With arithmetic, the science of numbers, our acquaintance with mathematics begins. One of the first Russian textbooks on arithmetic, written by L. F. Magnitsky in 1703, began with the words: “Arithmetic or numerator, is an art that is honest, unenviable, and conveniently understandable to everyone, most useful and most praised, from the oldest and newest, in different times who lived the finest arithmetic, invented and expounded. With arithmetic, we enter, as M. V. Lomonosov said, into the “gates of learning” and begin our long and difficult, but fascinating journey of knowing the world.
The word "arithmetic" comes from the Greek arithmos, which means "number". This science studies operations on numbers, various rules handling them, teaches you to solve problems that boil down to addition, subtraction, multiplication and division of numbers. Arithmetic is often imagined as some first step in mathematics, based on which it is possible to study its more complex sections - algebra, mathematical analysis, etc. Even whole numbers - the basic object of arithmetic - are referred when they are considered general properties and patterns, to higher arithmetic, or number theory. Such a view of arithmetic, of course, has grounds - it really remains the "alphabet of counting", but the alphabet is "most useful" and "comfortable".
Arithmetic and geometry are old companions of man. These sciences appeared when it became necessary to count objects, measure land, divide booty, keep track of time.
Arithmetic originated in the countries of the Ancient East: Babylon, China, India, Egypt. For example, the Egyptian papyrus Rinda (named after its owner G. Rinda) dates back to the 20th century. BC. Among other information, it contains expansions of a fraction into the sum of fractions with a numerator equal to one, for example:
The treasures of mathematical knowledge accumulated in the countries of the Ancient East were developed and continued by the scientists of Ancient Greece. Many names of scientists involved in arithmetic in the ancient world have been preserved for us by history - Anaxagoras and Zeno, Euclid (see Euclid and his "Beginnings"), Archimedes, Eratosthenes and Diophantus. The name of Pythagoras (VI century BC) sparkles here as a bright star. The Pythagoreans (disciples and followers of Pythagoras) worshiped numbers, believing that they contained 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 the positional number system; from al-Kashi (XV century), who worked at the Samarkand observatory Ulugbek, - decimal fractions.
Thanks to the development of trade and influence Eastern culture starting from the 13th century. increasing interest in arithmetic in Europe. One should remember the name of the Italian scientist Leonardo of Pisa (Fibonacci), whose work "The Book of the Abacus" introduced Europeans to the main achievements of the mathematics of the East and was the beginning of many studies in arithmetic and algebra.
Together 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. The Complete Arithmetic by the German mathematician M. Stiefel (early 16th century) already contains negative numbers and even the idea of the logarithm.
Around the 16th century the development of purely arithmetic questions flowed into the mainstream of algebra - as a significant milestone, one can note the appearance of the works of the French scientist F. Vieta, in which numbers are indicated by letters. Since that time, the basic arithmetic rules have been fully understood from the standpoint of algebra.
The basic object of arithmetic is the number. Natural numbers, i.e. the numbers 1, 2, 3, 4, ... etc., arose from counting specific items. 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 counted objects, to abstract from their shape, size, color, etc. Fibonacci already has a task: “Seven old women are going to Rome. Each has 7 mules, each mule carries 7 bags, each bag has 7 loaves, each loaf has 7 knives, each knife has 7 sheaths. How many? To solve the problem, you will have to put together old women, and mules, and bags, and bread.
The development of the concept of number - the appearance of zero and negative numbers, ordinary and decimal fractions, ways of writing numbers (numbers, symbols, number systems) - all this has a rich and interesting history.
“The science of numbers means two sciences: practical and theoretical. Practical studies numbers insofar as we are talking about countable numbers. This science is used in market and civil affairs. The theoretical science of numbers studies numbers in absolute sense abstracted by the mind from bodies and everything that can be counted in them. al-Farabi
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 arithmetic. Now, in our minds or on a piece of paper, we do only the most simple calculations, more and more often entrusting more complex computational work to microcalculators, which are gradually replacing such devices as abacus, adding machine (see Computer Science), slide rule. However, the operation of all computers - simple and complex - is based on the simplest operation - the addition of natural numbers. It turns out that the most complex calculations can be reduced to addition, only this operation must be done many millions of times. But here we are invading another area of mathematics that originates in arithmetic - computational mathematics.
Arithmetic operations on numbers have a variety of properties. These properties can be described in words, for example: “The sum does not change from a change in the places of the terms”, can be written in letters:, can be expressed in special terms.
For example, this property of addition is called a commutative or commutative law. We apply the laws of arithmetic often out of habit, without realizing it. Often students at school ask: “Why learn all these displacement and combination laws, because it’s so clear how to add and multiply numbers?” In the 19th century mathematics took an important step - it began to systematically add and multiply not only numbers, but also vectors, functions, displacements, tables of numbers, matrices and much more, and even just letters, symbols, without really caring about their specific meaning. And here it turned out that the most important thing is what laws these operations obey. The study of operations given on arbitrary objects (not necessarily on numbers) is already the domain of algebra, although this task is based on arithmetic and its laws.
Arithmetic contains many rules for solving problems. In old books you can find problems for the "triple rule", for "proportional division", for the "method of weights", for the "false rule", etc. Most of these rules are now obsolete, although the tasks that were solved with their help can by no means be considered obsolete. The famous problem about a pool that is filled with several pipes is at least two thousand years old, and it is still not easy for schoolchildren. But if earlier it was necessary to know a special rule to solve this problem, then today it is already junior schoolchildren learn to solve such a problem by entering the letter designation of the desired value. Thus, arithmetic problems led to the need to solve equations, and this is again the task of algebra.
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PYTHAGORAS
There are no written documents about Pythagoras of Samos, and according to later evidence, it is difficult to restore the true picture of his life and achievements. It is known that Pythagoras left his native island of Samos in the Aegean Sea off the coast of Asia Minor in protest against the tyranny of the ruler and already at a mature age (according to legend at 40 years old) appeared in the Greek city of Crotone in southern Italy. Pythagoras and his followers - the Pythagoreans - formed a secret alliance that played a significant role in the life of the Greek colonies in Italy. The Pythagoreans recognized each other by the star-shaped pentagon - the pentagram. The philosophy and religion of the East had a great influence on the teachings of Pythagoras. He traveled a lot in the countries of the East: he was in Egypt and in Babylon. There Pythagoras got acquainted with oriental mathematics. Mathematics has become part of his teachings, and the most important part. The Pythagoreans believed that the secret of the world was hidden in numerical patterns. The world of numbers lived a special life for the Pythagorean, numbers had their own special life meaning. Numbers equal to the sum of their divisors were perceived as perfect (6, 28, 496, 8128); pairs of numbers were called friendly, each of which was equal to the sum of the divisors of the other (for example, 220 and 284). Pythagoras was the first to divide numbers into even and odd, prime and composite, and introduced the concept of a figurative number. In his school were considered in detail Pythagorean triples natural numbers, in which the square of one was equal to the sum of the squares of the other two (see Fermat's great theorem). Pythagoras is credited with saying: "Everything is a number." By numbers (and he meant only natural numbers), he wanted to reduce the whole world, and mathematics in particular. But in the school of Pythagoras itself, a discovery was made that violated this harmony. It has been proven that is not a rational number, i.e. not expressed in terms of natural numbers. Naturally, the geometry of Pythagoras was subordinate to arithmetic, this was clearly manifested in the theorem that bears his name and later became the basis for the application of numerical methods in geometry. (Later, Euclid again brought geometry to the forefront, subordinating algebra to it.) Apparently, the Pythagoreans knew the correct solids: the tetrahedron, the cube, and the dodecahedron. Pythagoras is credited with the systematic introduction of proofs into geometry, the creation of planimetry of rectilinear figures, and the doctrine of similarity. The name of Pythagoras is associated with the doctrine of arithmetic, geometric and harmonic proportions, averages. It should be noted that Pythagoras considered the Earth to be a ball moving around the Sun. When in the 16th century the church began to fiercely persecute the teachings of Copernicus, this teaching was stubbornly called Pythagorean. |
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ARCHIMEDES
More is known about Archimedes, the great mathematician and mechanic, than about other scientists of antiquity. First of all, the year of his death is reliable - the year of the fall of Syracuse, when the scientist died at the hands of a Roman soldier. However, the ancient historians Polybius, Livy, Plutarch spoke little about his mathematical merits; from them information about the miraculous inventions of the scientist, made during the service of King Hieron II, has come down to our times. There is a famous story about the king's golden crown. Archimedes checked the purity of its composition with the help of the law of buoyancy he found, and his exclamation “Eureka!”, i.e. "Found!". Another legend tells that Archimedes built a system of blocks, with the help of which one person was able to launch the huge ship "Syracosia". The words of Archimedes uttered then became winged: “Give me a fulcrum, and I will turn the Earth.” The engineering genius of Archimedes manifested itself with particular force during the siege of Syracuse, a rich trading city on the island of Sicily. The soldiers of the Roman consul Marcellus were detained for a long time at the walls of the city by unprecedented machines: powerful catapults accurately fired stone blocks, throwing machines were installed in the loopholes, throwing out hail of cores, coastal cranes turned outside the walls and threw enemy ships with stone and lead blocks, hooks picked up ships and they threw them down from a great height, systems of concave mirrors (in some stories - shields) set fire to ships. In the History of Marcellus, Plutarch describes the horror that reigned in the ranks of the Roman soldiers: “As soon as they noticed that a rope or a log was showing from behind the fortress wall, they fled, shouting that Archimedes had also invented a new machine for their death” . The contribution of Archimedes to the development of mathematics is also enormous. Archimedes' spiral (see Spirals), described by a point moving in a rotating circle, stood apart from the numerous curves known to his contemporaries. The next kinematically defined curve, the cycloid, appeared only in the 17th century. Archimedes learned to find the tangent to his spiral (and his predecessors could only draw tangents to conic sections with it), found the area of its coil, as well as the area of the ellipse, the surface of the cone and the ball, the volumes of the ball and the spherical segment. He was especially proud of the ratio of the volume of the sphere and the cylinder described around it, which he discovered, which is 2:3 (see Inscribed and circumscribed figures). Archimedes also dealt a lot with the problem of squaring the circle (see Famous Problems of Antiquity). The scientist calculated the ratio of the circumference to the diameter (number) and found that it is between and. The method he created for calculating the circumference and area of a figure was an essential step towards the creation of differential and integral calculus, which appeared only 2000 years later. Archimedes also found an infinite sum geometric progression with denominator . In mathematics, this was the first example of an infinite series. An important role in the development of mathematics was played by his essay "Psammit" - "On the number of grains of sand", in which he shows how, using the existing number system, one can express arbitrarily big numbers. As a reason for his reasoning, he uses the problem of counting the number of grains of sand inside the visible universe. Thus, the then-existing opinion about the presence of mysterious "largest numbers" was refuted. |
Among the important concepts introduced by arithmetic, 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 many centuries.
One can clearly trace the "geometrization" of arithmetic: complicated rules and patterns expressed by formulas, become clearer if it is possible to represent them geometrically. An important role in mathematics itself and its applications is played by the reverse process - the translation of visual, geometric information into the language of numbers (see Graphical calculations). This translation is based on the idea of the French philosopher and mathematician R. Descartes on the definition of points on the plane by coordinates. Of course, this idea had already been used before him, for example, in maritime affairs, when it was necessary to determine the location of the ship, as well as in astronomy and geodesy. But it is precisely from Descartes and his students that the consistent use of the language of coordinates in mathematics comes. And in our time, when managing complex processes (for example, the flight of a spacecraft), they prefer to have all the information in the form of numbers, which are processed by a computer. If necessary, the machine helps a person to translate the accumulated numerical information into the language of the drawing.
You see that, speaking of arithmetic, we always go beyond its limits - into algebra, geometry, and other branches of mathematics.
How to delineate the boundaries of arithmetic itself?
In what sense is this word used?
The word "arithmetic" can be understood as:
an academic subject dealing primarily with rational numbers (whole numbers and fractions), operations on them, and problems solved with the help of these operations;
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 numerical systems (natural, integer, rational, real, complex numbers and their generalizations);
"formal arithmetic" - a part of mathematical logic (see. Mathematical logic), which deals with the analysis of the axiomatic theory of arithmetic;
"higher arithmetic", or number theory, an independently developing part of mathematics.
Mathematics begins with arithmetic. With arithmetic, we enter, as M. V. Lomonosov said, into the “gates of learning”.
The word "arithmetic" comes from the Greek arithmos, which means "number". This science studies operations on numbers, various rules for handling them, teaches you how to solve problems that boil down to addition, subtraction, multiplication and division of numbers. Arithmetic is often imagined as some first step in mathematics, based on which it is possible to 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 papyrus Rinda (named after its owner G. Rinda) dates back to the 20th century. BC e.
The treasures of mathematical knowledge accumulated in the countries of the Ancient East were developed and continued by the scientists of Ancient Greece. Many names of scientists involved in arithmetic in the ancient world have been preserved for us by history - 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 contained 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 the 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. increasing interest in arithmetic in Europe. One should remember the name of the Italian scientist Leonardo of Pisa (Fibonacci), whose work "The Book of the Abacus" introduced Europeans to the main achievements of the mathematics of the East and was the beginning of many studies in arithmetic and algebra.
Together 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. The Complete Arithmetic by the German mathematician M. Stiefel (beginning of the 16th century) already contains negative numbers and even the idea of taking a logarithm.
Around the 16th century the development of purely arithmetic questions flowed into the mainstream of algebra, as a significant milestone, one can note the appearance of the works of the French scientist F. Vieta, in which numbers are indicated by letters. Since that time, the basic arithmetic rules have been fully understood from the standpoint of algebra.
The basic object of arithmetic is the number. Natural numbers, i.e. the numbers 1, 2, 3, 4, ... etc., arose from counting specific items. 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 counted objects, 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 a change in the places of the terms”, can be written in letters: a + b \u003d b + a, can be expressed in special terms.
Among the important concepts introduced by arithmetic, 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 many centuries.
The word "arithmetic" can be understood as:
an academic subject dealing primarily with rational numbers (whole numbers and fractions), operations on them, and problems solved with the help of these operations;
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 numerical systems (natural, integer, rational, real, complex numbers and their generalizations);
"formal arithmetic" - a part of mathematical logic that deals with the analysis of the axiomatic theory of arithmetic;
"higher arithmetic", or number theory, an independently developing part of mathematics And
/Encyclopedic Dictionary of a Young Mathematician, 1989/
Popova L.A. 1
Koshkin I.A. 1
1 Municipal Budgetary Educational Institution "Education Center - Gymnasium No. 1"
The text of the work is placed without images and formulas.
Full version work is available in the "Files of work" tab in PDF format
Introduction
Relevance. Mental arithmetic is now gaining great popularity. Thanks to new teaching methods, children quickly learn new information, develop their creative potential learn to solve complex math problems mentally, without using a calculator.
Mental arithmetic is a unique method for developing the mental abilities of children from 4 to 16 years old, based on a system of mental counting. Learning by this technique, the child can solve any arithmetic problems in a few seconds (addition, subtraction, multiplication, division, calculation square root numbers) mentally faster than using a calculator.
Goal of the work:
Learn the history of mental arithmetic
Show how you can use the abacus when solving mathematical problems
To analyze what other alternative methods of calculation are that simplify the calculation and make it entertaining
Hypothesis:
Let's assume that arithmetic can be entertaining and easy, can be calculated much faster and more productively using mental arithmetic methods and various tricks.
Classes with Chinese accounts have a positive effect on memory, which is reflected in the assimilation educational material. This applies to memorizing poetry and prose, theorems, various mathematical rules, foreign words, that is, a large amount of information.
Research methods: Internet search, literature study, practical work on mastering the abacus, solving examples with the help of the abacus,
Study execution plan:
To study the literature of the history of arithmetic from the very beginning
Outline the principles of computing on the abacus
To analyze how mental arithmetic classes go, and draw conclusions from my classes
Find out the benefits and analyze the possible difficulties in the mental account
Show what other ways to calculate in arithmetic
Chapter 1. The history of the development of arithmetic
Arithmetic originated in the countries of the Ancient East: Babylon, China, India, Egypt. The name "arithmetic" comes from the Greek word "arithmos" - number.
Arithmetic studies numbers and operations on numbers, various rules for handling them, teaches you how to solve problems that reduce to addition, subtraction, multiplication and division of numbers.
The emergence of arithmetic is associated with the labor activity of people and with the development of society.
The importance of mathematics in Everyday life person. Without counting, without the ability to correctly add, subtract, multiply and divide numbers, development is unthinkable. human society. Four arithmetic operations, the rules of oral and written calculations, we study, starting with primary school. All these rules were not invented or discovered by any one person. Arithmetic originated from the daily life of people.
1.1 First counting devices
People have long tried to ease their account with the help of various means and devices. The first, most ancient "calculating machine" were 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 ancient merchants (Babylonian and other cities) made calculations using grains, pebbles and shells, which they began to lay out on a special board called an abacus.
The abacus's analogue ancient China there was a counting device "su-anpan", It is a small elongated box, divided along the length into unequal parts by partitions. Across the box are twigs on which balls are strung.
The Japanese did not lag behind the Chinese and, using their example, in the 16th century created their own counting device - the Soroban. It differed from the Chinese one in that there was one ball each in the upper compartment of the device, while in the Chinese version there were two.
Russian abacus first appeared in Russia in the 16th century. They were a board with parallel lines drawn on it. Later, instead of the board, they began to use a frame with wires and bones.
1.2 Abacus
Around the fourth century BC, the first counting device was invented. Its creator is the scientist Abacus, and the device was named after him. It looked like this: a clay plate with grooves into which stones were placed, denoting numbers. One groove was for units, and the other for tens.
Word "abacus" (abacus) means scoreboard.
Let's look at the modern abacus...
To learn how to use accounts, you need to know what they are.
Accounts consist of:
dividing line;
upper bones;
lower bones.
There is a center point in the middle. The top bones represent fives, and the bottom ones represent ones. Each vertical strip of bones, starting from right to left, denotes one of the digits of the numbers:
tens of thousands, etc.
For example, to postpone the example: 9 - 4=5, you need to move the top bone on the first line on the right (it means five) and raise the 4 lower bones. Then lower the 4 lower bones. So we get the required number 5.
Chapter 2. What is mental arithmetic?
mental arithmetic is a method of developing the mental abilities of children from 4 to 14 years old. The basis of mental arithmetic is the abacus score. It originated in ancient Japan over 2000 years ago. The child counts on the abacus with both hands, making calculations twice as fast. On the accounts, not only add and subtract, but also learn to multiply and divide.
mentality - it is the mental capacity of man.
During math lessons, only the left hemisphere of the brain develops, which is responsible for logical thinking, and the right is developed by such subjects as literature, music, drawing. There are special training techniques that are aimed at developing both hemispheres. Scientists say that those people who have fully developed both hemispheres of the brain achieve success. 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.
The use of an abacus makes the left hemisphere work - develops fine motor skills and allows the child to visually 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- and four-digit numbers.
In addition to solving examples without using notes and drafts, doing mental arithmetic allows you to:
improve academic performance in various subjects at school;
diversify from mathematics to music;
learn foreign languages faster;
become more proactive and independent;
develop leadership qualities;
be confident.
imagination: in the future, the link to the accounts is weakened, which allows you to make calculations in your mind, work with imaginary accounts;
the representation of the number is perceived not objectively, but figuratively, the image of the number is formed in the form of an image of combinations of bones;
observation;
hearing, method active listening improves listening skills;
concentration of attention, as well as the distribution of attention increases: simultaneous involvement in several types of thought processes.
Practicing mental arithmetic is not a direct training of mathematical skills. Quick counting is only a means and indicator of the speed of thinking, but not an end in itself. The purpose of mental arithmetic is the development of intellectual and creative abilities, and this will be useful for future mathematicians and humanities. However, one must be prepared for the fact that at the very beginning of training it will be necessary to put in enough effort, diligence, perseverance and attentiveness. There may be errors in the calculations - so do not rush.
Chapter 3. Classes in the school of mental arithmetic.
The entire program for the development of oral counting is built on the successive passage of two stages.
At the first of them, acquaintance and mastery of the technique of performing arithmetic operations using bones takes place, during which two hands are involved simultaneously. In his work, the child uses an abacus. This item allows him to absolutely freely subtract and multiply, add and divide, calculate the square and cube roots.
During the passage of the second stage, students are taught mental counting, which is performed in the mind. The child ceases to be constantly attached to the abacus, which also stimulates his imagination. The left hemispheres of children perceive numbers, and the right hemispheres perceive the image of knuckles. This is the basis of the method of mental counting. The brain begins to work with an imaginary abacus, while perceiving numbers in the form of pictures. The performance of the mathematical calculation is associated with the movement of the bones.
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.
For example, Brothers in mental arithmetic are two numbers, the addition of which gives five.
There are 5 brothers in total.
1+4 = 5 Brother 1 - 4 4+1 = 5 Brother 4 - 1
2+3 = 5 Brother 2 - 3 5+0 = 5 Brother 5 - 0
3+2 = 5 Brother 3 - 2
Friends in mental arithmetic are two numbers that add up to ten.
Only 10 friends.
1+9 = 10 Friend 1 - 9 6+4 = 10 Friend 4 - 6
2+8 = 10 Friend 2 - 8 7+3 = 10 Friend 7 - 3
3+7 = 10 Friend 3 - 7 8+2 = 10 Friend 8 - 2
4+6 = 10 Friend 4 - 6 9-1 = 10 Friend 9 -1
5+5 = 10 Friend 5 - 5
Chapter 4. My studies in mental arithmetic.
At the trial lesson, the teacher showed us the abacus abacus, briefly told us how to use them and the very principle of counting.
There was a mental warm-up at the lesson. And there were always breaks where we could have a little snack, drink water or play games. At home, we were always given sheets with examples, for independent work Houses. I also trained in a special program where examples were launched - they flashed on the monitor at different speeds.
At the very beginning of my training, I:
Get familiar with 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.
Over time I:
I learned to count two-stage examples with tens. Tens are located on the second needle from the far right. When counting with tens, we already use the thumb and forefinger of the left hand. Here the technique is the same as with the right hand: we raise it with a large one, we lower it with our index one.
In the 3rd month of study:
I used the abacus to solve examples of subtraction and addition with units and tens - three-stage.
Solve examples of subtraction and addition with thousandths - two-stage
Further:
Get to know the mind map. Looking at the card, I had to mentally move the knuckles and see the answer.
I worked out 2 hours a week and 5-10 minutes a day on my own for 4 months.
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First month of training |
fourth month |
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1. I count on the abacus 1 sheet (30 examples of 3 terms) |
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2. I mentally count 30 examples (5-7 terms each) |
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3. I am learning a poem (3rd quatrains) |
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4. Execution homework(mathematics: one problem, 10 examples) |
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Numbers arose from the need for counting and measuring and have undergone a long path of historical development. There was a time when people couldn't 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 recalculating 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 there are 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, man came to the next stage in the creation of natural numbers - to compare sets, they began to use intermediary sets: small pebbles, shells, fingers. These intermediary sets already represented the rudiments of the concept natural number, although at this stage the number was not separated from the counted objects: 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 items - by the words "the whole person". Only after a person has learned to operate with intermediary sets, has he established that common thing that exists, for example, between five fingers and five apples, i.e. when there was an abstraction from the nature of the elements of intermediary 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 pronounced. It was milestone 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, around 10-5 millennium BC. Over time, people learned not only to name numbers, but also to designate them, as well as perform actions on them. In general, the natural series of numbers did not arise immediately, the history of its formation is long. The stock of numbers that were used while keeping score increased gradually. Gradually, the concept of the infinity of the set of natural numbers also 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 a method for the 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. It became possible to study these numbers regardless of those. specific tasks in connection with which they arose. The theoretical science, which began to study numbers and operations on them, was called "arithmetic". The word "arithmetic" comes from the Greek arithmos, what does "number" mean? Therefore, arithmetic is the science of numbers. 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, the mathematicians of India, the countries of the Arab world and Central Asia made a great contribution to the development of arithmetic, and starting from the 13th century, 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 language 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 became the foundation of all mathematical science, on the state of which the strength of the entire building of mathematics depended. In this regard, there was a need for a strict logical substantiation of the concept of a natural number, for the systematization of what is connected with it. Since the mathematics of the 19th century turned to the axiomatic construction of its theories, an 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 theories created, the concepts of natural numbers and actions on them have become more abstract, but this always accompanies the process of generalization and systematization of individual facts. § 14. AXIOMATIC CONSTRUCTION OF A SYSTEM OF NATURAL NUMBERS As already mentioned, natural numbers are obtained by counting objects and by measuring quantities. But if during the measurement there appear numbers other than natural ones, then the calculation leads only to natural numbers. To keep count, you need a sequence of numbers that starts with one and that allows You may be interested | |
