# Studying quantity registration is very difficult

**Mathematical skills in kindergarten - promoting the concepts of quantities, digits and numbers**

Barbara Perras

"What has running backwards to do with arithmetic? Many people will not answer anything spontaneously now. Quite a lot - say developmental psychologists. After all, children whose brains have not learned such basic motor steps as running backwards and the associated spatial orientation can later find it difficult to understand numbers." And with children who cannot put their feet back because they do not have the protective control through the eyes, the sense of balance is certainly impaired - a problem that is affecting more and more children these days. " (Murphy-Witt 2000, p. 4).

We use the simplest numbers for counting, this started long before symbols like 1, 2, 3 existed. It is entirely possible to count without using numbers - for example, by counting with your fingers. "You can work out that you have 'two hands and one thumb camels' by bending as many fingers as you look at the camels. You don't need to know the concept of the number 'eleven' to get an overview, by the way to keep track of whether someone is stealing your camels. The next time you find that you seem to have only two hands of camels - so camels are missing a thumb. " (Stewart 2001, pp. 44f.).

"You can also count by making notches on pieces of wood or bones. Or you can count with the help of stamps - clay discs on which sheep are painted to count sheep, or those with pictures of camels to count camels. If the animals As you pass them, toss these tokens in a bag - one for each animal.

This use of symbols for numbers was developed about 5000 years ago when such tokens were wrapped in a clay envelope. Since it was annoying to break open the 'sound exchange' every time the contents were to be checked and then to make a new one, special markings were made on the outside of the exchange to indicate its contents. Eventually it was realized that there was really no need for stamps in the envelope: markings on clay tablets did the same. It's amazing how long it can take to see the obvious. But of course this is only evident today "(Stewart 2001, p. 44f.).

The ancient Romans already used finger support. Your numbers in dashes for 1, 2 and 3 or individual fingers, but also the V for five or the hand with five fingers show this very clearly.

"The next invention after the natural numbers were the fractions - the kind of numbers that we symbolize today with 2/3 (two thirds) or 22/7 (twenty-two sevenths or the equivalent of three whole and a seventh). You can't use fractions Count: Two-thirds of a camel may be edible, but cannot be counted. There are many other, more interesting, fractional operations that can be used for this. For example, if three brothers inherit two camels, you can imagine that each brother has two-thirds of a camel - a practical jurisprudence story so familiar that we forget how strange it is if we take it literally. " (Stewart 2001, p. 45).

Between 400 and 1200 AD, zero was accepted as a number. "If you find it strange that the zero was recognized as a number so late, then please do not forget that the" one "was not considered a number for a long time: it was believed that only several things could make a number. In Many history books have read that the key to the introduction of zero was the invention of a symbol for 'nothing.' This may have been the key to making arithmetic practical, but essential to mathematicians was the idea, a concept of a new kind of number, which represented the concrete idea of 'nothing'. Mathematics uses symbols, but it is just as little to be equated with those symbols as music with musical notation or language with the letters of the alphabet. " (Stewart 2001, p. 45).

Today the positive whole numbers 0, 1, 2, 3 ... are called natural numbers, the negative whole numbers are called whole numbers, and fractions are called rational numbers. Even the ancient Greeks knew how to use the minus sign. However, they believed that only one positive result was allowed. "It was not until the 18th century that negative numbers were recognized as real numbers." (Devlin 2002, p. 182).

Real numbers are of a more general nature and allow an adequate theory of limits. They are used to denote numerical measurement results of time and physical quantities such as length, temperature, weight, speed, etc.

Complex numbers are of an even more general kind. In contrast to the rational and real numbers, they cannot be interpreted as points on a straight line; they can instead be thought of as points on a plane. "Since complex numbers are not arranged in a straight line, it is impossible to say which of two complex numbers is the larger." (Devlin 2002, p. 183).

"So we have five number systems, each of which includes more than the previous one: the natural numbers, the whole numbers, the fractions (or rational numbers), the real and the complex numbers." (Stewart 2001, p. 47f.).

Mathematical "things" don't exist in the real world

"Mathematical processes are abstractions; therefore processes are no less 'things' than the 'things' to which they are applied. The reification of processes is an everyday occurrence." The number "two" is not a thing, but a process that occurs when you imagine two camels or two sheep bleating the symbols "1, 2" one after the other, as in roll call. "A number is a sequence of movements that was so thoroughly reified long ago that everyone imagines it as a thing. Just as plausible - though less familiar to most of us - is to think of an operation or a function as a thing For example, we could talk about the 'square root' as if it were a thing - and I'm not talking about the square root of a certain number, but the function itself. In this notion, the square root function is a kind of sausage machine: at one end you stuff one Number in, and the square root comes out at the other end. " (Stewart 2001, p. 49).

Ontogeny is the repetition of phylogeny

The structure of the human brain contains the evolutionary development of all brains. "At its core, it seems, the human brain is very similar to the brain of today's reptiles. The development of the fetal human brain completes all stages of evolution in miniature, and the more primitive layers are overlaid by new layers." (Ratey 2003, p. 16).

We can only think in terms of movement paths because we have no other! Therefore every child - also in mathematics - has to understand all the developmental steps of their ancestors one after the other and next to one another - building on and parallel to one another.

Apart from humans, however, only intensely trained apes can make the leap from the quantity sense to the number symbol. The human brain was not originally designed for arithmetic, it was built up evolutionarily with spatial orientation and the use of language. People were "tempted" to count by rhythms such as the two-pod when walking, the four-pound trot or the waltz when galloping. This development story explains to us why children can access algebra and mathematics in general through movement, rhythm, music and horse riding.

"Some animals have only one gait - a single standard rhythmic pattern for moving their limbs. The elephant, for example, can only walk at one step. If it is faster, the patterns of leg movement remain the same. Other animal species have many different gaits; Take the horse, for example. At low speeds horses walk at a step; at higher speeds they trot; at top speeds they gallop. The differences are fundamental: a trot is not just a fast walk, but a completely different type of movement. " (Stewart 2001, p. 120).

Most gaits show some degree of symmetry. When an animal jumps up, both the front and rear legs move simultaneously, maintaining the animal's bilateral symmetry. The left half of a camel, shifted by half a phase, can perform the same sequence of movements as the right half, i.e. after a time delay of half a period. "Thus the passageway has its own characteristic symmetry: 'Mirrors left and right, and shift the phase by half a period.' When walking around, you use exactly this kind of symmetry breaking: Despite your bilateral symmetry, you don't move both legs at the same time! There is an obvious advantage for bipedes not to do that: if you move both legs slowly at the same time, you will fall over. " (Stewart 2001, p. 120).

We distinguish the seven most common gaits (cf. Stewart 2001, p. 121):

- The trot is a diagonal gait, first the front left and back right touch the ground, then the front right and back left.
- When jumping, first the front legs hit the ground together, then the back legs.
- The pass gait as a lateral gait connects the movements of one side: the two left legs touch the ground, then the two right ones.
- The step represents a more complex, also rhythmic pattern: left front, right back, right front, left back.
- In the canter rotation, the front legs land on the ground almost simultaneously, but the right leg hits a little later than the left; then the hind legs come up almost simultaneously, this time the left touching the ground a little later than the right. In riding lessons this gallop is also called cross gallop and is undesirable because the horse is not bent in one line. This makes the movement uncomfortable and disharmonious for the sensitive rider.
- The transversal canter works in a similar way, but the order of the hind legs is reversed. The legs on the inside of the bend both base a little earlier than the outer ones. To the viewer, this looks as if the inner legs are reaching out farther than the outer ones.
- The canter is even more peculiar: first left in front, then right back, then the other two legs at the same time.
- The hop is a rare gait in which all four legs move at the same time. It is sometimes seen in young game.

"You can watch the pass walk in camels, the jump in dogs; cheetahs use the rotating canter to move at top speed. Horses are among the more flexible quadrupeds because, depending on the circumstances, they can walk, trot, transversal canter and canter The ability to change gaits comes from the dynamics of the CPGs (General Pattern Generator - central muscle generator). " (Stewart 2001, p. 121).

In addition to the mentioned rhythmic experiences while riding, the children have basic spatial experiences on their own body:

- next previous
- Left Right
- up down
- diagonal
- crossing movements
- Rotation around the spine

Development of the sense of quantity

Some animals already have a sense of quantity. When trained, they can correctly interpret a number of painted points, provided that it is not more than five. They can roughly cover larger amounts. "One, two, three, many, even more, very many" is a scheme according to which small children and many animals can record quantities (Mechsner 1999, p. 123).

Children need a clear imagination of the quantity, their co-training and further development. The long-held thesis that it is difficult for them to detach themselves from the visual material is ultimately due to the fact that this material was one-sided and did not "grow with it". So it was useless for schools. This problem was also demonstrated by Brazilian children who worked as salespeople: They "'can, for example, do excellent math, but hardly transfer their skills to other areas." According to Elsbeth Stern, mathematical competence only arises when children learn to disregard the surface features of a task and recognize their abstract structure. " (Mechsner 1999, p. 126).

The children need infinite time to deal with mathematical material. Your individual development window before starting school and the time factor make the natural sciences area one of the most important in kindergarten education.

Great research and material development work has been done by Maria Montessori in this area. Without our knowledge from modern brain research, it has fulfilled the most important criteria for working material:

- "Working material" because the child gains knowledge with the senses, not just looking at it with the sense of sight.
- Alternation between lengths, points and comprehensible spindles on the one hand as spatial orientation and quantity or muscle sense and comprehension of digits on the other.

Starting with the *numerical rods* - blue-red bars with sides 2.5 cm by 2.5 cm - in lengths of 10 dm, 9 dm to 1 dm, the Montessori mathematics material enables the child to proprioceptively perceive the arm span 1 meter to the finger span of a decimeter. *The sandpaper digits* let the child perceive the shapes of the digits with the fingertips using the sense of touch. In the third step, assigning digits to the numeric bars, the child connects digits with corresponding lengths.

The *Headstocks* lead back to proprioception: the children grasp the various numbers of wooden spindles in a pincer grip. For a small child's hand, 10 spindles can hardly be grasped. Impressions such as grasping the amount of 10 spindles with the hand, holding a decimeter between thumb and forefinger in the tweezer grip or spreading the arms a meter, leave significant impressions on the child. By promoting hand motor skills with all sensory perceptions, we not only promote language but also mathematical ideas in children.

*Digits and chips* offer a different way of assigning digits and quantities because, in contrast to fixed numbers, they are offered loose on a stick. The child counts the individual chips to the corresponding digits in ascending order as opposed to descending on the numeric bars. Due to the number of pieces, the child can pave a way between the chips with even numbers, while with odd numbers they have to take a detour (odd route). In this way, the insight into the divisibility of numbers by 2 is promoted. Transferred to movement games, I offer box hopping with both legs on the odd numbers and straddled legs on the even numbers.

This exercise corresponds to a basic human need for symmetry:

- It is a body with a spine, a stomach, etc., but with two sides: two legs, arms, hands, ears, eyes
- He has a brain - divided in half.

The child in motion always strives to coordinate and integrate both sides in order to then work out the opposites again, to find their own polarity and to find balance between them. In principle, even the toddler acts mathematically, physically, rhythmically.

I supplement the red numbers from digits and chips with blue ones for the even numbers, which will be exchanged in a later step. So I repeat the system of numerical bars, which always start with red on the left and represent the second, fourth, sixth, eighth and tenth decimeters in blue.

Promotion of the concept of numbers with rule games

In most rule games, promoting the concepts of quantities, numbers and digits is neglected. Preference is given to dice games that also include content from nature and environmental education (e.g. fruit basket, snowman). The boxes and game instructions describe exactly what the children (can or should) learn, and this is often adopted without criticism. Can I pursue many different goals at the same time and with the same importance?

A number concept can be sensibly built up with rule games. Starting with the well-known four first games (by Ravensburger), which are introduced with the color dice, because color perception is visually faster than form, quantity and number perception, this game can already look at a number dice - preferably one to three eyes - be converted. The color dice games can be expanded to include two-color combinations in the game "Dallifant" and three-color combinations such as "dice dwarves".

At the same time becomes the usual *Number domino *reduced to the numbers one to three, which have the same point arrangement as the dice. As an increase, this becomes three-sided *Trio dominoes *introduced with its triangular game pieces with the numbers zero to five. In addition to the changed position on the pieces, the points are shown differently in relation to each other and in relation to the shape of the piece. Little by little, the concept of numbers can be expanded to six on the basis of these games.

The children have fun scoring neutral cubes themselves - correctly and confused. Dice and dominoes can be easily assigned. The spinning cube with the usual points from 1 to 6, but on a rectangular basic shape instead of the usual square one, offers a modification option. The movements carried out, which should vary as widely as possible, are beneficial for learning success.

At "*Klipp-Klapp*"- a wooden box with the numbers 1 to 9 that can be folded down - the children assign the points on two dice to the corresponding digits. Example: The child rolls 2 and 5 points, it can turn down the two and the five or the seven. As an increase the children can add up the points and divide them as they wish, so with 2 and 5 they can also put down 3 and 4 or 6 and 1.

Card games can be used in parallel to the dice games. Unfortunately, we usually only think of the rather boring quartet or Schwarzer Peter. I personally like the game "*Forage*"Very good from Amigo. The child learns in a playful way to quickly recognize numbers up to six or more by means of cards: If there are exactly six points there is a cup, if the total is higher, the cards are worthless and must be discarded. The second mathematical ability of sorting carrots, nuts and apples is promoted rather incidentally. The natural aspect "What does a hamster eat" does not actually come into play. Hamstering is a game of chance, so that failure to recognize the amount is not "punished".

"*Elfer out*"Kindergarten children can already play. We have limited the numbers to 15. Another card game with 30 cards each (3 x the number 1, 3 x the number 2, etc. up to the number 10) in the colors yellow, blue and red we use it as a simple counting up and down game, starting with 1 or 10. Before "Elfer out" I recommend "*speed*" and "*Mau mouse*". With the latter, either the same number or the same color can be placed on the pile. With" Speed "there are three options: color, amount or symbol must be repeated on the card to be placed. If all three criteria come together, the child wins again your turn.

"*Who has 4?*", a Piatnik bring-along game, also enables you to record quantities of up to four regardless of their grouping. In addition to filing according to size from the smallest to the largest number or vice versa, it can be as *Memory* can be used, in which the first step is not about the quantities, but about the same symbols. As a second step, it can be agreed that there must always be five things on two cards. And last but not least, the game can be used as a language support question game in which the children have to ask another player for a certain card: "Do you have two apples?" If this player has this card, he must surrender it, and the questioner remains in turn and may ask further questions. If the person asked does not have this card, it is the next player's turn. The series of cards 1 to 4 may be discarded.

In addition to the card games, we are expanding the number range up to 9 with games such as

The simple cube

Bought model for the spatial position detection of points. Its square area is divided into 9 invisible squares, three squares each in 3 rows. In addition to the detection of quantities in dice games, this scheme also forms the basis for dominoes and my self-designed wooden stacking game. It maps the square number of 3 and contains the square number of 2. Thus it leads to the square numbers and to the cube numbers - the real and the complex numbers. In parallel, the child practices these numbers with pouring exercises.

The square and cubic chains and the deanom of Maria Montessori fit into this scheme, but also shape-laying games and the game "*Digit*"(from the Piatnik company).

The game Logeo

This game (by Nicola and Christoph Haas, Jakobs GmbH) continues this classification. As a moto pedagogue, I want both the cube exercises and the idea of Legeo to be holistic, i. H. with the whole body. To do this, we glued the grid shown above with Tesa tape and a dot grid with Decefix to the floor in our gym. So it is possible for the children to move and experience themselves as cube points or cones. They find their assigned location and can later transfer their experience to the small or fine motor area.

Our children already know the shapes of the game Logeo - three triangles, circles and squares in the three basic colors yellow, blue and red - from our geometric roller boards and the matching carpets in the basic colors (Perras-Emmer, Barbara: Geometry in Elementary School A challenge for the (exercise) kindergarten? Www.kindergartenpaedagogik.de/597.html).

So we can move the game from the table to the gym and convert it from an exercise game to a table game. In addition to recognizing spatial patterns and quantities, we establish cross-connections to graphomotor skills and geometry.

The first and easiest game variant starts with A1 to A5. The game pieces are shown in the grid of the template and must be placed in the correct place on the playing field from the top left according to the direction of writing. In series B, the place for the respective game piece is highlighted in gray, the piece is shown to the left of the grid. In the case of series C, the grid is no longer complete. Missing lines have to be added in the head and missing information replaced by combinations. The requirements increase with each series up to a special mix of all series. The game is also interesting for adults!

Expansion of the field of nine to 16 fields for the natural-colored wooden puzzle

On a square wooden plate (approx. 15 by 15 cm) there are four rows each with four small, also square wooden plates. All 16 parts are the same size. I know three different fruit motifs - we have apples in our kindergarten. This cannot be distinguished from its surroundings in terms of color, only the (three-dimensional) height reveals its shape.

"Rush hour" - spatial awareness on 6 x 6 fields

On a square plastic field, each with six small square hollows in six rows (about the same size as the wooden puzzle mentioned above), people play with colorful plastic cars that are one field wide and two to three fields long. The wheels of these small vehicles are fixed, but they can still be moved back and forth.

Forty template cards in four levels of difficulty show different starting positions with different numbers of vehicles, which are then simulated with the cars on the playing field. After this already very demanding task, the vehicles should be moved in such a way that the ice car "*Ice cream*"Gets free travel to the only exit from the field.

For advanced skiers, many steps have to be reversed after the ice cream truck has passed through, so the children experience special "mobility" within the play area. A very successful way to practice math skills within the number field! If you can't find the solution - it's on the back.

A similar game: Ask Mind

In a metal box (32 cm by 18 cm), which looks like a large crayon container, there are "wooden boats" with drill holes for 1, 2, 3, 4, 5 and 6 rolls in the colors blue, yellow and red - and green for the next level of difficulty. The starting position of different boat sizes and their different colored "occupants" is shown on template cards. As with rush hour, this picture now has to be recreated. The boats are held in place with magnets in the three or four "rows". All shuttles in a lane are placed next to each other without any gap. By swapping the boats within a row and / or swapping the rollers within a boat, the aim is to ensure that the three or four occupants, who are perpendicular to one another, have the same color.

For the simplified version of the game there are template cards on which the rolls in the boats are already correctly sorted so that only the boats have to be put in the correct order (further information at www.magnetspiele.com)

The transition to the tens

We borrow this from Maria Montessori, among others. Game of nine, sets of cards, bead sticks, etc. enable the child not only to count linearly but also to gain an insight into the decimal system. The child always has the opportunity to return to simpler games and to repeat them or to use the materials offered in their simplest way. The hundreds board gives an overview in the number range up to 100, and our knob board lets us follow this path with a rope.

The square chains already mentioned, which lead to the multiplication table, should also be mentioned. The feeling for the potencies can be easily introduced into movement while counting:

- 1: clap your hands, support linear counting
- 2: alternate stamping with the left and right leg
- 3: stamp left and right, pat on the left thigh
- 4: stamp left and right, pat on the left, then the right thigh.

I make sure that every odd number is "closed" with an even. I. E. the corresponding movement must *left and right *possible (snapping fingers, patting the shoulder, boxing sideways or upwards, etc.). Starting on the left means to familiarize the child with the writing and reading direction of our culture. Later movements that do not refer to the two-way system can be added (clapping, nodding, hopping on both legs). These games are a lot of fun for children, but we also sometimes notice that we are counting when climbing stairs or that we lose our balance if the size of the sidewalk tiles is not the way we walk.

Even if it is sometimes difficult for us: We need a culture of trust and not of mistrust ... - The child usually knows best where he is and what he wants to learn with. We just have to believe ourselves that a child wants to learn and prepare their surroundings accordingly.

"To teach details means to create confusion. To establish the relationship between things means to impart knowledge." (Montessori 2002, p. 126).

literature

Devlin, Keith: Patterns of Mathematics - Laws of Order of Mind and Nature. Heidelberg 2002

Mechsner, Franz: Thumbs up! GEO knowledge: thinking, learning, school 1/1999

Montessori, Maria: Cosmic Education. Freiburg, 5th edition 2002

Murphy-Witt, Monika: Playfully in balance. Freiburg im Breisgau, 4th edition, 2000

Perras, Barbara: The number range from 5 to 9 using a wooden ball game called AYA www.kindergartenpaedagogik.de/746.html

Perras-Emmer, Barbara: Geometry in primary school. A challenge for the (exercise) kindergarten? www.kindergartenpaedagogik.de/597.html

Ratey, John J .: The human brain. Munich 2003

Stewart, Ian: The Numbers of Nature - Mathematics as a Window to the World. Heidelberg 2001

Author

Barbara Perras, moto pedagogue, head of the Evang. Loderhof kindergarten in Sulzbach-Rosenberg

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