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The full title of the famous painting shown 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 this famous work, who Sergei Rachinsky was, and most importantly, you will 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 figure of the teacher has a real prototype - Sergei Aleksandrovich Rachinsky, botanist and mathematician, professor at Moscow University. Rural schoolchildren solve a 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 in 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 were a 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 the early period of 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.
The famous Russian artist Nikolai Petrovich Bogdanov-Belsky wrote a unique and incredibly life story in 1895. The work is called “Oral Account”, and in the full version “Oral Account. 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. Students solve an 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 or falsehood, no evil or 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 accustomed 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 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, 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 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 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 task depicted in the picture could not be offered to students of a standard primary school: the curriculum of one- and two-class primary public schools did not provide for the study of the concept of degree. However, Raczynski did not follow a typical 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 located:
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 simplifying 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. We are talking 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 a quadratic equation and is easy to solve. 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
In one of the halls of the Tretyakov Gallery you can see a famous painting by the artist N.P. Bogdanov-Belsky “Oral calculation”. It depicts a lesson in a rural school. The classes are taught by an old teacher. Village boys in poor peasant shirts and bast shoes crowded around. They are focused and enthusiastically solving the problem proposed by the teacher... The plot is familiar to many from childhood, but not many know that this is not the artist’s imagination and behind all the characters in the picture there are real people, painted by him from life - people whom he knew and loved, and the main character is an elderly teacher, a man who played a key role in the artist’s biography. His fate is surprising and extraordinary - after all, this man is a wonderful Russian educator, teacher of peasant children, Sergei Alexandrovich Rachinsky (1833-1902)
N.P. Bogdanov-Belsky "Oral calculation in the Rachinsky public school" 1895.
Future teacher S.A. Rachinsky.
Sergei Alexandrovich Rachinsky was born on the Tatevo estate, Belsky district, Smolensk province, into a noble family. His father Alexander Antonovich Rachinsky, a former participant in the December movement, was exiled to his family estate of Tatevo for this. Here, on May 2, 1833, the future teacher was born. His mother was the sister of the poet E.A. Baratynsky and the Rachinsky family closely communicated with many representatives of Russian culture. In the family, parents paid great attention to the comprehensive education of their children. All this was very useful to Rachinsky in the future. Having received an excellent education at the Faculty of Natural Sciences of Moscow University, he travels a lot, meets interesting people, studies philosophy, literature, music and much more. After some time, he writes several scientific papers and receives a doctorate and a professorship in botany at Moscow University. But his interests were not limited to scientific frameworks. The future rural teacher was engaged in literary creativity, wrote poetry and prose, played the piano to perfection, and was a collector of folklore - folk songs and handicrafts. Khomyakov, Tyutchev, Aksakov, Turgenev, Rubinstein, Tchaikovsky and Tolstoy often visited his apartment in Moscow. Sergei Alexandrovich was the author of the libretto for two operas by P.I. Tchaikovsky, who listened to his advice and recommendations and dedicated his first string quartet to Rachinsky. With L.N. Tolstoy Rachinsky had friendly and family relations, since the niece of Sergei Alexandrovich, the daughter of his brother, the rector of the Petrovsky (now Timiryazevsky) Academy Konstantin Aleksandrovich Rachinsky, Maria was the wife of Sergei Lvovich, Tolstoy’s son. The correspondence between Tolstoy and Rachinsky is interesting, full of discussions and disputes about public education.
In 1867, due to prevailing circumstances, Rachinsky left his professorship at Moscow University, and with it all the bustle of metropolitan life, returned to his native Tatevo, opened a school there and devoted himself to teaching and raising peasant children. A few years later, the Smolensk village of Tatevo becomes famous throughout Russia. Education and service to the common people will henceforth become his life’s work.
Professor of botany at Moscow University Sergei Aleksandrovich Rachinsky.
Rachinsky is developing an innovative, unusual for that time, system of teaching children. The combination of theoretical and practical studies becomes the basis of this system. During the lessons, children were taught various crafts needed by peasants. The boys learned carpentry and bookbinding. We worked in the school garden and apiary. Natural history lessons were held in the garden, field and meadow. The pride of the school is the church choir and icon-painting workshop. At his own expense, Rachinsky built a boarding school for children coming from far away and without housing.
N.P. Bogdanov-Belsky "Sunday reading of the Gospel at the Rachinsky public school" 1895. In the picture, second from the right is S.A. Rachinsky.
The children received a varied education. In arithmetic lessons, we not only learned how to add and subtract, but also mastered the elements of algebra and geometry, in an accessible and exciting way for children, often in the form of a game, making amazing discoveries along the way. It is precisely this discovery of number theory that is depicted on the school board in the painting “Mal Calculus.” Sergei Aleksandrovich gave the children interesting problems to solve, and they definitely had to be solved orally, in their heads. He said: “You can’t run to the field for a pencil and paper, you have to be able to count in your head.”
S. A. Rachinsky. Drawing by N.P. Bogdanov-Belsky.
One of the first to go to Rachinsky's school was the poor peasant shepherd Kolya Bogdanov from the village of Shitiki, Belsky district. In this boy, Rachinsky recognized the talent of a painter and helped him develop, taking full charge of his future artistic education. In the future, all the work of the Itinerant artist Nikolai Petrovich Bogdanov-Belsky (1868-1945) will be dedicated to peasant life, school and his beloved teacher.
In the painting “On the Threshold of School,” the artist captured the moment of his first acquaintance with Rachinsky’s school.
N.P. Bogdanov-Belsky "On the threshold of school" 1897.
But what is the fate of the Rachinsky public school in our time? Is the memory of Rachinsky preserved in Tatev, once famous throughout Russia? These questions worried me in June 2000, when I went there for the first time.
And finally, it is in front of me, spread out among green forests and fields, the village of Tatevo in Belsky district, the former Smolensk province, and nowadays classified as part of the Tver region. It was here that the famous Rachinsky school was created, which so influenced the development of public education in pre-revolutionary Russia.
At the entrance to the estate, I saw the remains of a regular park with linden alleys and centuries-old oak trees. A picturesque lake whose clear waters reflect the park. The lake of artificial origin, fed by springs, was dug under S.A. Rachinsky’s grandfather, St. Petersburg Chief of Police Anton Mikhailovich Rachinsky.
Lake on the estate.
And so I approach a dilapidated manor house with columns. Only the skeleton of the majestic building, built at the end of the 18th century, now remains. Restoration of the Trinity Church has begun. Near the church, the grave of Sergei Aleksandrovich Rachinsky is a modest stone slab with the Gospel words inscribed on it at his request: “Man will not live on bread alone, but on every word that comes from the mouth of God.” There, among the family tombstones, his parents, brothers and sisters rest.
A manor's house in Tatev today.
In the fifties, the landowner's house began to gradually collapse. Subsequently, the destruction continued, reaching its full apogee in the seventies of the last century.
Landlord's house in Tatev during Rachinsky's time.
Church in Tatev.
The wooden school building has not survived. But the school was preserved in another two-story brick house, the construction of which was planned by Rachinsky, but carried out soon after his death in 1902. This building, designed by a German architect, is considered unique. Due to a design error, it turned out to be asymmetrical - one wing is missing. Only two more buildings were built according to the same design.
The Rachinsky school building today.
It was nice to know that the school is alive, active and in many ways superior to the capital’s schools. In this school, when I arrived there, there were no computers or other modern innovations, but there was a festive, creative atmosphere; teachers and children showed a lot of imagination, freshness, invention and originality. I was pleasantly surprised by the openness, warmth, and cordiality with which the students and teachers, led by the school director, greeted me. The memory of its founder is cherished here. The school museum preserves relics related to the history of the creation of this school. Even the external design of the school and classrooms was bright and unusual, so different from the standard, official design that I had seen in our schools. These are windows and walls originally decorated and painted by the students themselves, and a code of honor invented by them hanging on the wall, and their own school anthem and much more.
Memorial plaque on the wall of the school.
Within the walls of the Tatev school. These stained glass windows were made by the school students themselves.
At the Tatev school.
At the Tatev school.
At the Tatev school today.
Museum N.P. Bogdanov-Belsky in the former manager's house.
N.P. Bogdanov-Belsky. Self-portrait.
All the characters in the painting “Oral Account” are painted from life and in them the residents of the village of Tatevo recognize their grandfathers and great-grandfathers. I want to talk a little about how the lives of some of the boys depicted in the picture turned out. Local old-timers who knew some of them personally told me about this.
S.A. Rachinsky with his students on the threshold of a school in Tatev. June 1891.
N.P. Bogdanov-Belsky "Oral arithmetic in the Rachinsky public school" 1895.
Many people think that the artist depicted himself in the boy depicted in the foreground of the picture - in fact, this is not so, this boy is Vanya Rostunov. Ivan Evstafievich Rostunov was born in 1882 in the village of Demidovo into a family of illiterate peasants. Only at the age of thirteen I entered the Rachinsky public school. Subsequently, he worked on a collective farm as an accountant, saddler, and postman. Lacking a mail bag, before the war he carried letters in a cap. Rostunov had seven children. They all studied at Tatev secondary school. Of these, one was a veterinarian, another was an agronomist, another was a military man, one was a livestock specialist’s daughter, and another daughter was a teacher and director of the Tatev school. One son died during the Great Patriotic War, and another, upon returning from the war, soon died from the consequences of injuries received there. Rostunov’s granddaughter until recently worked as a teacher at the Tatev school.
The boy standing on the far left in boots and a purple shirt is Dmitry Danilovich Volkov (1879-1966), who became a doctor. During the Civil War he worked as a surgeon in a military hospital. During the Great Patriotic War he was a surgeon in a partisan unit. In peacetime, he treated the residents of Tatev. Dmitry Danilovich had four children. One of his daughters was a partisan in the same detachment as her father and died heroically at the hands of the Germans. Another son was a participant in the war. The other two children are a pilot and a teacher. The grandson of Dmitry Danilovich was the director of the state farm.
The fourth from the left, the boy depicted in the picture is Andrei Petrovich Zhukov, he became a teacher, worked as a teacher in one of the schools created by Rachinsky and located a few kilometers from Tatev.
Andrei Olkhovnikov (second from the right in the picture) also became a prominent teacher.
The boy on the far right is Vasily Ovchinnikov, a participant in the first Russian revolution.
The boy, daydreaming and with his hand behind his head, is Grigory Molodenkov from Tatev.
Sergei Kupriyanov from the village of Gorelki whispers in the teacher’s ear. He was the most talented in mathematics.
The tall boy, lost in thought at the blackboard, is Ivan Zeltin from the village of Pripeche.
The permanent exhibition of the Tatev Museum tells about these and other residents of Tatev. There is a section dedicated to the genealogy of each Tatev family. Merits and achievements of grandfathers, great-grandfathers, fathers and mothers. The achievements of the new generation of students of the Tatev school are presented.
Peering into the open faces of today's Tatev schoolchildren, so similar to the faces of their great-grandfathers from the painting by N.P. Bogdanov-Belsky, I thought that maybe the source of spirituality on which the Russian pedagogue ascetic, my ancestor Sergei Alexandrovich Rachinsky so strongly relied, may not have completely died out.
Lesson objectives:
- development of observation abilities;
- development of thinking abilities;
- development of abilities to express thoughts;
- instilling interest in mathematics;
- touching the art of N.P. Bogdanov-Belsky.
DURING THE CLASSES
Learning is work that educates and shapes a person.
Four pages from the life of the painting
Page one
The painting “Oral Account” was painted in 1895, that is, 110 years ago. This is a kind of anniversary of the painting, 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 standing in front) have turned away from the board and are remembering something, or maybe counting. One boy whispers something into the ear of a man, apparently a teacher, while the other appears to be eavesdropping.
- Why are they wearing bast shoes?
- Why are there no girls here, only boys?
– Why do they stand with their backs to the teacher?
-What are they doing?
You probably already understood that students and a teacher are depicted here. Of course, the students’ costumes are unusual: some of the guys are wearing bast shoes, and one of the heroes of the picture (the one depicted 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 poorly; 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 primary school. Look at the picture: after all, only three of the students are wearing bast shoes, and the rest are in boots. Obviously, the guys are 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 into school. Studying was “not their business,” and not all of the boys studied.
Page two
This painting is called “Oral Counting”. Look how intently the boy depicted in the foreground of the picture is thinking. Apparently the teacher gave me a difficult task. But, probably, this student will soon finish his work, and there should be no mistakes: he takes mental calculation very seriously. But the student who whispers something in the teacher’s ear has apparently already solved the problem, but his answer is not entirely correct. Look: the teacher listens to the student’s answer carefully, but there is no approval on his face, which means the student did something wrong. Or maybe the teacher is patiently waiting 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 that stands in front: it’s immediately clear that he is the best student in the class.
What task did the teacher give them? Can't we solve it too?
- But try it.
I will write on the board the way you are used to writing:
(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, and the resulting amount divided by 365.
– That’s the problem (you can’t solve such an example quickly, especially in your head). Still, try to count verbally; I will help you in difficult places. Ten ten is 100, everyone knows that. Eleven multiplied by eleven is also not difficult to calculate: 11 10 = 110, and even 11 is 121 in total. 12 12 is also not difficult to calculate: 12 10 = 120, and 12 2 = 24, and the total will be 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, but these numbers still need to be added, and then the sum divided by 365. No, you won’t be able to calculate this yourself.
- We'll have to help a little.
– What numbers did you get?
– 100, 121, 144, 169 and 196 – many have counted this.
– 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 you divide by?
– Also at 365!
– How much do you get if the sum of the first three numbers is divided by 365?
- One! – everyone will already understand this.
– Now add up the remaining two numbers: 169 and 196. How much do you get?
– Also 365!
– Here’s an example, and a very simple one. It turns out there are only two!
- Only to solve it, you need to know well that the sum can not be divided 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 painting is called “Oral Counting”. It was written by the artist Nikolai Petrovich Bogdanov-Belsky, who lived from 1868 to 1945.
Bogdanov-Belsky knew his little heroes very well: he grew up among them and was once a shepherd. “...I am the illegitimate son of a poor little girl, that’s why Bogdanov, and Belsky became named after the district,” the artist said about himself.
He was lucky enough to get into the school of the famous Russian teacher Professor S.A. Rachinsky, who noticed the boy’s artistic talent 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 in people’s memory he remained, first of all, as an artist who was able to poetically and truly tell about smart rural children who greedily sought knowledge.
Who among us is not familiar with the paintings “At the School Door”, “Beginners”, “Essay”, “Village Friends”, “At the Sick Teacher”, “Voice Test” - these are the names of just a few of them. Most often the artist depicts children at school. Charming, trusting, focused, thoughtful, full of lively interest and always marked by natural intelligence - this is how Bogdanov-Belsky knew and loved peasant children, and who immortalized them in his works.
Page four
The artist depicted real-life students and a teacher in this picture. From 1833 to 1902 lived the famous Russian teacher Sergei Alexandrovich Rachinsky, a remarkable representative of Russian educated people of the century before last. 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 passing the exam” for the title of primary school teacher. Using his own funds, he opens a school for peasant children in the village of Tatyevo, Smolensk province, and becomes a teacher there. So, his students calculated so well orally that all visitors to the school were surprised. As you can see, the artist depicted S.A. Rachinsky together with his students at a lesson in oral problem solving. By the way, the artist himself N.P. Bogdanov-Belsky was a student of S.A. Rachinsky.
This picture is a hymn to the teacher and student.
