How Many Children Learned Mathematics from Kiselev's Textbooks?

A demographic estimate, with sources, of the readership of A.P. Kiselev’s textbooks across Russian and Soviet education — and a look at why one textbook held its place for so long.

In 1955 the seventeenth edition of A.P. Kiselev’s Arithmetic came off the presses of the Soviet state publisher. It was, at that point, the official and only school textbook of mathematics for fifth- and sixth-grade pupils across the Soviet Union — and it had held that position, with one major reworking, since 1938. The first edition of the book had appeared in 1884.

A textbook does not stay in classrooms across that span of years by accident.

This essay is a careful estimate of how many children passed through Russian and Soviet classrooms during the period in which Kiselev’s books were widely used and, for one substantial stretch, officially mandatory. The number is not a figure I can quote from any single source. It is built up from population data, education statistics, and the dates that bracket the textbook’s official status. The headline answer — tens of millions, plausibly around eighty million, with a defensible range of 60 to 100 million — is offered as an estimate, and below I show the demographic accounting that produces it. The story of how the books ended up there is more interesting than the number.

The four phases

The period in which Kiselev’s books had standard or near-standard status divides naturally into four phases, each with very different student populations and very different roles for his textbooks.

Phase one: Late Russian Empire, 1884–1917

A.P. Kiselev published his Systematic Course of Arithmetic for Secondary Educational Institutions in 1884. His Elementary Algebra followed in 1888. Both were quickly adopted across the gymnasia and the real schools — the two main types of secondary education in the empire — and both became dominant in their subjects.

By modern standards, however, secondary education in late imperial Russia was small. Total enrollment across all secondary types was about 150,000 in 1885 and approached 800,000 by 1914. With student turnover, the cumulative number of children who passed through classrooms using Kiselev during this 33-year stretch is on the order of 2 to 7 million — and the lower end is the safer figure absent year-by-year entry-cohort statistics. Modest in absolute terms, but in those classrooms Kiselev’s book was already the standard.

Phase two: Early Soviet, 1917–1938

The two decades after the revolution were a period of rapid expansion in education and considerable curricular experimentation. Mass literacy campaigns dramatically increased the reach of schooling. But the early Soviet education ministry — under several different names and several different leaders — experimented with alternative textbooks and competence-based curricula throughout the 1920s, and Kiselev did not have official monopoly status during that period.

By the early 1930s the experimental winds had blown out, and Kiselev’s books were once again dominant in practice if not yet officially. Cumulative unique users of Kiselev during this phase: roughly 10 to 20 million, with the caveat that adoption was uneven across regions and types of school throughout the 1920s, and that competing textbooks were in active use during the experimental years.

Phase three: The Khinchin and Barsukov era, 1938–1955

This is the core of the story.

In 1938 the eminent Soviet mathematician A.Ya. Khinchin published his reworking of Kiselev’s Arithmetic. In the same year, the pedagogue A.N. Barsukov published a reworking of Kiselev’s Elementary Algebra. Both reworkings were authorised, both kept Kiselev’s name on the cover, and both — by official decree — became the sole approved school mathematics textbooks for their grade ranges across the Soviet Union. Arithmetic for grades 5 and 6. Algebra for grades 6 through 10. No other textbook would do.

This was an extraordinary administrative fact. Across an empire of (in 1939) some 170 million people, every fifth-grader who studied arithmetic that year studied it from Kiselev’s pages. Every tenth-grader doing algebra had Kiselev’s Algebra on the desk.

The arithmetic of how many children that touched, over seventeen years, is straightforward. Annual cohorts entering grade 5 in this period ranged from about 2 million to about 5 million, depending on the year. Cohorts born during the war years 1941 through 1944 were sharply smaller — that is the demographic shadow of the war, visible in every Soviet population pyramid. But the cumulative number of unique children who reached grade 5 during the period of Kiselev’s sole-textbook status is roughly 40 to 60 million.

The cumulative number who reached grade 6, where Kiselev’s Algebra begins, is somewhat smaller because secondary completion rates were not yet 100%. By 1959, only about 36% of the working-age Soviet population had completed grade 7 or higher. Estimate: 30 to 50 million for Algebra in this phase.

The 17th edition of the reworked Arithmetic, published in 1955, marked the end. Newer official textbooks were on their way — “by no means always original and by no means always successful,” as N. Rozov would later write in his 2002 foreword to the re-edition of Kiselev’s Arithmetic. The names of their authors, he added, are today remembered only by historians of education.

Phase four: The post-replacement tail, 1955–1970s

Centrally-issued textbooks do not disappear from classrooms the moment a successor is approved. Kiselev’s books remained in active classroom use, especially in less-resourced regions, well into the 1960s and in pockets into the 1970s. Estimate: another 5 to 15 million children — the most speculative of the four phase ranges, since systematic data on continued usage after de-mandating is sparse. This phase is what pushes the overall midpoint from roughly 70 million to roughly 80 million; without the tail, the headline figure is correspondingly smaller.

The total

Adding the four phases:

Adding the four phases gives a range of roughly 60 to 100 million. The midpoint is around 80 million; without the speculative phase-four tail it is closer to 70 million. Across the rest of this essay I refer to the figure as plausibly around eighty million — never as a hard demographic fact, because it is not one. What the demographic accounting can defensibly support is a range, with phase three doing most of the heavy lifting and phases one, two, and four contributing meaningful but more uncertain numbers.

Why a textbook stays in classrooms for that long

The arithmetic of the readership is the easier question. The harder one is: how does a single textbook hold its place across two political regimes, a civil war, two world wars, the cultural revolution of the 1920s, the purges, the Stalinist consolidation of the 1930s, and the post-war reconstruction — without being displaced?

Part of the answer is that Kiselev’s books were unusually good at the thing a school mathematics textbook is supposed to do: turn a child into someone who can think mathematically. Generalities about quality are easy to write. Specifics are harder, and more useful. Two examples, both from the actual texts.

Children meet Euclid in the Arithmetic

Kiselev’s Arithmetic contains, in section 94 of the chapter on divisibility, the following passage:

It is easy to convince oneself that there exist infinitely many prime numbers. Indeed, suppose the contrary, that the number of primes is finite. Then there must exist a greatest prime; let it be a. To refute this assumption, imagine the new number N formed by the rule N = (2·3·5·7···a) + 1, that is, the product of all the primes up to a, plus one… The first term is divisible by every number in the list 2, 3, 5, …, a, while the second (the unit) is not divisible by any of them. Hence there is no greatest prime, and so the sequence of primes is infinite.

That is Euclid’s proof of the infinity of the prime numbers, which appears as Proposition 20 of Book IX of the Elements. It is one of the cleanest arguments in the whole of mathematics. And here it is, in a textbook for ten- and eleven-year-old children, presented in full and in plain language.

In a typical American or British arithmetic textbook of the same period — or, frankly, today — the topic “primes” would consist of the definition of a prime, a list of the first few, and perhaps a procedure for testing primality. The infinity of the primes would be asserted, if at all. The proof would not appear, and the argument that no list of primes can be completewould not be made.

Kiselev’s child reader is being treated as a participant in mathematics, not as a recipient of facts.

Signed arithmetic in the Algebra

A second example, from Kiselev’s Elementary Algebra. The standard modern presentation of the rules of signed arithmetic — that the product of two negatives is positive — is to state the rule and offer a memory aid. Kiselev does something different.

In section 27 he sets up a physical problem. A train at the station of Bologoye, which sits on the rail line between Saint Petersburg and Moscow. We let positive velocity mean toward Moscow; negative velocity, toward Saint Petersburg. We let positive time mean after noon; negative time, before noon. We let positive distance from Bologoye mean toward Moscow; negative, toward Saint Petersburg.

Now we work out four cases:

1. The train is going toward Moscow at +40 km/h and we ask where it will be in +3 hours. Answer: at +120 km. So (+40) × (+3) = +120.
2. The train is going toward Saint Petersburg at −40 km/h and we ask where it will be in +3 hours. Answer: at −120 km. So (−40) × (+3) = −120.
3. The train is going toward Moscow at +40 km/h and we ask where it was 3 hours ago. Three hours earlier the train had not yet reached Bologoye and was on the Saint Petersburg side of the station — at −120 km. So (+40) × (−3) = −120.
4. The train is going toward Saint Petersburg at −40 km/h and we ask where it was 3 hours ago. Three hours earlier the train had not yet reached Bologoye and was on the Moscow side of the station — at +120 km. So (−40) × (−3) = +120.

The four cases are concrete. They can be checked against a number line, which Kiselev draws. They are then generalised, in section 28, into the multiplication rule.

The point is not that “minus times minus is plus” because some external authority says so. It is that, if we want our rules to give consistent answers when applied to physical quantities that point in two opposite directions, this is what the rules must look like.

That is a different intellectual experience than memorising a sign chart. It is also a level of pedagogical care that survived seventeen editions and three regime changes.

The ratio of teaching effort to learning gain

There is a quiet observation buried in the demographics of this story. The tens of millions of children who learned mathematics from Kiselev did not learn it well because their teachers were uniformly excellent. They cannot have been. The Soviet teaching corps, especially after the wartime losses, was overstretched, often poorly trained, and was working with limited materials in classrooms of forty to fifty pupils.

What those teachers had was a textbook that did a great deal of the work for them. Every rule earned its place; every concept was built up from what came before; every reader, however young, was treated as capable of mathematical thought. A teacher who said open Kiselev to section 94 and read it carefully could trust that the page would do the explaining. That ratio — pedagogical density per page — is not unrelated to why the book outlived the Soviet Union itself in some classrooms.

The English edition

For most of its history, Kiselev’s mathematics has been inaccessible to English-speaking students. The Arithmetic and Algebra remained, in practice, locked behind the Russian.

I have spent the past year producing complete English editions of both. Kiselev’s Arithmetic is now available; Kiselev’s Algebra, Part I — the first half of the Elementary Algebra, covering preliminary concepts, relative numbers, monomials and polynomials, equations of the first degree, square roots, and quadratic equations — has just been released. The translations preserve Kiselev’s original section numbering, the structure of his exposition, and his pedagogical sequence. The typography is modern; the mathematics is Kiselev’s.

If your child, or your student, or you, would like to learn mathematics from the textbook by which generations of Russian and Soviet schoolchildren — by my estimate above, plausibly around eighty million of them — learned theirs, there is now no language barrier.

The books are available on Amazon and at valeman.gumroad.com.

— Valery Manokhin, PhD (Royal Holloway, University of London)

Sources and further reading

• Soviet population and census figures: All-Union censuses of 1926 (~147M), 1939 (~170.6M), and 1959 (~208.8M); standard demographic references including the Bolshaya Rossiyskaya Entsiklopediya.

The 60–100 million range is built up from these sources by standard demographic accounting; it is not a citation from any single published statistic.