What does the picture “oral arithmetic in a public school” say? Painting “Oral Account” by Bogdanov-Belsky

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Many have seen the picture "Oral calculation in 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 board in their minds. The first person to solve it reports the answer to the teacher in the ear, in a whisper, so that others do not lose interest.

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

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 a public school" S.A. Rachinsky". Sergei Rachinsky is 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 - in the middle of his life he abandoned all his affairs, went to his estate (Tatevo in the Smolensk province) and started a business there (of course , at his own expense) 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). Word classmate meant that children of three years of study form a single class, and one teacher teaches them all within one lesson. It was quite a tricky thing: 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 understandably, something that could be useful to a peasant in everyday life: simple letters, petitions, etc. Even at Rachinsky’s school, some manual labor, the 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 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 a gingerbread to the first child who solved the problem (in literally words, but he didn’t 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 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 as teachers primary school and priests, so he prepared children mainly for theological and teacher seminaries. There were also significant exceptions - first of all, the author of the picture himself, Nikolai Bogdanov-Belsky, whom Rachinsky helped 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 ecclesiastical 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 to teach him for political reasons, so the zemstvos pushed him into a corner as best they could. The law of God was taught by a parish priest who was 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 the 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 a unique school. This is a monument to Sergei Rachinsky, a unique personality and teacher, to the last representative that cohort of conservatives and patriots, to which it was not yet possible to include famous expression"patriotism is the last refuge of a scoundrel." 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, ordinary elementary school students could not 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.

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 the right decision. 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 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 and developed a unique teaching method mental arithmetic, instilling in 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, 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).

So the solution famous example Rachinsky can be produced 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 famous painting 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 are real people, painted by him from life - people whom he knew and loved, and most importantly actor- an elderly teacher, a man who played 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 Aleksandrovich Rachinsky was born in the Tatevo estate, Belsky district, Smolensk province in 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 comprehensive education for 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, gets acquainted with interesting people, studies philosophy, literature, music and much more. After a while he writes several scientific works and received a doctorate and a professorship in botany at Moscow University. But his interests were not limited to scientific frameworks. The future rural teacher was studying literary creativity, wrote poetry and prose, played the piano perfectly, was a collector of folklore - folk songs and handicrafts. Khomyakov, Tyutchev, Aksakov, Turgenev, Rubinstein, Tchaikovsky and Tolstoy often visited his apartment in Moscow. Sergei Alexandrovich was the author of the libretto for two operas by P.I. Tchaikovsky, who listened to his advice and recommendations and dedicated his first string Quartet. 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 Alexandrovich 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 from now on will become the work of his whole life.

Professor of botany at Moscow University Sergei Aleksandrovich Rachinsky.

Rachinsky is developing an innovative, unusual for that time, system of teaching children. A combination of theoretical and practical classes 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 form 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 saw the talent of a painter and helped him develop, completely taking over his future art education. In the future, the entire work of the Itinerant artist Nikolai Petrovich Bogdanov-Belsky (1868-1945) will be dedicated to peasant life, school and favorite 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 first went there.

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. Picturesque lake in clear waters which the park is reflected. 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. IN school museum they take care of relics associated with the history of the creation of this school. Even the external design of the school and classrooms was bright and unusual, so 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 former house manager

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 in Tatev high 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. Until recently, Rostunov’s granddaughter worked as a teacher at the Tatev school.

The boy standing on the far left in boots and a purple shirt is Dmitry Danilovich Volkov (1879-1966), who became a doctor. During Civil War worked as a surgeon in a military hospital. During the Great Patriotic War he was a surgeon in a partisan unit. IN Peaceful time treated 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.

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

"VERBAL COUNTING. AT S.A. RACHINSKY’S PEOPLE’S SCHOOL.”

The painting is done in oil on canvas and depicts a 19th century rural school during an arithmetic lesson.

A simple Russian class, children 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.

Students solve an interesting and complex example of solving fractions in their heads. 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.

The canvas depicts 11 children and only one boy quietly whispers in the teacher’s ear, perhaps the correct answer.

Standing nearby is a teacher, a real person, Sergei Aleksandrovich Rachinsky - a famous 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 to village children his skills and the basics of mathematical thinking.

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 Bogdanov-Belsky, himself a former student of Rachinsky, dedicated the painting to an episode from the life of the school with the creative atmosphere that reigned in the lessons, 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.

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

First way

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 way

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 way

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 + 122 + (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*122+2*22+2*12, 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.

in the series of two-digit numbers - the first five of its representatives - have an amazing property. The sum of the squares of the first three numbers in the 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...

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

Working in a rural school

Sergei Alexandrovich Rachinsky brought out to people:

Bogdanova I. L. - 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 royal family, a teetotaler pastor, a patriot-monarchist;

Sinev Nikolai Mikhailovich (December 10, 1906 - September 4, 1991) - doctor technical sciences(1956), professor (1966), Honored Worker of Science and Technology of the RSFSR. In 1941 - deputy chief designer for tank building, 1948-61 - head of the design bureau at the Kirov plant. In 1961-91 - Deputy Chairman of the USSR State Committee 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 the creation of a Russian rural school. Last May marked the 180th anniversary of the birth of this outstanding Russian man, a true ascetic, a tireless worker, a forgotten rural teacher 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 mentored in matters of writing.

By the way, the author of the above-mentioned painting, Nikolai Bogdanov-Belsky, came from poverty and was a student of Sergei Alexandrovich, who over thirty years, at his own expense, created about three dozen rural schools and, at his own expense, helped the brightest of his students to realize themselves professionally, who became not only rural teachers (about 40 people!) or professional artists (3 students, including Bogdanov), but also a teacher of the law for the royal children, a graduate of the St. Petersburg Theological Academy, Archpriest Alexander Vasiliev, and a monk of the Trinity-Sergius Lavra, like Titus (Nikonova).

Rachinsky built not only schools, but also hospitals in Russian villages; the peasants of Belsky district called him nothing less than “dear father.” Through the efforts of Rachinsky, temperance societies were recreated in Russia, uniting tens of thousands of people throughout the empire by the early 1900s.

Now this problem has become even more urgent, drug addiction has now grown into it. It is gratifying that the teetotaling path of the enlightener has again been picked up, that temperance societies named after Rachinsky are appearing in Russia again

Russian pedagogues and ascetics looked upon teaching as a holy mission, a great service to the noble goals of raising spirituality among the people.”

“The May Man” Sergei Rachinsky passed away on May 2, 1902. Dozens of priests and teachers, rectors of theological seminaries, writers, and scientists came to his funeral. In the decade before the revolution, more than a dozen books were written about Rachinsky’s life and work, and the experience of his school was used in England and Japan.

Surely, everyone who studied at school (especially in Soviet time), remember the picture from the textbook “Mathematics”, in which schoolchildren are trying to solve an example written on the board. Do you remember? I'm sure yes.

It was not very often that we were spoiled with some kind of in order to activate our attention and instill love for the subject. The majority asserted categorically: “You must study!” , “It’s your job,” etc.

But anyone (and even an adult, with a more conscious, so to speak, approach) will involuntarily ask the question: “Why should I study? WHY do I need this?

And here you can go in at least two ways. The first is to explain to the unconscious young creature the benefits of learning. And it immediately becomes clear that this is a dead-end move. Modern schoolchildren do not have guidelines and values in order to try and “tear their claws”, strain and deny themselves something. I’m not saying that there are no such children at all. There are enough of them, and among my students there are many such “conscious elements.” But basically, now they learn either under pressure or carelessly. And this is upsetting.

But at all times, and especially now, the question of motivating students to learn has been faced. And this article aims to awaken interest in mathematics using such techniques as mental calculation.

“How can this be done?” you ask.

“Very simple,” I will say in response.

Just look at the painting by the Russian artist N. P. Bogdanov-Belsky « Verbal counting. At the public school of S. A. Rachinsky."

Look what's on it. This is a 19th century village school. Moreover, it is real, not made up by the artist. And in the picture there is also a real person, Rachinsky Sergei Alexandrovich (1833 - 1902), noble origin. The name may not be familiar to most. However, he was a well-known personality in teaching circles at that time. He was a professor at Moscow University, a doctor of botany, a good writer, a corresponding member of the Imperial St. Petersburg Academy of Sciences, etc.

The merits of S.A. Rachinsky are sufficient: starting from the fact that in 1872 he created a school with a dormitory for peasant children, he himself taught painting and drawing there and educated many famous personalities, created the first textbook on “mental arithmetic” in Russia. But the most valuable thing for mathematics teachers is that he developed a unique method of teaching mental arithmetic.

His famous phrase: “You can’t run from the field for a pencil and paper. You have to decide mentally” speaks for itself. And you can't argue with that.

Rachinsky was reported to the emperor Alexander III So:

“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 leads, 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 in response from Nicholas II, the imperial words were heard in praise of the great philanthropist-teacher:

“The schools founded and led by you... became... a school of work, sobriety and good morals and a living model for all similar institutions. Close to my heart My concern for public education, which you serve worthily, prompts Me to express My sincere gratitude to you. I am with you, my kind Nikolai.”

So, what is depicted in the picture, which attracts attention if only because it depicts children? And not just frolicking or chasing a dog, playing hide and seek or stealing apples from a neighbor’s garden (how many similar scenes do we know from painting)?

Painting “Oral calculation. At the public school of S.A. Rachinsky"

On the artist's canvas N. P. Bogdanov-Belsky an episode from the life of the school was written out with the creative atmosphere that reigned in mathematics lessons, set by the teachers of the Tatev Rachinsky School.

A seemingly unsightly computational example is written on the board:

But how he interested the guys gathered at the blackboard!

Someone was thinking alone, someone was discussing their ideas with a group of classmates, someone was clinging to the teacher, supposedly asking for support and whispering their answer in his ear (“What if it’s wrong? What will the guys think then?”)

And it would seem that it won’t work out... and okay. This is just an example. “Just think...,” as the hero from the cartoon “In the Land of Unlearned Lessons” says.

And yet, schoolchildren think and think intensely. And the teacher sat down in the corner as an outside observer and... no, no. And I would like, perhaps, to suggest, to direct thoughts into the right direction. But that’s why an example is given: to figure it out, think it over slowly and give the correct answer. And the main thing is to perform all mental operations verbally.

I’m sure that if you offered modern kids such an example, most of them would immediately reach into their briefcases for calculators. Our people have forgotten how to think modern schoolchildren strain. And whoever wasn’t lazy (or didn’t have “brain crutches” at hand in time) would most likely consider this example “head-on”, i.e. would perform sequentially written actions. And thereby I would make my “life” more difficult.

But everything is much simpler and more interesting. See:

See, it's simple. And if you know the property of some numbers that the sum of the squares of three consecutive numbers is equal to the sum of the squares of the two consecutive numbers following them, then you could do without these calculations.

“This task is also good because it not only sharpens the brain, but is also suitable for many far-reaching generalizations,” said S.A. Rachinsky.

AND Rachinsky's problem also exists. But I will write about this later.

So, the main character today was the painting "". Recently, it was 195 years since the most famous mathematics lesson, which was taught at a peasant school in the Oleninsky district of the Smolensk province by Sergei Aleksandrovich Rachinsky. It was he who left the university department to become a rural teacher. And thanks to him, Russia received a lot prominent figures culture and art, among which were Tretyakov, Nikolai Stepanovich and the author of the painting discussed in this article, Nikolai Petrovich Bogdanov - Belsky.

What influence did he have on the formation of these two? legendary personalities S. A. Rachinsky, we will look at it in the next article. And at the same time, we will touch upon a topical topic for today: the influence of the teacher’s personality on the younger generation.

But if you were interested in getting acquainted with the personality of S.A. Rachinsky and the painting “Oral Account. At the folk school of S.A. Rachinsky” by artist N.P. Bogdanov-Belsky, click the buttons below and share this knowledge with your friends.