Mental Arithmetic

The lesson’s busiest activity cannot be seen directly

A boy presses his hand beneath his chin. Another reaches behind his head, and several pupils face a blackboard. Beside the teacher, one child leans close as if delivering an answer. The room is full of movement, yet the central activity is mental.

Nikolay Bogdanov-Belsky painted Mental Arithmetic in 1895. Its fuller title is usually translated as Oral Arithmetic in S. A. Rachinsky’s School. The original belongs to the State Tretyakov Gallery in Moscow, not the State Russian Museum.

The equation on the board gives the children a shared problem. Their different poses show that they are not arriving at its solution in the same way, or at the same moment.

Can you solve the problem on the board?

The written problem asks for a sum of five squares divided by 365:

(10² + 11² + 12² + 13² + 14²) ÷ 365

The answer is 2. The five squares are 100, 121, 144, 169 and 196. Together they total 730, which is twice 365.

The quickest route is to notice two equal groups. The first three squares give 100 + 121 + 144 = 365. The last two give 169 + 196 = 365. The numerator therefore contains two lots of the denominator, so the answer is visible without a long final division.

That small discovery changes the task from a laborious sum into a pattern. The children’s different poses can make us remember our own delay before noticing it, although the painting does not tell us precisely which method any individual pupil is using.

The picture thus offers a problem the viewer can share. We are not only looking at children being educated; for a moment, we can join the lesson.

A hand beneath the chin makes concentration visible

The foreground boy turns partly away from the blackboard. One arm crosses his body while the other supports the hand at his chin. His eyes do not meet ours in a relaxed portrait pose. He seems occupied by something that continues inside his head.

The body closes around that effort. Elbows draw inward, the shoulders gather and the feet stay planted. The painting gives thought a posture without pretending that we can see a complete train of reasoning.

Other pupils direct their bodies towards the board. This boy’s inward turn establishes a different relation to the same task. He is neither absent from the lesson nor obviously finished with it.

The answer is given quietly to one person

The seated teacher wears a dark formal coat, a white shirt and a bow tie. A pupil leans close to his ear. Their proximity makes the exchange feel personal despite the number of children around them.

The Tretyakov identifies the teacher as Sergei Alexandrovich Rachinsky, whose rural school at Tatevo Bogdanov-Belsky had attended. The painting is therefore connected with the artist’s own education and his lasting regard for the teacher.

Rachinsky’s presence does not dominate through size or a raised arm. He occupies the left-hand group, listening while the pupils work. The room’s activity grows around him rather than simply following a single demonstrative gesture.

A quietly delivered answer also leaves space for the others to continue thinking. The composition lets different stages of learning coexist: watching, calculating, testing an answer and waiting for a response.

Worn clothing does not define the limits of intelligence

The pupils’ clothing varies. Loose shirts, patched fabric, belts, boots and woven footwear bring the material circumstances of rural life into the room. The foreground boy’s shoes are as carefully observed as the blackboard.

Yet the lesson is not reduced to a display of deprivation. The children are engaged in a demanding shared task. Their clothes tell us something about daily conditions, while their poses insist on curiosity, effort and ability.

That distinction is central to the painting’s warmth. A viewer may first notice the modest schoolroom and worn garments, but the pupils’ concentration keeps redirecting attention towards what they can do.

This is a tribute to an education the painter knew

The Tretyakov’s account describes Rachinsky’s attention to mental calculation as a practical skill for rural life. Buying and selling could require quick arithmetic without paper at hand. The challenge also became a source of pleasure for his pupils.

The artist’s connection to the school prevents the scene from being merely a generic image of a classroom. It remembers a particular educational environment through people, gestures and the problem on the board.

The wall also carries a religious image and a sheet of musical notation. Arithmetic belongs within a wider programme of learning, not an empty room containing only numbers. The school’s modest scale does not make its intellectual or cultural life simple.

For a contrasting view of childhood and institutional care, see Fildes’s Applicants for Admission to a Casual Ward. One image gathers children around the possibility of learning; the other shows the uncertainty of obtaining a night’s shelter.

Questions about Mental Arithmetic

What is the answer to the blackboard problem?

Two. The numerator totals 730, and 730 divided by 365 equals 2.

Who is the teacher?

Sergei Alexandrovich Rachinsky, whose school Bogdanov-Belsky attended as a child.

Where is the original?

State Tretyakov Gallery, Moscow, inventory 1003, acquired by Pavel Tretyakov from the artist in 1896.

What are its medium and size?

Oil on canvas, 1895, approximately 107.4 × 79 centimetres.

Sources and image credit

• State Tretyakov Gallery: Mental Arithmetic, inventory 1003
• Yana Laguzinskaya, Tretyakov Gallery: Mental Arithmetic in detail
• Historical 1915 self-portrait, Luhansk Regional Museum, image record

Faithful public-domain reproduction of Mental Arithmetic, via Wikimedia Commons. Detail images are crops of the same historical painting; no reconstruction or upscaling.