Contents
Abstract
The author of this book, a professional mathematician, talks about his experience in teaching mathematics to preschoolers. The genre of the book is mixed: diary entries are interspersed with discussions about mathematics or psychology, observations of children and their reactions to what is happening serve as a source for new tasks, and those, in turn, allow deepening and developing ideas, as it were, outlined by a dotted line.
The book will be of interest to parents of preschoolers (as well as their grandparents), kindergarten teachers, primary school teachers, and in general to all those who are interested in the process of developing children’s intelligence.
Not to inform children of finally established truths, but to awaken their curiosity
First lesson and thoughts around
How it happened
… There are four members of our circle: my son Dima and three of his friends — Zhenya, Petya and Andryusha. Dima is the youngest, he is 3 years and 10 months old; the eldest is Andryusha, he will soon be five. We sit around the coffee table. Of course, I’m worried: how can I manage with all of them here? To begin with, I tell the children that we will do mathematics, and to maintain authority, I add that mathematics is the most interesting science in the world. This is where I get the question:
— What is science?
I have to explain:
“Science is when you think a lot.
“But I thought there would be tricks,” Andryusha says somewhat disappointedly. He was warned at home that Uncle Sasha would work with them today, and there would be tricks.
“There will also be magic tricks,” I say, and, folding the introduction, I get down to business.
Here is the first task. I put 8 buttons on the table. Without waiting for my instructions, the boys rush to count them together. Apparently, despite their young age, they already have some idea of what mathematics is: mathematics is when they count. When the noise has died down, I can formulate the problem itself: — Now put the same number of coins on the table. Now there are 8 more coins on the table. We put coins and buttons in two identical rows, opposite each other.
What more, coins or buttons? I ask.
The children look at me somewhat bewildered; They don’t get the answer right away.
— No one else.
“Equally, then,” I say. “Now look what I do.” And I push the row of coins so that it becomes longer. “Now what else?”
— Coins, more coins! the boys shout in unison. I suggest Petya count the buttons. Although we have already counted them four times, Petya is not at all surprised at my task and counts the number of buttons for the fifth time:
— Eight.
I suggest Dima count the coins. Dima counts and says:
Also eight.
— So eight? I emphasize with my voice. So they are equal?
— No, more coins! the boys say decisively.
In truth, I knew in advance that the answer would be just that. This task is only one of the countless series of tasks that the great Swiss psychologist Jean Piaget gave in his experiments to child test subjects … In his experiments, he established: young children do not understand what seems self-evident to us — if several objects are or move, then their number will not change. So, I knew in advance what the children would say. He knew, but for some reason did not prepare any reasonable reaction. What would you do, reader? What would you say to the children?
Unfortunately, the most common technique that almost all adults use in such a situation is to start telling the children something with all their might. “Well, how is it! — the adult says with mock surprise. How could there be more of them? After all, we did not add any new coins! After all, we just pushed them apart — that’s all. After all, before they were equally divided — you yourself said! So, they couldn’t get any bigger. Of course (highlight in voice), the coins and buttons are equally divided!”
Efforts are in vain — such pedagogy leads nowhere. More precisely, it leads to a dead end. First, don’t rely on your logic to convince the child of anything. He will learn logical structures even later than the law of conservation of the number of objects. Until this happens, logical reasoning will not seem convincing to him. Only the intonation of your voice is convincing. And she will show the child only that he was again not up to par and did something wrong. Children do not give up immediately, their common sense is not so easy to break. But if you sit properly, you can ensure that they stop relying on their own mind and observation, and will try to guess what an adult wants from them. In general, adults make many inexplicable demands on children: for some reason, you can’t draw on the wall; for some reason, you have to go to bed when the game is in full swing; for some reason you can’t ask: “When will this uncle leave?”. Something similar is happening now: although I can clearly see that there are more coins than buttons, for some reason it is supposed to be answered that they are equal in number. The attitude to mathematics as a kind of ritual in which certain spells must be pronounced in a certain order originates in school and lives up to the university, where it can be found even among mathematics students.
So what is there to do? Do not ask such questions at all, or what, if you can’t comment on the answer? On the contrary, it is necessary to ask questions. It is also very useful to exchange views:And you, Zhenya, what do you think? And you, Petya? And why? How many more coins?«You can even express your point of view on an equal basis with the rest, but very carefully and unobtrusively, providing all sorts of reservations like»it seems to me» and «maybe«. In other words, all one’s authority as an adult must be used not to assign to this authority the absolute power of the only correct judgment, but to convince the child of the importance and value of his own searches and efforts.
But it is even more interesting to push him into contradictions in his own point of view.
— And how many coins do you need to pick up in order to become even again?
— Take two coins.
We pick up two coins; We count: there are eight buttons, and six coins.
“Now what else?”
— Equal now.
Very well. I spread the coins wider again and ask the same question. Now it turns out that six coins are more than eight buttons.
— Why are there more of them?
Because you pushed them apart.
We again select two coins; then again. Finally, the picture becomes as shown in the figure:
At this point, a violent argument suddenly ensues. Some boys still believe that there are more coins, others suddenly «saw» that there are more buttons. Perhaps it’s time to take a break and move on to another task; let them think for themselves…
All these thoughts and ideas did not come to me right away, so in my story I ran ahead — both into my future thoughts and into my future studies. This task has repeatedly arisen with us in different guises. We had, for example, two armies that could not defeat each other in any way, because they had an equal number of soldiers. Then one of them parted, she had more soldiers, and she began to win. Seeing this, the second army spread even wider, etc. (You can end the story in accordance with your own imagination.) There was also Pinocchio, whom Alice the Fox and Basilio the Cat tried to deceive by pushing five gold coins and claiming that there were more of them.
I learned not to expect easy wins. All the same, before two or three years, children will not learn the law of conservation of the number of objects, no matter how you teach them. And most importantly, it is not necessary at all! I am sure that there is exactly as much benefit from this early knowledge as from premature birth. There is a time for everything, and one should not be ahead of events, including in the field of educating the intellect. (I admit that this point of view is expressed here in a somewhat demagogic form. But the arguments in its favor — and there are many of them — will be abundantly scattered throughout the subsequent text.) However, I repeat, all these thoughts were later. And then, in the first lesson, some kind of intuitive insight kept me from «explaining», and I just moved on to the next problem.
There are six matches on the table. I put various figures out of them and ask the guys in turn to count how many matches there are. Each time there are six of them …
No, I got too carried away with scholastic reasoning and began to write somehow in a clerical way. Let’s go back to the live children’s audience, let’s see how this happens in real life. Each new calculation result is met with a real explosion of delight and laughter. Already Andryusha and Zhenya are shouting that they will always get six. Now Dima rather impolitely tears the matches out of my hands in order to put together some kind of pretentious figure himself, and Petya, on the contrary, very politely asks if I can give him more matches. A little more — and their fun will grow into an uncontrollable childish rampage. We must somehow keep them, and listen carefully to Andryusha and Zhenya (“Why do you think there will always be six?”), and besides, not to miss new turns of thought: after all, Dima just folded a three-dimensional figure — a well. I draw everyone’s attention to her. This time, even Andryusha and Zhenya are no longer so firmly convinced that they will get six again. Counting matches is very difficult — the well is falling apart all the time. We restore it, count it again, it falls apart again … Finally, Dima gets seven! Everyone is slightly perplexed, but no one shows particularly strong surprise: seven is seven, although a little strange. Well, I’m probably repeating myself — well, I’ll repeat it, it doesn’t matter: my pedagogical task is not to communicate definitively established truths to children, but to awaken their curiosity.
The most remarkable result that I would like to count on, which, one might say, I dream of, is that one of the boys in a few days (or months) suddenly, on his own initiative, put the matches in a well and counted them — simply because it became interesting because I wanted to know how things really are. After all, this would be a small independent study!
Well, if this does not happen, then, hopefully, it will happen another time, with a different task. (In the future, I had a lot of evidence that this was the case more than once). Anyway, I confine myself to remarks like «How interesting!» and «wonderful!” – in the hope that this situation will be firmly stuck in their memory.
Download — Zvonkin A.K. «Kids and Mathematics».pdf