Lev Tolstoi is not only a great Russian writer, but also the author of an innovative method of teaching mathematics. Teacher, author of the telegram channel “Mathematics on the run. Why the writer’s pedagogical ideas were not accepted by his contemporaries, but today they sound more than relevant.
The program of teaching arithmetic, which Lev Tolstoi developed, was designed for three academic years. It included the study of four arithmetic operations with natural numbers (including multi-digit numbers), the study of decimal and ordinary fractions and operations with them, and the solution of text problems (sometimes very difficult). This set of skills in a modern school is obtained by children completing the 6th grade. At the same time, the school year in a rural school lasted on average 5-6 months, because during sowing and harvesting Tolstoi’s pupils had to work on a par with adults.
“The pupil may not understand because it is not yet his time. What you have beaten on in vain for hours becomes suddenly clear in a minute after a few moments.”
That is why modern teachers at universities often use Tolstoi’s phrases, and advise never rushing to do a college essay, if you are not ready, it is better to wait it out and then go back to your inner interpretations.
But despite the intensity of the program, Lev Tolstoi mentions more than once in his textbook how important it is to take the time to explain. Speaking out against rote learning and rote memorization, he tried to give students time to think through the material, to pass it through themselves, to relate it to their personal experiences and ideas. This often takes time. Tolstoi admitted that it makes no sense to repeatedly explain what remains incomprehensible to the child. Sometimes it is enough just to postpone the task so that after a few days or weeks it will finally make sense to the student. Half a century later, the Swiss psychologist Jean Piaget would come to similar conclusions in his research.
We do not study numbers, but actions.
In the 1860s, the method of the German educator August Wilhelm Grube was a huge success in Russia. In contrast to the theories of the day, he suggested that arithmetic be taught not through study of actions, but through the study of natural numbers. Grube called this process “Contemplation of Numbers. School instruction was based on a sequential study of each number in the first hundred. Children studied the properties of concrete numbers without perceiving arithmetic operations as abstract operations.
For example, this was what the teacher’s questions looked like in the lesson on the number 6:
How much must be added to 5 to get 6? And to 4? And to 3?
How many ones does the number 6 contain? How many twos? Threes?
By how many numbers is 6 greater than 4? А 3? А 2?
What is half of 6? And a third?
In the next lesson, the number 7 was similarly analyzed. And in the next lesson, the number 8. And so lesson after lesson, for several years. This way of studying arithmetic was widespread, its follower in Russia was the teacher V.A. Evtushevsky.
Leo Tolstoy repeatedly criticized this methodology: “They write arithmetic either for themselves alone or for imaginary children, brought up from childhood outside any idea of number.”
“Mathematics has the task not of teaching numbering, but of teaching the methods of human thought in calculating.”
Lev Tolstoy’s methodology, on the contrary, was based not on the study of numbers, but on the study of actions with them. At the same time pupils learned to make calculations with small numbers in mind or with the help of accounts, quite quickly moving on to the multi-digit numbers. Tolstoy knew that children come to school already having a basic understanding of numbers and a natural understanding of arithmetic operations. Very often, ignoring the strict mathematical terms, he formulated the rules using words that his students intuitively understood: “1404 – 565. Fold hundreds out of hundreds. Not enough, decompose one thousand into hundreds and write it down this way: 14 hundredths 4 – 5 hundredths 65.”
Lev Tolstoi tried to ensure that in the study of arithmetic there was nothing artificial, that the problems solved in class, supplemented and enriched the experience of students outside the school.
Down with the rules and definitions
Lev Tolstoi was zealously criticized by his fellow teachers for his lack of rigor and clearly articulated rules. Tolstoy believed: the child must identify mathematical patterns on their own, on the basis of personal experience and after repeated exercises. This approach is in much accord with modern theories of developmental learning.
Even before showing students the simple techniques of addition and subtraction in columns, Tolstoy spends much time coaching them to perform the actions, beginning with the higher digit, in writing and orally.
“Avoid the message of definitions and general rules. The shorter the way by which you teach a student to do an action, the worse he will understand and know the action.”
Lev Tolstoi was categorically against memorizing anything in mathematics, including the multiplication table. He taught children to multiply by successively doubling numbers, and then, if necessary, subtracting the original number. In this way they gained multiplication experience, and the results were gradually memorized without special effort.
Explaining addition and subtraction of common fractions (in Tolstoi’s school it was the third year material), he never showed a strict algorithm for finding the common denominator of fractions, allowing students each time to find it on their own, using the selection and the already accumulated knowledge of the properties of numbers. Tolstoi believed that by accumulating experience, a child could come up with the idea that the common denominator, for example, could simply be the product of two denominators.
At the head of everything is the number system.
Unlike other curricula of the late 19th century and even modern textbooks, Tolstoy paid enormous attention to the concept of the number system. His students wrote down numbers not only in the usual Arabic numerals, but also in Roman and Slavic numerals (the letters of the ancient Russian alphabet were used as numerals in this system).
In addition, from the very first lessons they were already practicing putting down numbers on Russian accounts. In the second volume of Tolstoi’s “Arithmetic,” a separate chapter is devoted to positional systems of notation with different bases – the usual decimal, binary, ternary, and others.
He taught children to write numbers in different notation systems and perform all arithmetic operations with them.
For example, by removing extra knuckles and leaving only one on each spoke, he had a tool for binary arithmetic. Moreover, he arranged the counters (in the lessons and in the illustrations in the textbook) not vertically, but horizontally.
Leo Tolstoy’s “Arithmetic” textbook
It turned out that the higher digits of the number were to the left of the lower, as it happens when writing numbers. Only at the beginning of the study of arithmetic, Tolstoy suggested that students pronounce numbers not quite as usual: not forty, but four tens, not one hundred and thirteen, and one hundred, one tenth and three prime (three units). This allowed students to see numbers and gain a deeper understanding of the decimal number system. This approach allowed for an easy transition to the study of decimals. Tenths, hundredths, and thousandths were immediately perceived by students similarly as digits smaller than ones.
The text of the problems should be understandable
The text problems that Tolstoy offers in his “Arithmetic” can safely be called outstanding. Firstly, their content reflects those everyday situations that his students had already encountered or should have encountered in adulthood. Secondly, all of the tasks in the textbook are quite complex. They are accompanied by examples of reasoning, which allow a student to understand the logic of the solution.
He intentionally did not include in the textbook elementary problems, the content of which is far-fetched, and artificial simplicity leads to a stupor in children. But there are not many problems in the textbook – this is just a sample list. Tolstoi assumed that every teacher can independently compose a large number of similar problems. To begin with tasks with small numbers, which a student could perform in mind, and then for independent exercises to gradually give those that require written calculations. In addition, Tolstoi proposes, studying a particular type of problems, swap known and unknown.
Each of Tolstoi’s problems is a text that is skillful and verified both methodologically and linguistically.
“Five brothers divided the inheritance equally after their father. There were three houses in the inheritance. The three houses could not be divided, the older three brothers took them. And the younger ones were given money for that. The older ones each paid 800 rubles to the younger ones. The smaller ones divided the money among themselves, and then all the brothers got equal. Were the houses worth a lot?”
Lev Tolstoi’s “Arithmetic” was not recognized in his lifetime. But we, unknowingly, come to the same ideas that were already expressed by him 150 years ago. After all this, would it be fair to classify Lev Tolstoi as a pure humanist? Or would it be fair to say that he wasn’t a humanist at all, but a man of pronounced mathematical ability? No.
The truth is that our basic understanding of arithmetic is conditioned not by our predisposition to the subject, but by our life experience. Mathematics is all around us. But it is very important to see in everyday situations not a set of individual numbers, but algorithms and actions, mathematical laws that we understand and feel, even if we cannot articulate them rigorously. Lev Tolstoi, while knowing and promoting these truths, did not forget another, equally important component of successful mathematics. Let us, too, not forget that “in order for the pupil to learn well, he must learn willingly.