to myself. My first French teacher, dear Rosa Naumovna, seemed greatly surprised when she learned that her subject would become my life’s work.
*****
But let's return to 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 for all this he had a scientific 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 deep religious person. And at the same time he translated the famous materialist 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 inspired him to the cause of public education. Even in his youth, Tolstoy was fascinated by the ideas of Jean-Jacques Rousseau; the Great Enlightener was his idol. He, for example, wrote a wonderful pedagogical work “Emil or on education.” I not only read it, but wrote from it coursework At the institute. To tell the truth, Rousseau, it seemed to me, put forward ideas in this work that were more than original. 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 fruits characteristic of another. He mixes and confuses climates, elements, seasons. He mutilates his dog, his horse, his slave. He turns everything upside down, distorts everything, loves ugliness, the monstrous. He doesn’t want to see anything the way nature created it, not excluding man: he needs to train a man, like a horse for an arena, he needs to remake him in his own way, just as he uprooted a tree in his garden.”
And in his declining years, Tolstoy tried to put into practice the wonderful idea outlined above. He wrote textbooks and manuals. He wrote the famous "ABC" and also wrote children's stories. Who doesn’t know the famous Filipp or the story about the bone.
*****
As for Rachinsky, here, as they say, two kindred souls met. So much so that, inspired by Tolstoy’s ideas, Rachinsky left Moscow and returned to his ancestral village of Tatevo. And built according to example famous writer with my own money, a school and a hostel for gifted village children. And then he completely became the ideologist of church and parish schools in the country.
This activity of his in the field of public education was noticed at the very top. Read what Pobedonostsev wrote about him to Emperor Alexander III:
“You will 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 on his estate, in the most remote forest wilderness of the Belsky district of the Smolensk province, and lives there forever. for more than 14 years, working from morning to night for the benefit of the people. He inhaled completely new life into a whole generation of peasants... He truly became a benefactor of the area, having founded and led, with the help of 4 priests, 5 public schools, which now represent a model for the entire land. This is a wonderful person. He gives everything he has and all the resources of his estate to this cause, limiting his needs to the last degree.”
And here is what Nicholas II himself writes to Sergei Rachinsky:
“The schools founded and led by you, being among the parochial ones, became a nursery for educated leaders in the same spirit, a school of labor, sobriety and good morals, and a living model for all similar institutions. Close to My heart is the concern for public education, whom you serve worthily, prompts Me to express My sincere gratitude to you. I am with you, my kind Nikolai.”
In conclusion, having gathered the courage, I want to add a few words of my own 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 trying to prolong its existence. And above all, to find something to eat. Both herbivores and carnivores. Both the biggest and the smallest. All! And the person too. But a person has a great many such possibilities. The choice of activities is enormous. That is, activities 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 provide complete satisfaction for the soul. The vast majority of all other things come down to 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. This is our human universe. Anyone gets settled in life as best they 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. WITH capital letters. I know what I'm talking about. Since I’m already in this topic long years. A teacher is an earthly cross, a calling, torment, and joy all together. Without all this there is no teacher. And there are plenty of them, even among those who have work book in the profession column it is written - teacher.
And you have to prove your right to be a teacher every day, from the very second you cross the threshold of the classroom. And sometimes this is not so easy. Don’t think that beyond this threshold only happy moments of your life await you. And you also don’t have to 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 classroom space is populated entirely by angel-like, disembodied cherubs. These cherubs can sometimes bite like that. And how painful it is too. This nonsense needs to be thrown out of your head. Just the opposite, you need to 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 clearly remember one such “cherub” when I first appeared in class during an internship. I was warned. There is one boy there. Not very simple. And God will help you cope with it.
How much time has passed, but I still remember it. If only because he had some kind of strange surname. Noak. That is, I knew that the PLA is the People's Liberation Army of China. But here... I went in and instantly identified this asshole. This sixth grader, sitting at the last desk, put one of his legs on the table when I appeared. Everyone stood up. Except him. I realized that this Noak wanted to immediately tell me and everyone else in this manner who was their boss here.
Sit down, children,” I said. Everyone sat down and began to wait with interest for the continuation. Noak's leg remained in the same position. I approached him, not yet knowing what to do or what to say.
Why are you just going to sit for the entire lesson? Very uncomfortable position! - I said, feeling a wave of hatred rise in me towards this impudent person who intended to disrupt my first lesson in my life.
He didn’t answer anything, turned away and made a forward movement with his lower lip as a sign of complete contempt for me. And he even spat towards the window. And then, no longer realizing what I was doing, I grabbed him by the collar and kicked him in the ass out of the classroom and into the corridor. Well, he was still young and hot. There was an unusual silence in the class. As if it were completely empty. Everyone looked at me in shock. “Yes,” 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 a long journey of my teaching.
Paths 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 there was a reason! “You sow here what is reasonable, good and eternal, and henbane grows - thistle,” he once said in his hearts. And I 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. You cannot express yourself to the fullest. Take it - be patient. It is a great duty and a great honor to be a teacher. And this is exactly how Sergei Aleksandrovich Rachinsky understood it, who of his own free will placed himself in prison for his entire life sentence. chalkboard.
P.S. If you still tried to solve this equation on the board, then the correct answer will be 2.
Many have seen the picture “Mental arithmetic in a public school.” The end of the 19th century, a public school, a blackboard, an intelligent teacher, poorly dressed children, 9–10 years old, enthusiastically trying to solve a problem written on the blackboard in their minds. The first person to decide tells the answer to the teacher 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 =???
Crap! Crap! Crap! 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, but our children were taught so poorly?!
Don't rush to be indignant. Take a closer look at the picture. Don’t you think that the teacher looks too intelligent, somehow like a professor, and is dressed with obvious pretension? Why in school class such a high ceiling and an expensive stove with white tiles? Is this really what village schools and their teachers looked like?
Of course, they didn't look like that. The painting is called "Oral arithmetic in the public school of S.A. Rachinsky." Sergei Rachinsky - professor of botany at Moscow University, a man with certain government relations(for example, a friend of the Chief Prosecutor of the Synod Pobedonostsev), a landowner - in the middle of his life he abandoned all his affairs, went to his estate (Tatevo in the Smolensk province) and started there (of course, at his own expense) an experimental public school.
The school was one-class, which did not mean that they taught there for one year. In such a school they taught for 3-4 years (and in two-year schools - 4-5 years, in three-year schools - 6 years). The word one-class meant that children of three years of study form a single class, and one teacher teaches them all within one lesson. It was a rather tricky business: while the children of one year of study were doing some kind of written 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 did not fit together well. Firstly, Rachinsky considered the basis of education for the people to be teaching the Church Slavonic language and the Law of God, and not so much explanatory as consisting in memorizing prayers. Rachinsky firmly believed that a child who knew a certain number of prayers by heart would certainly grow up to be a highly moral person, and the very sounds of the Church Slavonic language would already have a moral-improving effect. To practice the language, Rachinsky recommended that children hire themselves out to read the Psalter over the dead (sic!).
Secondly, Rachinsky believed that it was useful and necessary for peasants to quickly count in their heads. Rachinsky had little interest 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. Squaring, as 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, and they were not taught theoretical grammar at all. The main thing was to learn to read and write fluently, albeit in clumsy handwriting and not very competently, but clearly, something that could be useful to a peasant in everyday life: simple letters, petitions, etc. Even at Rachinsky’s school, some manual labor was taught, children sang in chorus, and that was where all the 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 maintenance work for themselves and the school. Rachinsky, who had no family, spent all his time with the children from early morning until late evening, and since he was a very kind, noble person and sincerely attached to children, his influence on his students was enormous. By the way, Rachinsky gave the first child who solved the problem a carrot (in the literal sense of the word, he didn’t have a stick).
School classes themselves took 5–6 months a year, and the rest of the time Rachinsky individually studied with older children, preparing them for admission to various educational institutions of the next level; the primary public school was not directly connected with others educational institutions and after it it was impossible to continue training without additional preparation. Rachinsky wanted to see the most advanced of his students become primary school teachers and priests, so he prepared children mainly for theological and teacher seminaries. There were also significant exceptions - first of all, this was the author of the picture himself, Nikolai Bogdanov-Belsky, whom Rachinsky helped to get into Moscow school painting, sculpture and architecture. But, oddly enough, leading peasant children along the main road educated person- gymnasium / university / civil service- Rachinsky did not want to.
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 the certain influence of Rachinsky's ideas, the religious department decided that the zemstvo school would be of no use - the liberals would not teach children anything good - and in the mid-1890s they began to develop their own independent network of parochial schools.
In some ways, parochial schools were similar to Rachinsky's school - they had a lot of Church Slavonic language and prayers, and other subjects were correspondingly reduced. But, alas, the advantages of the Tatev school were not passed on to them. The priests had little interest in school affairs, ran the 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 did not like the parochial school, because they realized that they hardly taught anything useful there, and they were of little interest in prayers. 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 original pedagogy, designed for the deep involvement and enthusiasm of the teacher, immediately dies during mass reproduction, falling into the hands of uninterested and lethargic people. But for that time it was a big bummer. Parochial schools, which by 1900 made up about a third of primary public schools, turned out to be disliked by everyone. When, starting in 1907, the state began to send elementary education a lot of money, there was no question of passing subsidies to church schools through the Duma; almost all the funds went to the zemstvo residents.
The more widespread zemstvo school was quite different from Rachinsky’s school. To begin with, the Zemstvo people considered the Law of God completely useless. It was impossible to refuse his teaching, according to 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 underpaid and ignored, with corresponding results.
Mathematics in the zemstvo school was taught worse than in Rachinsky, and in a smaller volume. The course ended with operations with simple fractions and non-metric system of measures. The teaching did not go as far as exponentiation, so ordinary elementary school students simply would not understand the problem depicted in the picture.
The zemstvo school tried to turn the teaching of the Russian language into world studies, through the so-called explanatory reading. The technique consisted in the fact that while dictating an educational text in the Russian language, the teacher also additionally explained to the students what was said in the text itself. In this palliative way, Russian language lessons also turned into geography, natural history, history - that is, into all those developmental subjects that had no place in the short course of a one-grade school.
So, our picture depicts not a typical, but 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 a problem on the board? Only straight forward: 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 counting techniques, omitting all arithmetic and algebraic transformations that required calculations on paper.
For some reason, the picture shows only boys, while all the materials show that Rachinsky taught children of both sexes. What this means is unclear.
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Full title famous painting which is pictured above: " Verbal counting. At the public school of S. A. Rachinsky " This painting by the Russian artist Nikolai Petrovich Bogdanov-Belsky was painted in 1895, and now hangs in the Tretyakov Gallery. In this article you will learn some details about it. famous work, who Sergei Rachinsky was, and most importantly - get the correct answer to the task shown on the board.
Brief description of the painting
The painting depicts a 19th-century rural school during an arithmetic lesson. The teacher figure has real prototype— Sergei Aleksandrovich Rachinsky, botanist and mathematician, professor at Moscow University. Rural schoolchildren decide very interesting example. It is clear that it is not easy for them. In the picture, 11 students are thinking about the problem, but it seems that only one boy has figured out how to solve this example in his head, and quietly speaks his answer into the teacher’s ear.
Nikolai Petrovich dedicated this painting to his school teacher Sergei Aleksandrovich Rachinsky, who is depicted on it in the company of his students. Bogdanov-Belsky knew the characters in his film very well, since he himself had once been in their situation. He was lucky enough to get into the school of the famous Russian teacher Professor S.A. Rachinsky, who noticed the boy’s talent and helped him get an art education.
About Rachinsky
Sergei Alexandrovich Rachinsky (1833-1902) - Russian scientist, teacher, educator, professor at Moscow University, botanist and mathematician. Continuing the endeavors of his parents, he taught at a rural school, even though the Rachinskys - noble family. Sergei Alexandrovich was a man of diverse knowledge and interests: in the school art workshop, Rachinsky himself taught painting, drawing and drawing classes.
IN early period In his teaching career, Rachinsky searched in line with the ideas of the German teacher Karl Volkmar Stoy and Leo Tolstoy, with whom he corresponded. In the 1880s, he became the main ideologist of the parochial school in Russia, which began to compete with the zemstvo school. Rachinsky came to the conclusion that the most important practical need of the Russian people is communication with God.
As for mathematics and mental arithmetic, Sergei Rachinsky left as a legacy his famous problem book “ 1001 mental arithmetic problems ", some tasks (with answers) from which you can find at.
Read more about Sergei Alexandrovich Rachinsky on his biography page.
Solution to the example on the board
There are several ways to solve the expression written on the board in Bogdanov-Belsky’s painting. By following this link you will find four different solutions. If at school you learned squares of numbers up to 20 or up to 25, then most likely the task on the board will not cause you much difficulty. This expression is equal to: (100+121+144+169+196) divided by 365, which ultimately equals 730 divided by 365, which is “2”.
In addition, on our website in the “” section you can meet Sergei Rachinsky and find out what “” is. And it is the knowledge of these sequences that allows you to solve the problem in a matter of seconds, after all.
known to many. The painting depicts a late 19th century village school during an arithmetic lesson while solving fractions in one's head.
The teacher is a real person, Sergei Aleksandrovich 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 dormitory for peasant children, developed a unique method of teaching mental arithmetic, instilling in the village children his skills and the basics of mathematical thinking. Bogdanov-Belsky, himself a former student of Rachinsky, dedicated his work to an episode from the life of the school with the creative atmosphere that reigned in the lessons.
However, for all the fame of the picture, few who saw it delved into the content of that " difficult task", which is depicted on it. It consists of quickly finding the result of a calculation by mental calculation:
| 10 2 + 11 2 + 12 2 + 13 2 + 14 2 |
| 365 |
The talented teacher cultivated mental counting in his school, based on the masterly use of the properties of numbers.
The numbers 10, 11, 12, 13 and 14 have an interesting feature:
10 2 + 11 2 + 12 2 = 13 2 + 14 2 .
Indeed, since
100 + 121 + 144 = 169 + 196 = 365,
Wikipedia suggests 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 = 2·365.
In my opinion, it’s too tricky. It's easier to do it differently:
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,
| 730 | = 2. |
| 365 |
The above reasoning can be carried out orally - 12 2 , of course, you need to remember, double the products of the squares of binomials to the left and right of 12 2 are mutually destroyed and they can not be counted, but 5·144 = 500 + 200 + 20 - not difficult.
Let’s use this technique and verbally find the sum:
48 2 + 49 2 + 50 2 + 51 2 + 52 2 = 5 50 2 + 10 = 5 2500 + 10 = 12510.
Let's complicate it:
84 2 + 87 2 + 90 2 + 93 2 + 96 2 = 5 8100 + 2 9 + 2 36 = 40500 + 18 + 72 = 40590.
Rachinsky series
Algebra gives us a means of asking the question about this interesting feature of a series of numbers
10, 11, 12, 13, 14
more generally: is this the only series 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 required 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.
Opening the brackets and making simplifications, we get:
x 2 - 10x - 11 = 0,
where
x 1 = 11, x 2 = -1.
There are, therefore, two series of numbers that have the required property: the Raczynski series
10, 11, 12, 13, 14
and a row
2, -1, 0, 1, 2.
Indeed,
(-2) 2 +(-1) 2 + 0 2 = 1 2 + 2 2 .
Two!!!
I would like to finish with the bright and touching memories of the author of the author’s blog, V. Iskra, in the article About the squares of two-digit numbers and not only about them...
Once upon a time, around 1962, our “mathematician”, Lyubov Iosifovna Drabkina, gave this task to us, 7th graders.
At that time I was very interested in the newly appeared KVN. I was rooting for the team from the Moscow region town of Fryazino. The “Fryazinians” were distinguished by their special ability to use logical “express analysis” to solve any problem, to “pull out” the most tricky issue.
I couldn't do the math quickly in my head. However, using the “Fryazin” method, I figured that the answer should be expressed as an integer. Otherwise, this is no longer an “oral count”! This number could not be one - even if the numerator had the same 5 hundreds, the answer would be clearly greater. On the other hand, he clearly didn’t reach the number “3”.
- Two!!! - I blurted out, a second ahead of my friend, Lenya Strukov, the best mathematician in our school.
“Yes, indeed two,” Lenya confirmed.
- What did you think? - asked Lyubov Iosifovna.
- I didn’t count at all. Intuition - I answered to the laughter of the whole class.
“If you didn’t count, the answer doesn’t count,” Lyubov Iosifovna made a pun. Lenya, didn’t you count either?
“No, why not,” Lenya answered sedately. I had to add 121, 144, 169 and 196. I added numbers one and three, two and four in pairs. It is more comfortable. It turned out 290+340. The total amount, including the first hundred, is 730. Divide by 365 and we get 2.
- Well done! But remember for the future - in a series of double-digit numbers - the first five of its representatives have an amazing property. The sum of the squares of the first three numbers in the series (10, 11 and 12) is equal to the sum of the squares of the next two (13 and 14). And this sum is equal to 365. Easy to remember! So many days in 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, even more pictures. The topics were very different - the defense of Rostov, space... In the central passage, under the Engels intersection (now Bolshaya Sadovaya) - Voroshilovsky made a whole panorama about the main stages life path Soviet man- maternity hospital - kindergarten- school, prom...
In one of the “school” paintings one could see a familiar scene - the solution to a problem... Let’s call it like this: “Rachinsky’s problem”...
...Years passed, people passed... Cheerful and sad, young and not so young. Some remembered their school, while others “used their brains”...
The master tilers and artists, led by Yuri Nikitovich Labintsev, did a wonderful job!
Now " Rostov miracle"temporarily unavailable." Trade came to the fore - literally and figuratively. Still, let’s hope that in this common phrase the main word is “temporarily”...
Sources: Ya.I. Perelman. Entertaining algebra (Moscow, “Science”, 1967), Wikipedia,