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Plutarch
Once a mathematician (not a teacher, but a real scientist!) began to lead a home mathematical circle.
With children aged 3-5, he did not cram the multiplication table and counting from 1 to 9 and vice versa, but was engaged in the development of scientific thinking.
Interesting? What came of it, he told in the book (Zvonkin A.K., «Kids and mathematics. Home circle for preschoolers»). I read it in one breath and share it with you.
For those looking for quick, practical advice, go to the section «For those who want to do math with their children» and beyond. Who cares what the book is about — see the fragment of the table of contents as an illustration.
How it was…
«Those same six matches that still remained on the table after the previous task are laid out in a row. I ask you to attach a button to each match. Standard question:
— Which is more — matches or buttons?
— Equally.
“So there are as many buttons as there are matches,” I summarize.
I take all the buttons into my fist and ask them to tell me how many buttons are hidden in my fist. Characteristically, no one makes the slightest attempt to count the matches. And why, exactly? After all, they ask about buttons — which means that you need to count buttons. Dima, as a man with me on the shortest leg, is trying to unclench my fist, others ask in surprise:
How can we count them?
I am laughing:
— Of course, it is impossible to count — the buttons are hidden. But try to guess somehow.
Then a real flurry of riddles falls upon me, most often based on nothing. Everyone shouts something different; at the same time, only Zhenya shouts the correct answer. I try to listen to him, ask why, but he retreats. In general, shyness often interferes with a wife. … With Andryusha — another problem. He is a very purposeful boy, and in our classes he clearly lacks motivation. The next time I proposed the same task in a different arrangement — there were no longer buttons with matches, but soldiers with guns, then they left, but the guns remained, and now the scout needs to find out how many soldiers there were — that’s when he was the first to guess that can you count the guns».
For four years, a “math circle” gathered every week: a caring mathematician dad studied with his four-year-old son and his peers.
“Every week” is perhaps an exaggeration. There were gaps due to departure, illness or employment. Once the children got bored and classes were interrupted for three weeks.
When the boys went to school, the youngest daughter grew up and dad resumed the circle in the “girlish” version. For two years, the girl Zhenya and her friends studied mathematics.
Kolya Butee Vasya, or What is Mathematics?
The traditional understanding of mathematics by adults for children is counting and geometric shapes. As a result, «doing math» comes down to memorization — «remember, Vasya: this is a circle, this is a square, this is a triangle» — and poor Vasya does not understand why you need to know this. With regard to the account, the situation is even simpler. This process was amusingly and accurately described by Zvonkin:
«Let’s take the place of a child and try to learn arithmetic ourselves … but only in Japanese! So, here are the first ten numbers for you: ´iti, ni, san, si, go, r´oku, s ´iti, x´ati, ku, ju. The first task is to learn this sequence by heart. You will see that it is not so easy. When you finally succeed, you can proceed to the second task: try to learn how to count in reverse as well, from ju to ichi. If this already succeeds, let’s start calculating. How much dignity will be added to rock? And take it away from the city? And hati divided by si? Now let’s solve the problem. Mother bought a kull of apples at the market and gave each of these children an apple; how many apples does she have left? A very difficult but obligatory condition is not to translate into Russian, even in your mind. … But now you have finally learned how to run count within ju. How much time did it take you? A week? Month?»
You trained mechanical memory, but what is the mathematical meaning here, you still need to figure it out.
What then is the science of mathematics? What is the main thing in it, if not numbers and figures?
Maths:
- this is the law of conservation of quantity
- it’s the beauty of symmetry
- it’s a probability theory
- this is the idea of a schematic image (from drawings and pictograms to hieroglyphs and to alphabetical writing)
- this is combinatorics
- and not only.
Here is a fun task for the development of thinking (transitivity):
«Kolya is more beautiful than Vasya, and Vasya is more beautiful than Tanya. Who is the best of all?
Ask the kids this riddle. They will probably give the correct answer, but they will ask with curiosity: what does “butee” mean. Let me tell you a secret: this word is meaningless, it is given to express the degree of some abstract quality in the heroes of the task.
The fundamental position of mathematics (however, relating to science in general), known to everyone:
- it proof of their statements.
Science does not tolerate unfoundedness. Read on.
Were we watching the monkeys or were the monkeys looking at us?
The children were at the zoo, excitedly say that they were taken to «look at the monkeys.» Papa never misses an opportunity to give food for thought: “It wasn’t you who were taken to look at the monkeys, it was you who were brought to the monkeys to look at you.” A dispute ensued.
«Such an insinuation on my part could not but provoke a strong protest, but they did not immediately find what to oppose to it.
We were looking at them.
This argument is easy to break down:
— Well, think about it, look! They were looking at you too.
The second argument was much more serious:
We can walk where we want, but monkeys can’t. They are in a cage. But I also found something to object to this.
— No, you don’t go where you want. For example, you are not allowed to walk inside the cage. And the monkeys are not allowed outside. It’s just that there is a grate, and the monkeys go where they want, on one side of the grate, and you on the other.
So we argued for some time, and suddenly Dima exclaimed joyfully, as if catching me in a trick:
— Oh, folder! After all, we are doing math again!
An interesting evolution… At the very first lesson of the circle, the children raced to count the buttons laid out on the table. Then they just imagined mathematics in this way — this is when they count. Now mathematics has become for them something like a logic game in the style of Lewis Carroll».
I also think that mathematics is the most interesting science, my son and I did it a lot and with pleasure. I am not an expert, but for me mathematics is solving puzzles, and a cube of ueka, and the basics of reading drawings, and logical puzzles.
Train — teach — develop thinking
«Az, beeches, lead …» repeated in every way like a spell — this is study in the understanding of teachers and schoolchildren of the eighteenth century. Cramming, coaching — that’s what we call it today.
Classes for school preparation, which are taught by a good teacher, are most often studies with elements of the game. Children memorize the rule and apply it in tasks. Children learn to learn, this is a useful skill for life.
Such an approach does not solve the task of developing thinking. This is not how researchers, scientists, creative people are brought up. But as?
Questions are more important than answers
An excellent memory will allow a person to become an “encyclopedist”, to accumulate gigabytes of knowledge. Such a person will be likened to a closet in which information is stored (well, if it is systematized). Intelligence is something more. The intellectual easily operates with information, creatively processes it, quickly finds unexpected, non-trivial solutions.
Transferring knowledge to a child is important, but not enough. If you want to develop your intellect, train it.
A living example: «Petya, what do you think is more in the world — squares or quadrangles?»
Petya, without hesitation, assures that there are more squares, hotly pointing at the windows, napkins and cubes strewn on the floor.
Step on the throat of your song! Silent! Do not rush to explain why there are still more quadrilaterals.
This is one of the problems of the type «the whole is always greater than the part.» An elementary rule for adults, but mankind has been going to the knowledge of this axiom for hundreds of years. Do not give a ready-made, «overcooked» answer. Your child will get more, much more, if he discovers this truth for himself in a week, a month, a year or more!
Why are questions more important than answers?
- The question makes it possible to «break your head» — this is the very development of the intellect, which is difficult to organize otherwise
- The “think” question inspires the child with the value of independent research and personal opinion.
- Self-reflection, without neglect from adults, increases self-respect and self-confidence.
What questions to ask? Suppose the baby does not yet know what a quadrilateral is and how a square stands out. Ask like this:
— Who is more: boys or children?
— Who is more in the world: birds or sparrows?
— What do you think, what is more in the world: food or porridge?
It doesn’t matter how simple or sophisticated the concepts are used. You can ask about boys, or you can ask about tetrahedra. Important food for the mind.
The reader will ask: how long to be silent? If the next day the child does not come up with the correct answer, can you already tell him the “correct answer”?
- «You can’t leave a baby wandering in the darkness of ignorance!»
Do not be afraid. Let the child forget about the task, but you don’t forget. From time to time return to her in a «new guise», ask about people, then about figures, then about apples. Independent attempts to think will not go unnoticed for the child.
As for the “forgotten tasks”, joyful discoveries await you when your son suddenly comes running: “Dad, it seems that I told you wrong about the squares!” – and clearly explain why.
Another way to «wake up» the child’s independent thinking is … a mistake made by an adult.
A.K. writes: «Suddenly, another idea comes to mind: I start making mistakes on purpose. Petya is the first to notice this; the happiness of children has no end».
Noticing the mistake of an adult, children become much more attentive, develop critical thinking.
Results?
The children have already grown up. It is interesting to understand how the mathematical circle influenced them. The guys became mathematicians, engineers? No, from the boyish company only Dima, the son of Zvonkin, chose mathematics as a profession. He defended his thesis, got a position as a researcher at the University of Paris. He worked with schoolchildren a lot, organized the Tournament of Cities in Paris — an international mathematical Olympiad.
The second boy, Petya, graduated from the Institute of Oriental Languages of Moscow State University, spent a year in Japan, works as a translator. One of the organizers of the parish youth club, publishes a youth magazine.
Andryusha graduated from the State Financial Academy with a degree in World Economy. Works in the Moscow branch of one of the largest Western banks with «financial instruments» — securities, currencies, derivative contracts.
Separately, Zvonkin studied mathematics with girls. Only one of them connected life with mathematics.
Daughter Zhenya graduated from the university in Paris, department of film studies. She teaches film studies to students and teachers of lyceums, translates, makes subtitles for films, participates in the organization of film festivals. Filmed short films, was a member of the youth jury at the Cannes Film Festival; represented contemporary French cinema in Russia, Kazakhstan and Kyrgyzstan; dances in a semi-professional swing troupe.
Her friend Sanya graduated from the Faculty of History and Philology of the Russian State University for the Humanities; she works a lot with children at the Protestant church, at the time of writing the book «Kids and Mathematics» she was working on opening a private school.
Dina lives in America; She graduated from the Faculty of Mathematics at Brandeis University and works as a teacher of mathematics. Edits math textbooks. Seriously fond of ceramics: participated in several exhibitions.
Not mathematicians grew up, just talented people. Are your parents lucky? Maybe they day after day, taking care of children, did so that they were lucky?
For those who want to do math with their children
Make a lesson plan. In the book of A.K. Zvonkin is given more than a dozen mathematical and logical tasks for kids. The indicative plan of the first lesson was made by the reader Julia.
Examples of tasks for kids:
- fourth extra
«Hare, squirrel, fox and suitcase. What is redundant here?”- you will find many such tasks in manuals for teaching children. With their help, you will teach children to guess patterns (this facet of mathematical thinking, unfortunately, is ignored by school pedagogy).
- Geometric puzzles (many varieties are sold)
- Approach to probability theory:
The opaque bag contains two yellow and two black dice. It’s a dark closet that contains a pair of yellow and a pair of black boots. To go on a visit, you must definitely get a pair (you can’t wear shoes of different colors). But due to the fact that the closet is dark, you have to get the shoes at random, one by one. The guys take turns pulling the cubes from the bag and remember how many cubes they had to pull out before getting a pair. Discuss with what number of drawn cubes it is impossible to obtain a one-color pair (a); (b) you can, but not necessarily; (c) you will definitely get a pair.
- Challenges for ingenuity.
«The Boy in the Elevator» One little boy who lived on the 16th floor was allowed by his mother to ride the elevator himself. But he drove in a strange way: when he went down, he drove from 16 to the 1st floor, and when he went up, for some reason he only reached the 8th floor, and then he walked. How can you explain this?
- Symmetry
For symmetry problems Zvonkin A.K. used the game «Mosaic». Having depicted the axis of symmetry in the middle, explain to the children that on the right is a mirror. Lay out different figures on the left, and ask the children to lay out on the right with a mosaic “what will be reflected in the mirror”.
Tip: use bright, beautiful aids. The author noticed that the children ask to continue their studies only when they practice with colorful mosaics. At first, this seriously upset him: it’s a shame to think that the kids liked to play with mosaics, and not do mathematics. In the next lesson, he was convinced that it was not the game. The children did not agree to just “play with the mosaic”, they asked for tasks. As Zvonkin writes:Children need a full-fledged intellectual and aesthetic pleasure.».
For those who don’t like math
One of the author’s most unexpected pieces of advice: don’t do math if you don’t love it!
Mom, who hates mathematics, let the teacher leave the opportunity to reveal to the children the secrets of this exact science.
If a mother and her daughter draw, do needlework, gardening, and a father goes hiking, instills an interest in wildlife, teaches them to learn its laws through simple physical experiments, this will give a lot to children, including for the development of thinking.
A year after the completion of the circle with the boys, Zvonkin began classes with his daughter and her girlfriends. Daughter Zhenya was not at all interested in mathematics. Then why? I give the floor to the author of the book:
“Why did I start a circle with Zhenya at all, if she was not interested in mathematics? Well, shame on you, dear reader, to ask such questions! No, seriously, jokes aside. Communication with parents is vital for a child. Even negative communication, such as swearing and punishment, is better than no communication at all.
Moments of meaningful communication with children, that is, joint participation in some reasonable business, are extremely rare, and their value cannot be measured. This is what happened with Zhenya. She enthusiastically studied at the circle — not because of mathematics, but because it was her father who gave her his time and attention, was engaged with her, and was doing something serious, what he used to do with his older brother.
Other parents were not mathematicians, their desire to open the world to children was embodied differently. The father of the boy Zhenya went with the guys to the forest, together they climbed the rope through the ravines, “hunted deer”, navigated the map, dug out some caves, arranged hiding places in them. One of the mothers studied English with the children, the second — music.
- And what could you, the reader, teach your child and his peers?
Organizational issues
Whether you lead a math class or teach girls how to embroider, decide to allocate special time for classes. Get together once a week for your venture — it will soon become a pleasant habit. Children love rituals and traditions. If at the beginning of the lesson you ring a bell or put on a short piece of music, the sound will set the kids up for intellectual work.
Keep a diary! This is useful for summing up and evaluating the results of classes for a month, six months or a year. In addition, ten years later, when the children are older, diary entries will be exciting reading for both you and the children.
Serious regular classes are ideally conducted before the child goes to school. Schoolwork takes a lot of time and energy of the child, at school they are taught to certain rules, the discrepancy between which and yours can confuse a first-grader.
Exciting and educational at the same time scientific experiments for kids. Look for books with titles such as Entertaining Chemistry, Physics, Astronomy, Science Experiments with Children. I recommend the manual from my childhood «Physics for Kids» by Leonid Sikoruk.
Zvonkin A.K., in addition to classes, read the book of Zhitomirsky and Shevrin «Geometry for Kids» to children.
In addition to the content of the classes, prepare the material base: paper, pens, pencils, etc. There should be an abundance of materials so that the lesson is not disrupted by a children’s fight for glue or scissors.
Finish the cycle of classes beautifully. For example, solemnly present diplomas. As a result of conducting a mathematical circle, which I talked about, the children were given diplomas, issued on a printed form, into which the text was printed: “this diploma was given to Dima Zvonkin for the fact that he studied mathematics for a whole year and became very smart».
At the end of the article, I will list the principles that I consider important:
— do what you love with your children, with interest and desire,
– create your own good traditions,
– approach the issue creatively: there are many different ways to discover the world,
— encourage the child to think, do not always give him ready-made answers.
If you were interested, read Zvonkin’s book, he does not teach children mathematics, he teaches them to think!
Download — Zvonkin A.K. «Kids and Mathematics».pdf