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 blackboard 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 the public school of 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, for own account) 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 complex 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, 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 the first child who solved the problem a carrot (in the literal sense of the word, he didn’t have a stick).
Sami school activities occupied 5–6 months of the 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, 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 path of an educated person is a 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 a certain influence of Rachinsky's ideas, the religious department decided that the zemstvo school would be of no use - 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 primary education a lot of money, there was no talk 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 an underpaid and neglected parish priest, 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 teaching the Russian language into world studies, through the so-called explanatory reading. The technique was that by dictating educational text in Russian, the teacher also further 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, the last representative of that cohort of conservatives and patriots, which could not yet be included 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 methods of counting, omitting all arithmetic and algebraic transformations, requiring 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.
Full name famous painting which is pictured above: " Oral 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 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, that 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, because:
10 2 +11 2 +12 2 = 13 2 +14 2 = 365
Humor and parody interpretations
Nowadays, schoolchildren not only solve some of Rachinsky’s popular problems, but also write essays based on the painting “Oral Calculus. At the public school of S. A. Rachinsky,” which could not but affect the desire of schoolchildren to joke about the work. The popularity of the painting “Oral Reckoning” is reflected in the numerous parodies of it that can be found on the Internet. Here are just a few of them:
photo clickable
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 is there such a high ceiling and an expensive stove with white tiles in the school classroom? 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 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 complex mathematical operation studied in his school.
And finally, Rachinsky was a proponent 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 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 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 other educational institutions and after it it was impossible to continue education 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, the author of the picture himself, Nikolai Bogdanov-Belsky, whom Rachinsky helped to get into the Moscow School of Painting, Sculpture and Architecture. But, oddly enough, Rachinsky did not want to lead peasant children along the main path of an educated person - gymnasium / university / public service.
Rachinsky wrote popular pedagogical articles and continued to enjoy a certain influence in the capital's intellectual circles. The most important was the acquaintance with the ultra-influential Pobedonostsev. Under 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 allocate a lot of money to primary education, there was no question of passing subsidies to church schools through the Duma; almost all the funds went to the zemstvo 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 teaching 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, 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, 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“Oral counting. At the public school of S. A. Rachinsky."
Nikolay Bogdanov-Belsky. Oral 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, the children are dressed in peasant clothes: bast shoes, pants 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 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 shows a village school late XIX century during an arithmetic lesson while solving fractions in your head. Teacher - 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 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, Raczynski did not follow a typical training course; he was sure of excellent mathematical abilities most peasant children and considered it possible to significantly complicate the mathematics program.
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 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. 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 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