Rus Articles Journal

Occupations by mathematics of

Part 1. From 3 to 4 years.

the Serial account and figures with

Yes generally, we began

, as well as all - considered fingers. The serial account to 10byl is acquired very quickly by two years. We knew designations of figures in 1 g 4 m. Learned figures according to pictures: cat, dog, letter A, horse, figure 3. Vmalenky age all the same what information on the picture to perceive - why and not this?

it would be possible for

earlier, but I knew nothing about early development izanimatsya with the child purely intuitively.

of Fraction

So till 3,5 years we also got stuck on the serial account, did not begin zanimatsyamuzyky yet. And study in music duration of notes, and so and call them - whole, a half, quarter, the eighth. So we had to begin with fractions. Cut apple, tore paper to parts, just drew - did not understand. In kontsekonts, the daughter told: mother, the eighth is a short sound - THAT, the quarter is a one sound from two short, it is necessary to play a note and to manage to sing THAT is THAT, the half is when you manage to sing THAT is THAT, THAT - THAT. Also reflected for a long time. I surprised with such train of thought showed no sign and I speak: well, and whole? Sashagovorit: in mind it is difficult to count it. I speak: then draw. Drawing of byltaky:

Each stick is THAT. And only eight THAT! - whole note. Sasha was a takgorda the success that two days tried to explain to all garden this teoriyudroby, however, as I understood without results.

the Teacher of music on the following occupation tried to cut apple too. Sasha told also her about the theory - and they decided to cut nothing, to akhlopat in palms - one cotton - the eighth note, two cottons - a quarter.

It appeared that it is much simpler to three-year-olds to put, than to divide p.

Having come in the difficult way to simple truth, we decided to be engaged in addition.


Business was so. I decided that all the same what to put if only nachatprotsess. Suggested Sasha to put books. I speak: there are two books, and voteshche one. How many books? I receive the answer: one book about ducks, one about mice and one about Petya - a cockerel. Approximately the same answer of a yapoluchil and about cups: one with a lodge, one with a floret and one with the aunt, very awful. Then I understood that if at first to teach the child to addition, and then to grouping of objects of a sign, then it is better to begin skladyvatyodinakovy things. We had no calculating sticks, but there were children`s dominoes, the large size, about 7õ14 cm, with points - peas. Convenient it onookazatsya also by the fact that on one card - dominoes two pictures with goroshka visually.

On it “didactic material“ the account to twelve was acquired by

instantly, minutes for 15. The following stage I decided to show what can be written down etiprimer on paper.

we Began

so: I drew a rectangle, divided it into two parts, risovalagoroshka in the top part and in the lower part and suggested to count narisovannyegoroshka together. We with Sasha at the same time kept saying following “literaturnoyeproizvedeniye“: we draw dominoes, with halves it, from above we put figure 2, an avniza 4. If we count together, then we will receive exactly 6! (Here boundless poledlya imaginations, but the child went bananas just from the repeating beginning: we risuyemdomino, with halves it.)

At once under drawing I wrote down this example by means of figures, however, especially without drawing Sashino attention to this process, very quickly.

Further by means of the same drawings began to complicate a task: we draw dominoes, with halves it, from above we put figure 2, and below - we do not know! How many to nadodopisat that 5 turned out?

Agree, it already practically the equation! 2 + x = 5. Below under risunkomya and wrote down, having explained that when we do not know something, we write instead of an etogobukva x.

having Fairly trained examples on addition, Sasha, without reflecting, the vydayetpravilny answer - 3! Means, x =3, I transport.

the Next two days Sasha had a good time the fact that she drew to herself examples itself - and! - below under drawing wrote down everything in figures! That was kakpo - to the present! There was no doll or the hare who did not learn to byreshat examples on addition and the equations.


I Am afraid of p that at a subtraction explanation I will have same problems as prislozheniya of books and cups before explaining a subject subtraction, I to reshilaobjyasnit to the child that such grouping of objects of a sign.

About a week left at us on learning to group objects to popriznaka a form, color, the size and the general properties. (Rates are not vunderkindsky, we will directly tell.) About 50 - 70 objects were required (from those which, certainly, are at home at everyone). Having spread out these objects on a sofa (more true, having piled them), we spread on a chair objects of a sign “red“, to a popriznak “rectangular“, by sign “small“, by sign “toy“ it. In passing “invented the bicycle“ - thought up game “What subject superfluous“ on some sign.

Actually, a subject this infinite, we something group all life, we classify by various signs. Later we began to group to popriznaka “taste“ (sweet, bitter, sour, salty), “began to smell“ (is or not), “temperature“ (ice, cold, warm, hot, burning), “volume“ (volume, flat, linear); began to unite in groups of a plant (a grass, flowers, bushes, trees), fruits (vegetables, fruit, berries), animals (a bird, fish, amphibious, mammal). By four years it became odnoyiz loved by that. Besides, if the person is able to group objects, he can irasskazat about anything. For example, Sasha could characterize in a medvedyasleduyushchy way: the bear is a wildlife, an animal, mammal, brown color, the big size, an irregular symmetric shape, volume, warm, wild - lives in the wood, usually not predatory, falling asleep for winter. - “to characterize“ it, and does not mean at all that verses, fairy tales about mishku the drawn Winnie - Down passed by it, “ratsio“ did not win and not the bylotaky purpose it is perfect.

Game 2 - we defined, than, for example, the cat differs from a crow, and chemon are similar; except repetition of the gained knowledge also very much a razvivayetfantaziya.

Game 3 - someone thinks of a subject, and to pytayetsyaugadat the second for signs that it.

Subtraction and division equally I tell


After this retreat about how we passed a subject “subtraction“ (sometimes still we call subtraction “taking away“ - you will do nothing).

Can tell

that everything passed on “hurrah“ and without problems, groups were acquired if we “took away“ books, then their contents at the same time was not retold. Everything was good, did not begin to take away candies by means of a klassicheskoyzadachka yet: you have six candies, you gave me two how many at you remained? Sashaskazala, that it is so dishonest, she will give me three candies because so to a budetporovn. It was necessary to agree. But here the father came from work. Again it is dishonest, Sasha declared, - it is necessary and to share with the father. Well, we will share, I say, but so far we will solve a problem! - Unfair! - Sasha declared, burst into tears and went to a kseba to the room.

Had to change

a problem so: you have six candies, you gave two to mother, a dvepapa how many at you remained? This problem of complaints from the child did not cause and was quickly solved.

This case suggested to me an idea that division equally for is easier to trekhletkivosprinimat, than subtraction. Because it is FAIR. And when utebya eight balls, and two burst, it is UNFAIR and therefore takiyezhalostlivy problems do not want to be solved. Subtraction is almost vsegdanekhoroshy action because from it losses and injustice. Therefore izzadachek on subtraction went off the following with a bang: “From ten tasks Sasha uzhereshit eight on the five. How many still it was necessary to solve tasks on the five that all tasks were solved on the five?“ .


Needed to pass multiplication that a classical set of four deystviybyl assembled.

Nothing it is cleverer than

, than “three rows on four trees“ or “the area of a subject“ to me to a vgolov did not come. Sasha came to the rescue and told: let`s draw to iskladyvat!

In general, by means of drawings business always goes easier. Drew those zheneschastny three rows on four trees, saw that if to put 4+4+4, toeto just and will be correct. The mechanism of multiplication was understood, and it imennoto what I also tried to obtain.

If to the child processes of four actions and an osnovygruppirovaniye are clear to

, then by means of drawings he will solve any problem from a nachalnoyshkola.

One more thought which appeared at me after classes with mine rebenkom:mozhno, of course, to teach the child to solve till 4 years examples with hundreds, it is a poymetmekhanizm, but will not be able to imagine, for example 90 machines. After 20 (tvoipalchik and mine) for the three-year-old goes “much“ even if he also knows that the tamdalsha, does not have understanding.

Part 2. From four to five years.

When to Sasha was four years old, I bought the textbook for 1 class to a pomatematika. I was very disappointed as for my child nichegonovy was not there. We counted it methodically all in a month and kupilidrugy, for the 2nd class. There were some new concepts, but even with an ikhobjyasneniye in 2 months the textbook was passed (already not so methodically, on desyatyodinakovy we did not solve problems). After that we with Sasha, having conferred, decided not to fool about any more, and to be engaged in what to us budetinteresno.

Hundreds, tens, units.

the serial account to hundred was acquired by

By four years somehow by itself, kakstikh. But understanding - what is it - hundred - did not come. On pomoshchprishet the train of firewood...

In each car on ten thick logs. If in the train 2 cars, skolkobreven it is lucky? The answer was, as always exhaustive: trains from two to the vagonoyena happens. Well, I speak. The train from ten cars. How many logs in pervykhdvukh cars? Twenty, - practically without reflecting Sasha answers. Correctly, and in three? Includes the serial account, considers. Thirty! And you know how it is written thirty? I write. Fine, well done! And in four? Again the vklyuchayetporyadkovy account, but considers about itself, feels a dirty trick. Forty! I write - 40. Av to five? Mother reflected, I will write, and you will tell, well? Writes 50! I call - “fifty“. You hear - five - desit, that is five tens! Asks: and then - six - desit? Seven - desit? - Yes! - Eight - desit! Nine - desit! Ten - desit! - But you counted up to hundred, were there devyatdesyat desyatdesit? - No. - And as was, remember! - (considers) - ninety! (writes 90), (considers) - hundred. And how to write hundred? - I show - From three-figures?! - Yes! 90 - these are nine tens, and is 100 10 tens!

All evening goes, become silent, probably, considers. In the morning: mother, and in two-trains - 20 tens, it it is possible to write so - 200? - It is possible! - And nazyvatsyabudt as? - Two hundred, that is two hundred. - And in three cars - there is so much, trist! - Well, almost correctly, only not trist, and three hundred.

Here so we in two days “promoted“ from the account to 20 till 1000. Of course, mistakes still were, generally with names of numbers, but the main thing - understanding came.

Due to the studying of hundreds - tens - units the sleduyushcheeuprazhneniye is useful to make

681 are six hundred, eight tens and one unit, or: six trains, eight cars on a side-track and one - lonely brevnyshko which was not located. (-was not located, but not from where - nibud it dropped out.)

That is in spite of the fact that everything is acquired, periodically to repeat, represent.

the Account in a column.

After the understanding of large numbers came, the question as remove to be how to consider problems and examples with hundreds rose. It was necessary to show, kakproiskhodit addition and subtraction in a column. (Five primerovv a column for warm-up) we do this “gymnastics“ before each occupation by mathematics iseychas (I will not tell that occupations at us happen very regularly). Of course, everything was impaired a little by the father who from school is at odds with mathematics, chistyygumanitary - taught to use the calculator (destroyed magic - it appears, it is possible quicker, more simply and without reflecting, “and in general, everyone durakspravitsya!) . But on the calculator we check then whether correctly it turned out. I consider that the account is a good exercise for brains.

Addition and subtraction in a column was acquired by

, as well as at school, from a pomoshchyyualgoritm (at addition - we write three, we remember one; at subtraction - otshest it is impossible to take away eight, we will occupy ten, and we will take away otshestnadtsat). By the way, in connection with the account in a column, it was necessary to acquire, chtotaky “algorithm“ - a decision order, sequence of actions. I vypisalaalgoritm on a leaf, and at first it was before eyes in vremyaresheniye of examples.

At the solution of examples in a column arises quite difficult for a chetyrekhletkizadach, for example, from 17 to take away 9. In relation to this case to prishlosyobjyasnit as it is possible to simplify this example.


(It is meant that the child well knows structure of numbers to 10.)

Was given an example: 17 - 9. What to us just to take away from 17? Correctly, 7. And the skolkena has enough from 7 to 9? Two. Well. Means, we will take away nine not at once, asnachat 7, then 2. We write: 17 - 9 = 17 - 7 - 2 = 10 - 2 = 8.

Usually the child well imagines

of the Inequality what number from two is more, akaky is less if properly helped to present it a bolshiyechisl. If is not present - it does not matter, it is possible to begin with small numbers, in passing pomogayapredstavlyat big.

28 + x <50. Let`s solve at first the equation 28 + x = 50. (what should be made, chtobyiz two cars and eight not located logs to receive five tselykhvagon? It is necessary to add two whole cars and two a brevnyshka in a polupustomvagonchik, that is 22!) . Received x = 22. And that it turned out not 50, to an amensha, it is necessary how many to add? (reflected) Well, it is more or less than 22? - I encourage. - It is less! - Sasha rejoices. And it is possible to add nothing at all, all the same will be less! - Correctly, we will write down x <22. That is if mydobavy any number of cars and brevnyshka smaller, than 22 (at least 0!) at the correct decision naspoluchitsya.


When mastered inequalities, it is possible to pass to functions right there. We did not make it because we very much were fond of geometry.


Ya I share opinion that it is better to give to the child of knowledge more, than it is less. He will not apprehend that to it it is inaccessible and so. Therefore my explanations for the stroilisena as at school - at first all definitions, then some signs, potomsvoystvo on everything.

We took

a geometrical figure, for example, a triangle and tried izuchity thoroughly.


- the square, means three corners. Let`s say that a triangle - an etoplosky geometrical figure which has three corners. What else does it have? Parties. How many them? Rub. How it is possible to call the parties in a different way? Pieces. Whether there can be some parties of a triangle parallel? No. Aperpendikulyarnymi? Yes. In this case the triangle will be nazyvatsyapryamougolny. Whether there can be at a triangle two or three right angles? No. Corners which there is less direct - sharp it is more direct - stupid. If vseugla are equal, then and all parties are equal and the triangle will be nazyvatsyaravnostoronny if two corners are equal, then and two parties are equal, and treugolnikravnobedrenny. I show how to measure the parties by a ruler, we study risovatrazny triangles. Whether there can be a rectangular triangle isosceles? (Yes) And equilateral? (no). Whether there can be one party more, than summadvukh others. I give concept of perimeter, the grounds, heights, the areas (dlyaety the leaf in a section is more convenient). Did not reach only Pythagorean theorem of isinus - cosines though it would be possible. There was no wish to enter concepts of legs, hypotenuses, square numbers, and trigonometry - all - still rather early a dlyavospriyatiya. (At least, for my child).

Quadrangles sorted everything together (a square, a rhombus, a rectangle, a trapeze, a parallelogram), looked for differences and similarities, that very much helped, to a chtoprezhda were engaged in grouping. Came to a conclusion that each square - a rhombus, a rectangle and parallelogram, but not on the contrary. That is the square - an etoperesecheniye of sets of rhombuses and rectangles, and a rectangle is chastmnozhestvo of parallelepipeds. Considered the area and perimeter of figures.

Besides together were sorted by a circle and an oval and understood that each circle - an oval, Nona on the contrary. Sorted concepts of radius, diameter, a chord, the center (centers).

was passed Further to volume figures, but so far only at the level of definitions - it is rather heavy as we are not able to draw in stereometry yet.

Long examples and problems in several actions.

It is one more of subjects which we passed. To explain here not prishlosnichy except that process is possible that it is possible to write down so to ireshat examples and tasks in parts.

Here, generally, and all. The system does not apply for anything as, however, and on a svoyusistemnost. Perhaps, to somebody she will help to explain to the malyshamkaky - nibud a subject if the understanding does not come in a different way. Thanks is zavnimany, it was very pleasant that you read everything up to the end.