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We can all count

We can all count

The good news for parents who do not have major mathematical achievements is that love for mathematics depends not so much on heredity, but on the quality of education and the positive attitude of parents to the question. But for a start, the most adult needs to realize what this science is studying for, what it is for. Mathematics is not a simple recalculation and dry numbers, it is also an idea of ​​space and time, size and quantity.

For example, if you look at a leaf of a tree or butterfly wings, you can see mathematical proportions, symmetry in them. In addition, mathematics allows you to systematize objects, to see the logical laws in nature and in life.

Unfortunately, often mathematical education begins with the fact that children, like parrots, repeat after adult “1, 2, 3, 4, 5, 6”, memorize whole series of numbers, not understanding their meaning. The kid smartly counts up to a hundred, but if you ask him to bring in, say, five candies, he freezes in bewilderment. The fact is that the child is not aware of the relationship between the number of any items and the number that means this number.

If, for example, you show children 6 tokens and ask how many they are most likely to count and answer correctly, but do children understand that 6 is not just a number that corresponds to token number 6, do they understand that there are 6 tokens! How to clearly convey to the child the meaning of the bill?

Many teachers explain that 3 is not the same as 1, 2, 3 (the usual enumeration), but 1, 1 and again 1. For example, the teacher uses a book in which the picture shows the objects of one, the other and the third species (for example, one machine, two bears, three dolls). On the right side, using the flap system, 1, 2 and 3 fingers appear.

To say which item is shown in the picture in triplicate, the child points to 3 dolls. Why? Because there is one doll drawn, another and one more.

With this method, the child begins to understand what the number 3 is. If the child thinks that 3 is just a component of the list 1, 2, 3, this knowledge is hardly useful to him for performing mathematical operations.

We can all count

As one of the founders of the theory of developmental education, V.V. Davydov, teaching any science, it is necessary to explain why we perform certain actions. So, in the workbook in mathematics, children are given the task to paint five of the painted Christmas trees.

Some children carefully and diligently paint all the Christmas trees and are genuinely surprised to hear that the task is done incorrectly. The fact is that children do not understand the meaning of the task that is obvious to us: for them, the point is to color, not count.

According to Davydov, the basis of the educational process is the so-called “learning task”, that is, the task that forces the student to seek and apply a common way to solve all the tasks of a given class, that is, all such tasks. For example, a teacher at a lesson invites children to determine which is higher in height — a classroom door or a window.

The task cannot be solved directly practically, since the door cannot be brought to the window. We need to look for another solution.

First you need to figure out how to solve this type of problem in general. In this case, there are ideas of measurement and the need for measurement, comparison of numbers.

It is extremely important to ask the children what they have learned in the lesson or lesson. Most likely, it turns out that some of the children learned to fold, and some – to add two to one Christmas tree.

It is clear that in the latter case the children solved only a particular task and are unlikely to be able to transfer the mode of action to another similar one.

French novelist Anatole France wrote that “you can only learn with fun …”. Indeed, motivation plays a huge role in any training.

Therefore, arithmetic can be mastered playfully. For example, with the smallest you can start with finger games:

One of the means for teaching mathematics can be games where a cube is used, because they need to say how many points a cube falls in and perform the number of actions corresponding to the number of points.

Another great game is dominoes. The child no longer needs to count “1, 2, 3.” when he sees 3 objects: he is used to three points as 3, not 1, 2, and 3, he can immediately say that there are 3 forks on the table, and not one, two and three.

He realizes that 3 consists of 1, 1 and 1, or 2 and 1. This is not just a list of characters.

Here are five kittens.
One is gone and there is none.
Well, no it – and no.
Four kittens left.
Here are four kittens.
Single nightly sometimes
Climbed a tree –
there are three kittens left.
But somewhere squeaked
little mouse thinly,
the kitten heard –
left two kittens.
One of them with a ball
disappeared in the door without a trace
and the cleverest one
remaining, last:
to lap up for five
became the milk from the bowl.
(A child, following the example of an adult, alternately bends one finger at a time.)

We can all count

You can use, for example, this game:

A rabbit on the way met another rabbit; two little rabbits became friends. (An adult, uttering the word “two”, simultaneously shows two fingers to a child. Thus, 2 means not just a word / number, but also a number of objects.)
Two little rabbits on the way meet another rabbit. (The adult extends one more finger.) The three little rabbits became friends.

In this game, the spoken number is related to the number. At the same time, the numbers add up and the child begins to realize that they are getting bigger.

You can come up with games for older children or take advantage of those that are collected, for example, in the book of Igor Sukhin “800 new logical and mathematical puzzles”.

“Who has more?”
In this game you can play together and three. For the game you will need a cube with dots. As a counting material, you can use buttons, cones, nuts, etc. Put the buttons (nuts) in a vase or box.

Now roll the dice in turn. What number falls, so much and take from a vase of objects.

When the vase is empty, calculate who has more.

“Find a toy”
A child driving out of the room. At this time, hiding a toy.

Then the child is explained where to find her: “You have to stand in front of the table and go three steps forward, two to the left, etc.”. When children are well oriented, the tasks can be complicated – to give a scheme, not a description of the location of the toy.

According to the scheme, children must determine where the hidden object is located.

At the same author you can find funny puzzles with characters going through the whole book, for example:

1. Dwarf Zabyvalka returned from fishing pleased.
– How many fish caught? – asked comrades.
– I will not say. But I’ll eat both myself.
How many fish did Zabyvalka catch?

2. Today I fed two animals unknown to me and counted that in total they have 6 legs. But I don’t remember how many paws the first animal had and how many the second one had, ”said Zabyvalka to his comrades. – There is nothing to think about!

Each has 3! – laughed.

We can all count

Remy Brisso, a researcher in the field of cognitive psychology, believes that if you teach mathematics as an endless list of abstract rules, patterns that need to be applied “because it’s like nothing else,” you can cause an aversion to science. It is important to show where this or that rule comes from, how it appeared in people’s heads, its genetic roots. He is echoed by the author of the book “Mathematics, not like …” Alexander Zvonkin.

He writes that the theory of probability arose from people’s observations of random, unpredictable phenomena of the surrounding world. And it is precisely such observations that can be carried out with children, using, for example, games with a dice. In the classroom of the mathematical circle, organized by him for preschoolers, he tried to emphasize the probabilistic nature of children’s observations.

For example, instead of a cube, children were offered a crooked polyhedron so that they could see how the game becomes “unfair”: some numbers fall out more often than others. The meaning of such observations and research is that the child should make his own independent discovery, play the role of a researcher.

For example, in one of the Czech schools, children painted a portrait of an average student in a class. To do this, they needed to make a whole series of mathematical measurements, find out their height, weight, foot length, etc.

The child’s research position is also manifested when, with the help of an adult, he encounters contradictions in his own point of view. An example of this methodical technique is water experiments in which the child is asked to guess whether this or that object will sink. Children usually think that if the object is small, it will float, but if it is large, it will sink.

The child is offered to experiment with various objects himself, as a result of which he is surprised to notice that the small pin does not float, as he supposed, but drowns! At this moment there arises a difficult, but interesting task for children’s thinking. This example is rather from the field of physics.

If we talk about mathematics, then Alexander Zvonkin uses the same technique, studying with children the well-known phenomena of Jean Piaget, who discovered that children hardly understand the principle of conservation. For example, if two equal rows of objects lie in front of a child, he sees that the number of objects in both rows is the same.

However, if one of the rows is moved apart, without adding anything or diminishing, the child claims that there are more items in one row. Zvonkin several times removes items in a row and again pushes the row.

As a result, some children begin to realize that equality in the number of objects does not depend on the spatial increase of the series.

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