Research paper "Does mental arithmetic develop the mental abilities of a child"? From the history of the origin of the concept of a natural number The main theorem of arithmetic.

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Editorial preface: Of more than 500 thousand clay tablets found by archaeologists during excavations in Ancient Mesopotamia, about 400 contain mathematical information. Most of them are deciphered and allow you to get a fairly clear idea of the amazing algebraic and geometric achievements of the Babylonian scientists.

Opinions differ about the time and place of birth of mathematics. Numerous researchers of this issue ascribe its creation to various peoples and date it to different eras. The ancient Greeks did not yet have a single point of view on this matter, among whom the version was especially widespread that geometry was invented by the Egyptians, and arithmetic was invented by Phoenician merchants, who needed such knowledge for trading calculations.

Herodotus in History and Strabo in Geography gave priority to the Phoenicians. Plato and Diogenes Laertius considered Egypt to be the homeland of both arithmetic and geometry. The same is the opinion of Aristotle, who believed that mathematics originated from the availability of leisure among the priests there. This remark follows the passage that in every civilization, practical crafts are first born, then arts that serve pleasure, and only then sciences aimed at cognition.

Evdem, a student of Aristotle, like most of his predecessors, also considered Egypt to be the birthplace of geometry, and the reason for its appearance was the practical needs of land surveying. In its improvement, geometry passes, according to Evdem, three stages: the emergence of practical skills in land surveying, the emergence of a practically oriented applied discipline and its transformation into a theoretical science. Apparently, the first two stages Evdem attributed to Egypt, and the third - to Greek mathematics. True, he nevertheless admitted that the theory of calculating areas arose from the solution of quadratic equations that were of Babylonian origin.

The historian Josephus Flavius ("Ancient Judea", v. 1, ch. 8) has his own opinion. Although he calls the Egyptians the first, he is sure that they were taught arithmetic and astronomy by the forefather of the Jews Abraham, who hid in Egypt during the famine that befell the land of Canaan. Well, the Egyptian influence in Greece was strong enough to impose on the Greeks a similar opinion, which, with their light hand, has been circulating in historical literature to this day. Well-preserved clay tablets covered with cuneiform texts found in Mesopotamia dating from 2000 BC. and before 300 AD, testify to both a slightly different state of affairs and what mathematics was like in ancient Babylon. It was a rather complex fusion of arithmetic, algebra, geometry, and even the beginnings of trigonometry.

Mathematics was taught in scribal schools, and each graduate had a fairly serious amount of knowledge for that time. Apparently, this is what Ashurbanipal, the king of Assyria in the 7th century, speaks about. BC, in one of his inscriptions, reporting that he had learned to find

"Complex reciprocals and multiply."

To resort to calculations, life forced the Babylonians at every step. Arithmetic and simple algebra were needed in housekeeping, when exchanging money and making payments for goods, calculating simple and compound interest, taxes and the share of the harvest handed over to the state, temple or landowner. Mathematical calculations, moreover, rather complex ones, required large-scale architectural projects, engineering work during the construction of an irrigation system, ballistics, astronomy, astrology. An important task of mathematics was to determine the timing of agricultural work, religious holidays, and other calendar needs. How high in the ancient city-states between the Tigris and Euphrates rivers were the achievements in what the Greeks would later call so surprisingly precisely μαθημα ("knowledge"), it is possible to judge the decoding of Mesopotamian clay cuneiforms. By the way, among the Greeks, the term μαθημα initially denoted a list of four sciences: arithmetic, geometry, astronomy and harmonics, he began to denote mathematics itself much later.

In Mesopotamia, archaeologists have already found and continue to find cuneiform tablets with records of a mathematical nature, partly in Akkadian, partly in Sumerian, as well as reference mathematical tables. The latter greatly facilitated the calculations that had to be done on a daily basis, therefore, a number of decrypted texts quite often contains the calculation of percent. The names of the arithmetic operations of the earlier, Sumerian period of Mesopotamian history have survived. So, the operation of addition was called "accumulation" or "addition", when subtraction was used the verb "pluck", and the term for multiplication meant "eat".

It is interesting that in Babylon they used a more extensive multiplication table - from 1 to 180,000 than the one that we had to learn at school, i.e. designed for numbers from 1 to 100.

In Ancient Mesopotamia, uniform rules for arithmetic operations were created not only with integers, but also with fractions, in the art of operating with which the Babylonians were significantly superior to the Egyptians. In Egypt, for example, operations with fractions for a long time continued to remain at a primitive level, since they knew only aliquots (i.e., fractions with a numerator equal to 1). Since the time of the Sumerians in Mesopotamia, the number 60 was the main counting unit in all economic affairs, although the decimal number system was also known, which was in use among the Akkadians. Babylonian mathematicians widely used the sixagesimal positional (!) Counting system. On its basis, various calculating tables were compiled. In addition to multiplication tables and tables of reciprocals, with the help of which division was performed, there were tables of square roots and cubic numbers.

Cuneiform texts devoted to solving algebraic and geometric problems indicate that Babylonian mathematicians were able to solve some special problems, including up to ten equations with ten unknowns, as well as certain varieties of cubic equations and equations of the fourth degree. At first, quadratic equations served mainly purely practical purposes - measuring areas and volumes, which was reflected in the terminology. For example, when solving equations with two unknowns, one was called "length" and the other was called "width". The work of unknown persons was called "square". As it is now! In problems leading to a cubic equation, there was a third unknown quantity - "depth", and the product of three unknowns was called "volume". Later, with the development of algebraic thinking, the unknown began to be understood in a more abstract way.

Geometric drawings were sometimes used to illustrate algebraic relationships in Babylon. Later, in Ancient Greece, they became the main element of algebra, while for the Babylonians, who thought primarily algebraically, drawings were only a means of visualization, and the terms "line" and "area" were most often understood as dimensionless numbers. That is why there were solutions to problems where the "area" was added to the "side" or subtracted from the "volume", etc.

Of particular importance in ancient times was the accurate measurement of fields, gardens, buildings - the annual floods of rivers brought a large amount of silt, which covered the fields and destroyed the boundaries between them, and after the recession of water, land surveyors, by order of their owners, often had to re-measure the allotments. In the cuneiform archives, there are many such land surveying maps compiled over 4 thousand years ago.

Initially, the units of measurement were not very accurate, because the length was measured with fingers, palms, elbows, which are different for different people. The situation was better with large values, for the measurement of which they used a reed and a rope of a certain size. But here, too, the measurement results often differed among themselves, depending on who was measuring and where. Therefore, different measures of length were adopted in different cities of Babylonia. For example, in the city of Lagash the "elbow" was equal to 400 mm, and in Nippur and Babylon itself - 518 mm.

Many of the surviving cuneiform materials were teaching aids for Babylonian schoolchildren, in which solutions to various simple problems that were often encountered in practical life were given. It is not clear, however, whether the student solved them in his head or did preliminary calculations with a twig on the ground - only the conditions of mathematical problems and their solution are written on the tablets.

The main part of the mathematics course at school was occupied by the solution of arithmetic, algebraic and geometric problems, in the formulation of which it was customary to operate with specific objects, areas and volumes. On one of the cuneiform tablets, the following problem has been preserved: "How many days can it take to make a piece of fabric of a certain length, if we know that so many cubits (a measure of length) of this fabric are made every day?" The other shows tasks related to construction work. For example, "How much soil is required for a fill with known dimensions, and how much soil should each worker drag if the total number is known?" or "How much clay should each worker prepare to build a wall of a certain size?"

The student also had to be able to calculate coefficients, calculate totals, solve problems of measuring angles, calculating the areas and volumes of rectilinear figures - this was a common set for elementary geometry.

The names of geometric figures that have survived from Sumerian times are interesting. The triangle was called "wedge", the trapezium was called the "bull's forehead", the circle was "hoop", the capacity was designated by the term "water", the volume was "earth, sand", the area was called "field".

One of the cuneiform texts contains 16 problems with solutions that relate to dams, shafts, wells, water clocks, and earthworks. One problem is supplied with a drawing related to a circular shaft, another considers a truncated cone, determining its volume by multiplying the height by half the sum of the areas of the upper and lower bases. Babylonian mathematicians also solved planimetric problems using the properties of right-angled triangles, formulated by Pythagoras later in the form of a theorem on the equality in a right-angled triangle of the square of the hypotenuse to the sum of the squares of the legs. In other words, the famous Pythagorean theorem was known to the Babylonians at least a thousand years before Pythagoras.

In addition to planimetric problems, they also solved stereometric ones, associated with determining the volume of various kinds of spaces, bodies, drawing plans of fields, areas, individual buildings, but usually not to scale, was widely practiced.

The most significant achievement of mathematics was the discovery of the fact that the ratio of the diagonal to the side of a square cannot be expressed as a whole number or a simple fraction. Thus, the concept of irrationality was introduced into mathematics.

It is believed that the discovery of one of the most important irrational numbers - the number π, which expresses the ratio of the circumference of a circle to its diameter and equal to an infinite fraction = 3.14 ..., belongs to Pythagoras. According to another version, for the number π, the value 3.14 was first proposed by Archimedes 300 years later, in the 3rd c. BC. Another one, the first to calculate it was Omar Khayyam, this is generally 11-12 centuries. It is only known for certain that this ratio was first designated by the Greek letter π in 1706 by the English mathematician William Jones, and only after the Swiss mathematician Leonard Euler borrowed this designation in 1737, it became generally accepted.

The number π is an ancient mathematical riddle; this discovery should also be sought in Ancient Mesopotamia. Babylonian mathematicians were well aware of the most important irrational numbers, and the solution to the problem of calculating the area of a circle can also be found in the decoding of cuneiform clay tablets of mathematical content. According to these data, π was taken equal to 3, which, however, was quite enough for practical land surveying purposes. Researchers believe that the sexagesimal system was chosen in Ancient Babylon for metrological reasons: the number 60 has many divisors. The sexagesimal notation of integers did not become widespread outside Mesopotamia, but in Europe until the 17th century. both the sixagesimal fractions and the usual division of a circle by 360 degrees were widely used. The hour and minutes, divisible into 60 parts, also originate in Babylon. Remarkable is the ingenious idea of the Babylonians to use the minimum number of digital characters to write numbers. The Romans, for example, did not even think that the same number can denote different values! To do this, they used the letters of their alphabet. As a result, a four-digit number, for example, 2737, contained as many as eleven letters: MMDCCXXXVII. And although in our time there are extreme mathematicians who will be able to divide LXXVIII into a column by CLXVI or multiply CLIX by LXXIV, it remains only to pity those inhabitants of the Eternal City who had to perform complex calendar and astronomical calculations using such mathematical balancing act or calculated large-scale architectural projects and various engineering objects.

The Greek number system was also based on the use of letters of the alphabet. At the beginning, Greece adopted the Attic system, which used a vertical bar to denote the unit, and for the numbers 5, 10, 100, 1000, 10000 (in fact, it was a decimal system) - the initial letters of their Greek names. Later, in about the 3rd century. BC, the Ionic number system was widespread, in which 24 letters of the Greek alphabet and three archaic letters were used to designate numbers. And to distinguish numbers from words, the Greeks put a horizontal line above the corresponding letter.

In this sense, the Babylonian mathematical science stood above the later Greek or Roman, since it is she who owns one of the most outstanding achievements in the development of number notation systems - the principle of positionality, according to which the same number sign (symbol) has different meanings depending on whether places where it is located.

By the way, the Egyptian number system was inferior to the Babylonian and contemporary to it. The Egyptians used a non-positional decimal system, in which the numbers from 1 to 9 were indicated by the corresponding number of vertical bars, and individual hieroglyphic characters were introduced for successive powers of the number 10. For small numbers, the Babylonian number system was basically similar to the Egyptian one. One vertical wedge-shaped line (in the early Sumerian tablets - a small semicircle) meant one; repeated the required number of times this sign served to record numbers less than ten; to designate the number 10, the Babylonians, like the Egyptians, introduced a new symbol - a wide wedge-shaped sign with a point directed to the left, resembling an angle bracket in shape (in early Sumerian texts - a small circle). Repeated the appropriate number of times, this sign served to denote the numbers 20, 30, 40 and 50.

Most modern historians believe that ancient scientific knowledge was purely empirical. With regard to physics, chemistry, natural philosophy, which were based on observations, it seems to be true. But the idea of sensory experience as a source of knowledge is faced with an insoluble question when it comes to such an abstract science as mathematics operating with symbols.

The achievements of Babylonian mathematical astronomy were especially significant. But whether the sudden leap lifted the Mesopotamian mathematicians from the level of utilitarian practice to a vast knowledge, allowing the use of mathematical methods to predict the positions of the Sun, Moon and planets, eclipses and other celestial phenomena, or the development proceeded gradually, we, unfortunately, do not know.

The history of mathematical knowledge in general looks strange. We know how our ancestors learned to count on fingers and toes, made primitive numerical records in the form of notches on a stick, knots on a rope, or lined with pebbles. And then - without any transitional link - suddenly information about the mathematical achievements of the Babylonians, Egyptians, Chinese, Hindus and other ancient scientists, so solid that their mathematical methods stood the test of time until the middle of the recently ended II millennium, i.e., for more than three thousand years ...

What is hidden between these links? Why did the ancient sages, in addition to their practical meaning, revered mathematics as sacred knowledge, and gave the names of gods to numbers and geometric shapes? Is it only behind this that there is a reverent attitude towards Knowledge as such?

Perhaps the time will come when archaeologists will find answers to these questions. In the meantime, we wait, let's not forget what the Oxford resident Thomas Bradwardin said 700 years ago:

"He who has the shamelessness to deny mathematics should have known from the very beginning that he will never enter the gate of wisdom."

Acquaintance with mathematics begins with arithmetic. With arithmetic, we enter, as MV Lomonosov said, into the "gates of learning."

The word "arithmetic" comes from the Greek arithmos, which means "number." This science studies actions on numbers, various rules for dealing with them, teaches how to solve problems that reduce to addition, subtraction, multiplication and division of numbers. Arithmetic is often thought of as a first step in mathematics, on the basis of which one can study its more complex sections - algebra, mathematical analysis, etc.Arithmetic originated in the countries of the Ancient East: Babylon, China, India, Egypt. For example, the Egyptian Rinda papyrus (named after its owner G. Rinda) dates back to the 20th century. BC NS.

The treasures of mathematical knowledge accumulated in the countries of the Ancient East were developed and continued by the scientists of Ancient Greece. History has preserved many names of scientists who were engaged in arithmetic in the ancient world - Anaxagoras and Zeno, Euclid, Archimedes, Eratosthenes and Diophantus. The name of Pythagoras (VI century BC) sparkles here as a bright star. The Pythagoreans worshiped numbers, believing that they contain all the harmony of the world. Individual numbers and pairs of numbers were assigned special properties. The numbers 7 and 36 were in high esteem, at the same time attention was paid to the so-called perfect numbers, friendly numbers, etc.

In the Middle Ages, the development of arithmetic is also associated with the East: India, the countries of the Arab world and Central Asia. From the Indians came to us the numbers that we use, zero and positional number system; from al-Kashi (XV century), Ulugbek - decimal fractions.

Thanks to the development of trade and the influence of oriental culture since the XIII century. interest in arithmetic is growing in Europe as well. It is worth remembering the name of the Italian scientist Leonardo of Pisa (Fibonacci), whose work "The Book of the Abacus" introduced Europeans to the main achievements of mathematics in the East and was the beginning of many studies in arithmetic and algebra.

Along with the invention of printing (mid-15th century), the first printed mathematical books appeared. The first printed book on arithmetic was published in Italy in 1478. In the "Complete arithmetic" by the German mathematician M. Stiefel (early 16th century) there are already negative numbers and even the idea of taking logarithms.

From about the XVI century. the development of purely arithmetic questions merged into the mainstream of algebra, as a significant milestone can be noted the appearance of the works of the scientist from France F. Vieta, in which numbers are indicated by letters. From that time on, the basic arithmetic rules were finally realized from the standpoint of algebra.

The main object of arithmetic is number. Natural numbers, i.e. the numbers 1, 2, 3, 4, ... etc., arose from the account of specific objects. Many millennia passed before man learned that two pheasants, two hands, two people, etc. can be called the same word "two". An important task of arithmetic is to learn to overcome the specific meaning of the names of the objects being counted, to be distracted from their shape, size, color, etc. In arithmetic, numbers are added, subtracted, multiplied, and divided. The art of quickly and accurately performing these operations on any numbers has long been considered the most important task of arithmetic.Arithmetic operations on numbers have a variety of properties. These properties can be described in words, for example: "The sum does not change from the change of places of the terms", can be written in letters: a + b = b + a, can be expressed in special terms.

Among the important concepts that arithmetic introduced, proportions and percentages should be noted. Most of the concepts and methods of arithmetic are based on comparing various relationships between numbers. In the history of mathematics, the process of merging arithmetic and geometry took place over the course of many centuries.

The word "arithmetic" can be understood as:

an academic subject that deals mainly with rational numbers (whole numbers and fractions), actions on them and the tasks solved with the help of these actions;

part of the historical building of mathematics, which has accumulated various information about calculations;

"Theoretical arithmetic" - a part of modern mathematics that deals with the construction of various number systems (natural, whole, rational, real, complex numbers and their generalizations);

"Formal arithmetic" - a part of mathematical logic that analyzes the axiomatic theory of arithmetic;

"Higher arithmetic", or number theory, an independently developing part of mathematicians and

/ Encyclopedic Dictionary of a Young Mathematician, 1989 /

Municipal autonomous educational institution

secondary school № 211 named after L.I. Sidorenko

Novosibirsk

Research:

Does mental arithmetic develop a child's mental abilities?

Section "Mathematics"

The project was completed by:

Klimova Ruslana

student of 3 "B" class

MAOU SOSH No. 211

named after L.I. Sidorenko

Project Manager:

Vasilyeva Elena Mikhailovna

Novosibirsk 2017

Introduction 3

2. Theoretical part

2.1 History of arithmetic 3

2.2 First devices for counting 4

2.3 Abacus 4

2.4 What is mental arithmetic? 5

3. Practical part

3.1 Classes at the school of mental arithmetic 6

3.2 Lesson 6 Conclusions

4. Conclusions on the project 7.8

5. List of used literature 9

1. INTRODUCTION

Last summer, my grandmother and my mother watched the program “Let them talk”, where a 9-year-old boy, Daniyar Kurmanbaev from Astana, was counting in his mind (mentally) faster than a calculator, while doing manipulations with the fingers of both hands. And in the program they talked about an interesting method of developing mental abilities - mental arithmetic.

It struck me, and my mother and I became interested in this technique.

It turned out that in our city there are 4 schools where they teach mentally counting tasks and examples of any complexity. These are "Abakus", "AmaKids", "Pythagorka", "Menar". Classes in schools are not cheap. My parents and I chose a school so that it was close to home, the classes were not very expensive, so that there was real feedback about the teaching program, as well as certified teachers. In all respects, the Menard school was suitable.

I asked my mother to enroll me in this school, as I really wanted to learn how to count quickly, improve my school performance and discover something new.

The method of mental arithmetic is more than five hundred years old. This technique is an oral counting system. Mental arithmetic is taught in many countries of the world - in Japan, the USA and Germany, Kazakhstan. In Russia, they are just beginning to master it.

Objective of the project: to find out:

mental arithmetic does a child's mental abilities develop?

Project object: student 3 "B" class MAOU secondary school № 211 Klimova Ruslana.

Subject of study: mental arithmetic is a system of oral counting.

Research objectives:

Learn how learning in mental arithmetic is done;

Understand whether mental arithmetic develops the thinking abilities of a child?

Find out if you can learn mental arithmetic on your own at home?

2.1 HISTORY OF ARITHMETICS

In every case, you need to know the history of its development.

Arithmetic originated in the countries of the Ancient East: Babylon, China, India, Egypt.

Arithmetic studies numbers and actions on numbers, various rules for dealing with them, teaches them to solve problems that reduce to addition, subtraction, multiplication and division of numbers.

The name "arithmetic" comes from the Greek word (arithmos) - number.

The emergence of arithmetic is associated with the labor activity of people and with the development of society.

The importance of mathematics in human everyday life is great. Without counting, without the ability to correctly add, subtract, multiply and divide numbers, the development of human society is unthinkable. We study four arithmetic operations, the rules of oral and written calculations, starting from elementary grades. All of these rules were not invented or discovered by any one person. Arithmetic originated from the everyday life of people.

Ancient people obtained their food mainly by hunting. The whole tribe had to hunt for a large animal - a bison or an elk: alone, you cannot cope with it. To prevent the prey from leaving, it had to be surrounded, well, at least this way: five people on the right, seven behind, four on the left. You can't do without counting! And the leader of the primitive tribe coped with this task. Even in those days when a person did not know words like "five" or "seven", he could show numbers on his fingers.

The main object of arithmetic is number.

2.2 FIRST ACCOUNTING DEVICES

People have long tried to make it easier for themselves with the help of various means and devices. The first, most ancient "calculating machine" was the fingers and toes. This simple device was quite enough - for example, to count the mammoths killed by the entire tribe.

Then there was trade. And the ancient traders (Babylonian and other cities) made calculations using grains, pebbles and shells, which they began to spread on a special board called an abacus.

The analogue of the abacus in ancient China was the "su-anpan" calculating device, in ancient China - the Japanese abacus called "soroban".

Russian abacus first appeared in Russia in the 16th century. They were a board with parallel lines drawn on it. Later, instead of a board, they began to use a frame with wires and bones.

2.3 ABACUS

Word "abacus" (abacus) means the counting board.

Let's take a look at the modern abacus ...

To learn how to use accounts, you need to know what they are.

Accounts consist of:

dividing strip;
upper bones;
lower bones.

In the middle is the center point. The upper bones represent fives and the lower ones represent ones. Each vertical stripe of pits, starting from right to left, denotes one of the number digits:

tens of thousands, etc.

For example, to postpone the example: 9 - 4 = 5, on the first line on the right, move the upper bone (it denotes a five) and pick up 4 lower bones. Then lower the 4 lower bones. So we get the required number 5.

The mental abilities of children develop through the ability to count in the mind. To train both hemispheres, you need to constantly engage in solving arithmetic problems. After a short time, the child will already be able to solve complex problems without using a calculator.

2.4 WHAT IS MENTAL ARITHMETICS?

Mental arithmetic Is a method of developing the mental abilities of children from 4 to 14 years old. The basis of mental arithmetic is abacus counting. The child counts on the abacus with both hands, doing the calculations twice as fast. In abacus, children not only add and subtract, but also learn to multiply and divide.

Mentality - it is the thinking ability of a person.

During math lessons, only the left hemisphere of the brain develops, which is responsible for logical thinking, and the right hemisphere develops such subjects as literature, music, drawing. There are special teaching techniques that are aimed at developing both hemispheres. Scientists say that success is achieved by those people who have fully developed both hemispheres of the brain. Many people have a more developed left hemisphere and a less developed right.

There is an assumption that mental arithmetic allows you to use both hemispheres, performing calculations of varying complexity.
Using the abacus makes the left hemisphere work - it develops fine motor skills and allows the child to clearly see the counting process.
Skills are trained gradually with the transition from simple to complex. As a result, by the end of the program, the child can mentally add, subtract, multiply and divide three- or four-digit numbers.

Therefore, I decided to go to classes at the school of mental arithmetic. Since I really wanted to learn how to quickly learn poetry, develop my logic, develop purposefulness, and also develop some qualities of my personality.

3.1 LESSONS AT THE SCHOOL OF MENTAL ARITHMETICS

My mental arithmetic lessons were held in classrooms equipped with computers, a television, a whiteboard, and a large teacher's abacus. On the wall near the classrooms there are teacher's diplomas and teacher's certificates, as well as patents for the use of the international method of mental arithmetic.

In a trial lesson, the teacher showed us abacus abacus to me and my mother, briefly told how to use them and the principle of counting itself.

The training is structured like this: once a week I worked for 2 hours in a group of 6 people. In the lessons we used abacus (abacus). Moving the bones on the abacus with our fingers (fine motor skills), we learned to perform arithmetic operations physically.

The lesson was attended by a mental warm-up. And there were always breaks during which we could have a little snack, drink water or play games. At home, we were always given sheets with examples for independent work at home.

In 1 month of study, I:

got acquainted with the accounts. I learned how to use my hands correctly when counting: with the thumb of both hands we raise the knuckles on the abacus, with the index fingers we lower the knuckles.

In the 2nd month of study, I:

learned to count two-step examples with tens. There are tens on the second spoke from the far right. When counting with tens, we are already using the thumb and forefinger of the left hand. Here is the same technique as with the right hand: raise the big one, lower the index one.

In the 3rd month of study, I:

I solved examples of subtraction and addition with ones and tens on the abacus - three-stage.

Solved examples of subtraction and addition with thousandths - two-step

In the 4th month of study:

I got acquainted with the mental map. Looking at the map, I had to mentally move the knuckles and see the answer.

Also in the classroom on mental arithmetic I trained to work on a computer. There is a program installed where the number of numbers for counting is set. The frequency of their display is 2 seconds, I watch, remember and count. While I count on the accounts. Give 3, 4 and 5 numbers. The numbers are still unambiguous.

In mental arithmetic, more than 20 formulas are used for calculations (close relatives, help from a brother, help from a friend, etc.) that need to be remembered.

3.2 CONCLUSIONS ON THE LESSON

I did 2 hours a week and 5-10 minutes a day on my own for 4 months.

First month of training	Fourth month
1. I count 1 sheet on the abacus (30 examples)

2. I count mentally 1 sheet (10 examples)

3.Learning a poem (3 quatrains)
	20-30 minutes
4. Doing homework (math: one problem, 10 examples)
40-50 minutes

4. CONCLUSIONS ON THE PROJECT

1) I was interested in logic puzzles, puzzles, crosswords, games for finding the differences. I became more assiduous, attentive and collected. My memory has improved.

2) The purpose of mental mathematics is to develop the child's brain. By doing mental arithmetic, we develop our skills:

We develop logic and imagination by performing mathematical operations, first on a real abacus, and then imagining the abacus in our mind. And also solving logical problems in the classroom.

We improve concentration of attention by performing arithmetic counting of a huge number of numbers on imaginary abacus.

Memory improves. After all, all pictures with numbers, after performing mathematical operations, are stored in memory.

Thinking speed. All "mental" mathematical operations are performed at a speed that is comfortable for children, which is gradually increased and the brain "accelerates".

3) In the classroom at the center, teachers create a special play atmosphere and sometimes children, even against their will, are included in this exciting environment.

Unfortunately, such an interest in classes cannot be realized when teaching on your own.

There are many video courses on the Internet and on the YouTube channel with which you can understand how to count on an abacus.

You can learn this technique on your own, but it will be very difficult! First, it is necessary for mom or dad to understand the essence of mental arithmetic - to learn how to add, subtract, multiply and divide by themselves. Books and videos can help them in this. The video tutorial demonstrates at a slow pace how to work with abacus. Of course, videos are preferable to books, since everything is clearly shown on it. And then they explained it to the child. But the adults are very busy, so this is not an option.

It's hard without a teacher-instructor! After all, the teacher in the class monitors the correctness of the work of both hands, corrects, if necessary. It is also extremely important - the correct setting of the counting technique, as well as the timely correction of the wrong skills.

The 10-level program is designed for 2-3 years, it all depends on the child. All children are different, some are given quickly, while others need a little more time to master the program.

Our school now also has classes in mental arithmetic - this is the "Formula Aykyu" center at the MAOU Secondary School No. 211 named after L.I. Sidorenko. The method of mental arithmetic in this center was developed by Novosibirsk teachers and programmers, with the support of the Department of Education of the Novosibirsk region! And I started attending classes at school, as it is generally convenient for me.

For me, this technique is an interesting way to improve my memory, increase concentration and develop my personality traits. And I will continue to do mental arithmetic!

And maybe my work will attract other children to classes in mental arithmetic, which will affect their academic performance.

Literature:

Ivan Yakovlevich Depman. History of arithmetic. A guide for teachers. Second edition, revised. M., Education, 1965 - 416 p.

Depman I. The world of numbers M. 1966.

A. Benjamin. Secrets of Mental Mathematics. 2014 .-- 247 p. - ISBN: N / A.

“Mental arithmetic. Addition and subtraction ”Part 1. Textbook for children 4-6 years old.

G.I. Glazer. History of Mathematics, Moscow: Education, 1982 .-- 240 p.

Karpushina N.M. "Liber abaci" by Leonardo Fibonacci. Magazine "Mathematics in school" No. 4, 2008. Popular science department.

M. Kutorgi "On the accounts of the ancient Greeks" ("Russian Bulletin", vol. SP, p. 901 et seq.)

Vygodsky M.L. "Arithmetic and algebra in the ancient world" M. 1967.

ABACUSxle - seminars on mental arithmetic.

UCMAS-ASTANA- articles.

Internet resources.

Numbers arose from the need for counting and measurement and have undergone a long history of development.

There was a time when people did not know how to count. To compare finite sets, a one-to-one correspondence was established between these sets or between one of the sets and a subset of another set, i.e. at this stage, a person perceived the number of objects without recounting them. For example, about the size of a group of two objects, he could say: "The same number of hands a person has", about a set of five objects - "as many as fingers on a hand." With this method, the compared sets had to be simultaneously visible.

As a result of a very long period of development, a person came to the next stage of creating natural numbers - to compare sets, they began to use intermediary sets: small pebbles, shells, fingers. These intermediary sets were already the rudiments of the concept of a natural number, although at this stage the number was not separated from the objects to be counted: it was, for example, about five pebbles, five fingers, and not about the number "five" in general. The names of intermediary sets began to be used to determine the number of sets that were compared with them. So, among some tribes the number of a set, consisting of five elements, was denoted by the word "hand", and the number of a set of 20 objects - by the words "the whole person."

Only after a person learned to operate with mediating sets did he establish the common that exists, for example, between five fingers and five apples, i.e. when there was a distraction from the nature of the elements of the mediating sets, the idea of a natural number arose. At this stage, when counting, for example, apples, "one apple", "two apples", etc. were not listed, but the words "one", "two", etc. were spoken. This was the most important stage in the development of the concept of number. Historians believe that this happened in the Stone Age, in the era of the primitive communal system, approximately in the 10-5th millennium BC.

Over time, people learned not only to name numbers, but also to designate them, as well as to perform actions on them. In general, the natural series of numbers did not appear immediately, the history of its formation is long. The stock of numbers that were used to keep track of them increased gradually. Gradually, the idea of the infinity of the set of natural numbers has developed. So, in the work "Psammit" - the calculus of grains of sand - the ancient Greek mathematician Archimedes (III century BC) showed that a series of numbers can be continued indefinitely, and described the method of formation and verbal designation of arbitrarily large numbers.

The emergence of the concept of a natural number was the most important moment in the development of mathematics. Now it is possible to study these numbers regardless of those. specific tasks in connection with which they arose. Theoretical science, which began to study numbers and actions on them, was called "arithmetic". The word "arithmetic" comes from the Greek arithmos, which means "number". Hence, arithmetic is the science of number.

Arithmetic originated in the countries of the Ancient East: Babylon. China. India and Egypt. The mathematical knowledge accumulated in these countries was developed and continued by the scientists of Ancient Greece. In the Middle Ages, mathematicians from India, the countries of the Arab world and Central Asia made a great contribution to the development of arithmetic, and from the 13th century onwards, European scientists.

The term "natural number" was first used in the 5th century. Roman scientist A. Boethius, who is known as a translator of the works of famous mathematicians of the past into Latin and as the author of the book "On an Introduction to Arithmetic", which until the 16th century was a model for all European mathematics.

In the second half of the 19th century, natural numbers turned out to be the foundation of the entire mathematical science, on the state of which the strength of the entire building of mathematics depended on. In this regard, it became necessary to rigorously substantiate the concept of a natural number, to systematize what is associated with it. Since the mathematics of the 19th century moved on to the axiomatic construction of their theories, the axiomatic theory of the natural number was developed. The set theory, created in the 19th century, also had a great influence on the study of the nature of the natural number. Of course, in the created theories, the concepts of the natural number and actions on them received a greater abstractness, but this is always accompanied by the process of generalization and systematization of individual facts.

§ 14.AXIOMATIC CONSTRUCTION OF THE SYSTEM OF NATURAL NUMBERS

As already mentioned, natural numbers are obtained when counting objects and when measuring quantities. But if, when measuring, numbers appear that are different from natural numbers, then counting leads only to natural numbers. To keep score, you need a sequence of numbers that starts with one and which allows

What is arithmetic? When did humanity start using and working with numbers? Where are the roots of such ordinary concepts as numbers, addition and multiplication, which a person made an inseparable part of his life and worldview? Ancient Greek minds admired such sciences as geometry as the most beautiful symphonies of human logic.

Perhaps arithmetic is not as deep as other sciences, but what would have happened to them if a person had forgotten the elementary multiplication table? The logical thinking we are accustomed to, using numbers, fractions and other tools, was not easy for people and for a long time was inaccessible to our ancestors. In fact, before the development of arithmetic, no area of human knowledge was truly scientific.

Arithmetic is the ABC of Mathematics

Arithmetic is the science of numbers, with which any person begins acquaintance with the fascinating world of mathematics. As MV Lomonosov said, arithmetic is the gateway to learning, which opens the way for us to the knowledge of the world. But he is right, is it possible to separate the knowledge of the world from the knowledge of numbers and letters, mathematics and speech? Perhaps in the old days, but not in the modern world, where the rapid development of science and technology dictates its own laws.

The word "arithmetic" (Greek "arithmos") of Greek origin, means "number". She studies numbers and everything that may be associated with them. This is the world of numbers: various actions on numbers, number rules, solving problems that involve multiplication, subtraction, etc.

The main object of arithmetic

The basis of arithmetic is an integer, the properties and patterns of which are considered in higher arithmetic or In fact, the strength of the entire building - mathematics - depends on how correct the approach is taken in considering such a small block as a natural number.

Therefore, the question of what arithmetic is can be answered simply: it is the science of numbers. Yes, about the familiar seven, nine and all this diverse community. And just as good and mediocre poems cannot be written without an elementary alphabet, even an elementary problem cannot be solved without arithmetic. That is why all sciences have advanced only after the development of arithmetic and mathematics, being before that just a set of assumptions.

Arithmetic is a phantom science

What is arithmetic - natural science or phantom? In fact, as the ancient Greek philosophers reasoned, neither numbers nor figures exist in reality. This is just a phantom that is created in human thinking when considering the environment with its processes. Indeed, nowhere around do we see anything like this that could be called a number, rather, a number is the way of the human mind to study the world. Or maybe this is a study of ourselves from the inside? Philosophers have been arguing about this for many centuries, so we do not undertake to give an exhaustive answer. One way or another, arithmetic managed to take its positions so firmly that in the modern world no one can be considered socially adapted without knowing its foundations.

How the natural number appeared

Of course, the main object that arithmetic operates on is a natural number, such as 1, 2, 3, 4, ..., 152 ... etc. Natural number arithmetic is the result of counting common objects such as cows in a meadow. Still, the definition of "a lot" or "a little" once ceased to suit people, and they had to invent more perfect counting techniques.

But a real breakthrough happened when human thought reached the point that one and the same number “two” can denote 2 kilograms, and 2 bricks, and 2 parts. The fact is that you need to abstract from the forms, properties and meaning of objects, then you can perform some actions with these objects in the form of natural numbers. This is how the arithmetic of numbers was born, which further developed and expanded, occupying more and more positions in the life of society.

Such in-depth concepts of number as zero and negative numbers, fractions, denoting numbers by numbers and in other ways, have a rich and interesting history of development.

Arithmetic and the practical Egyptians

The two most ancient companions of man in the study of the world around him and in solving everyday problems are arithmetic and geometry.

It is believed that the history of arithmetic originates in the Ancient East: in India, Egypt, Babylon and China. So, the Rinda papyrus of Egyptian origin (named so because it belonged to the owner of the same name), dating back to the XX century. BC, besides other valuable data, contains the decomposition of one fraction into the sum of fractions with different denominators and a numerator equal to one.

For example: 2/73 = 1/60 + 1/219 + 1/292 + 1/365.

But what is the point of such a complex decomposition? The fact is that the Egyptian approach did not tolerate abstract thinking about numbers, on the contrary, the calculations were made only for a practical purpose. That is, the Egyptian will be engaged in such a thing as calculations, solely in order to build a tomb, for example. It was necessary to calculate the length of the rib of the structure, and this forced the person to sit down at the papyrus. As you can see, the Egyptian progress in calculations was caused, rather by massive construction, rather than love for science.

For this reason, the calculations found on the papyri cannot be called reflections on the topic of fractions. Most likely, this is a practical template that helped in the future to solve problems with fractions. The ancient Egyptians, who did not know the multiplication tables, performed rather long calculations, decomposed into many subtasks. Perhaps this is one of such subtasks. It is easy to see that calculations with such blanks are very laborious and unpromising. Perhaps for this reason, we do not see the great contribution of Ancient Egypt to the development of mathematics.

Ancient Greece and philosophical arithmetic

Many knowledge of the Ancient East was successfully mastered by the ancient Greeks, known for lovers of abstract, abstract and philosophical reflections. Practice interested them no less, but the best theorists and thinkers are hard to find. This was beneficial to science, since it is impossible to delve into arithmetic without breaking it off from reality. Of course, you can multiply 10 cows and 100 liters of milk, but you won't be able to get very far.

Deep-minded Greeks have left a significant mark on history, and their works have come down to us:

Euclid and the Beginnings.
Pythagoras.
Archimedes.
Eratosthenes.
Zeno.
Anaxagoras.

And, of course, the Greeks who turned everything into philosophy, and especially the successors of the Pythagorean work, were so carried away by numbers that they considered them the mystery of the harmony of the world. The numbers have been so studied and researched that special properties were attributed to some of them and their pairs. For example:

Perfect numbers are those that are equal to the sum of all their divisors, except for the number itself (6 = 1 + 2 + 3).
Friendly numbers are such numbers, one of which is equal to the sum of all divisors of the second, and vice versa (the Pythagoreans knew only one such pair: 220 and 284).

The Greeks, who believed that science should be loved and not be with it for the sake of profit, achieved great success by exploring, playing and adding numbers. It should be noted that not all of their research was widely used, some of them remained only "for beauty."

Eastern thinkers of the Middle Ages

In the same way, in the Middle Ages, arithmetic owes its development to Eastern contemporaries. The Indians gave us the numbers that we actively use, such a concept as "zero", and the positional version familiar to modern perception. From Al-kasha, who worked in Samarkand in the 15th century, we inherited without which it is difficult to imagine modern arithmetic.

In many ways, Europe's acquaintance with the achievements of the East became possible thanks to the work of the Italian scientist Leonardo Fibonacci, who wrote the book "The Book of the Abacus", introducing Eastern innovations. It became the cornerstone of the development of algebra and arithmetic, research and scientific activities in Europe.

Russian arithmetic

And finally, arithmetic, which found its place and took root in Europe, began to spread to the Russian lands. The first Russian arithmetic came out in 1703 - it was a book about arithmetic by Leonty Magnitsky. For a long time, it remained the only teaching manual in mathematics. It contains the initial moments of algebra and geometry. The numbers used in examples by the first Russian textbook of arithmetic are Arabic. Although Arabic numerals have been encountered before, in engravings dating from the 17th century.

The book itself is decorated with images of Archimedes and Pythagoras, and on the first sheet there is an image of arithmetic in the form of a woman. She sits on the throne, under her is written in Hebrew the word denoting the name of God, and on the steps that lead to the throne are inscribed the words "division", "multiplication", "addition", etc. One can only imagine what meaning was betrayed such truths that are now considered commonplace.

The 600-page textbook covers both the basics like the addition and multiplication table and applications to the science of navigation.

It is not surprising that the author chose images of Greek thinkers for his book, because he himself was captivated by the beauty of arithmetic, saying: "Arithmetic is a numerator, there is art that is honest, unenviable ...". This approach to arithmetic is quite justified, because it is precisely its widespread introduction that can be considered the beginning of the rapid development of scientific thought in Russia and general education.

Difficult primes

A prime number is a natural number that has only 2 positive divisors: 1 and itself. All other numbers, apart from 1, are called composite. Examples of primes: 2, 3, 5, 7, 11, and all others that have no other divisors, except for the number 1 and itself.

As for the number 1, it is on a special account - there is an agreement that it should be considered neither simple nor compound. A simple at first glance, a prime number conceals many unsolved secrets within itself.

Euclid's theorem says that there is an infinite set of primes, and Eratosthenes invented a special arithmetic "sieve" that eliminates non-prime numbers, leaving only prime numbers.

Its essence is to underline the first uncrossed number, and then cross out those that are multiples of it. We repeat this procedure many times - and we get a table of prime numbers.

Basic theorem of arithmetic

Among the observations about primes, the main theorem of arithmetic must be mentioned in a special way.

The main theorem of arithmetic says that any integer greater than 1 is either prime, or it can be decomposed into the product of primes up to the order of the factors, and in a unique way.

The main theorem of arithmetic is proved rather cumbersome, and its understanding no longer looks like the simplest foundations.

At first glance, prime numbers are an elementary concept, but they are not. Physics also once considered the atom to be elementary, until it found the whole universe inside it. The excellent story "The First Fifty Million Prime Numbers" by mathematician Don Tsagir is devoted to prime numbers.

From "three apples" to deductive laws

What can truly be called the reinforced foundation of all science is the laws of arithmetic. Even in childhood, everyone is faced with arithmetic, studying the number of legs and arms of dolls, the number of cubes, apples, etc. This is how we study arithmetic, which then goes into more complex rules.

Our whole life acquaints us with the rules of arithmetic, which have become the most useful for the common man of all that science gives. The study of numbers is "baby arithmetic" that introduces a person to the world of numbers in the form of numbers in early childhood.

Higher arithmetic is a deductive science that studies the laws of arithmetic. Most of them are known to us, although we may not know their exact wording.

The law of addition and multiplication

Any two natural numbers a and b can be expressed as the sum a + b, which will also be a natural number. The following laws apply to addition:

Commutative, which says that the sum does not change from the permutation of the terms in places, or a + b = b + a.
Associative, which says that the sum does not depend on the way the terms are grouped in places, or a + (b + c) = (a + b) + c.

The rules of arithmetic, such as addition, are some of the elementary ones, but they are used by all sciences, not to mention everyday life.

Any two natural numbers a and b can be expressed in the product a * b or a * b, which is also a natural number. The same commutative and associative laws apply to a product as to addition:

a * b = b * a;
a * (b * c) = (a * b) * c.

Interestingly, there is a law that combines addition and multiplication, also called the distributive or distributive law:

a (b + c) = ab + ac

This law actually teaches us to work with parentheses, opening them, thereby we can work with more complex formulas. These are the very laws that will guide us through the bizarre and complex world of algebra.

The law of arithmetic order

Human logic uses the law of order every day, checking clocks and counting bills. And, nevertheless, it needs to be formalized in the form of specific formulations.

If we have two natural numbers a and b, then the following options are possible:

a is b, or a = b;
a is less than b, or a< b;
a is greater than b, or a> b.

Of the three options, only one can be fair. The basic law that governs order says: if a< b и b < c, то a< c.

There are also laws that relate order to multiplication and addition: if a< b, то a + c < b+c и ac< bc.

The laws of arithmetic teach us to work with numbers, signs and brackets, turning everything into a harmonious symphony of numbers.

Positional and non-positional calculus systems

We can say that numbers are a mathematical language, on the convenience of which a lot depends. There are many systems of calculus, which, like the alphabets of different languages, differ from each other.

Let us consider the number systems from the point of view of the influence of the position on the quantitative value of the figure at this position. So, for example, the Roman system is non-positional, where each number is encoded by a certain set of special characters: I / V / X / L / C / D / M. They are equal, respectively, to the numbers 1/5/10/50/100/500 / 1000. In such a system, the figure does not change its quantitative definition, depending on what position it is in: first, second, etc. To get other numbers, you need to add the base ones. For example:

DCC = 700.
CCM = 800.

A more familiar number system for us using Arabic numerals is positional. In such a system, the digit of the number determines the number of digits, for example, three-digit numbers: 333, 567, etc. The weight of any digit depends on the position at which this or that digit is located, for example, the digit 8 in the second position has a value of 80. This is typical for the decimal system, there are other positional systems, for example, binary.

Binary arithmetic

Binary arithmetic works with a binary alphabet, which consists of only 0 and 1. And the use of this alphabet is called the binary number system.

The difference between binary arithmetic and decimal arithmetic is that the significance of the position on the left is no longer 10, but 2 times. Binary numbers have the form 111, 1001, etc. How are such numbers to be understood? So, consider the number 1100:

The first digit on the left is 1 * 8 = 8, remembering that the fourth digit, which means that it needs to be multiplied by 2, we get position 8.
The second digit is 1 * 4 = 4 (position 4).
The third digit is 0 * 2 = 0 (position 2).
The fourth digit is 0 * 1 = 0 (position 1).
So, our number 1100 = 8 + 4 + 0 + 0 = 12.

That is, when switching to a new digit on the left, its significance in the binary system is multiplied by 2, and in the decimal - by 10. This system has one drawback: it is too large an increase in the digits that are necessary to write numbers. Examples of representing decimal numbers in the form of binary numbers can be found in the following table.

The decimal numbers in binary are shown below.

Both octal and hexadecimal systems are used.

This cryptic arithmetic

What is arithmetic, "twice two" or the unexplored mysteries of numbers? As you can see, arithmetic may seem simple at first glance, but its non-obvious lightness is deceiving. Children can also study it together with Aunt Owl from the cartoon "Baby arithmetic", or you can immerse yourself in deeply scientific research of an almost philosophical nature. In history, she went from counting objects to worshiping the beauty of numbers. Only one thing is known for sure: with the establishment of the basic postulates of arithmetic, all science can rely on its strong shoulder.