Chinese or Japanese multiplication. Magic mathematics, or how the Japanese multiply How the Japanese multiply explanation

For a long time, Asian countries, in particular Singapore and Japan, top the lists of Program for International Student Assessment (PISA) test results. What is the secret of their mathematical success? Are mathematical geniuses born or made? To understand the issues, let's look at approaches to teaching mathematics in Asian countries.

How mathematics is taught in Japan

At the age of 7-8, Japanese children begin to learn a rhyming multiplication table called kuku, “ku” meaning “nine” in Japanese. Japanese children learn the table by heart and then recite it quickly at school and at home. There are even special competitions for second-graders on the speed of reproducing tables. In order to win, schoolchildren are forced to train long and hard with a stopwatch.

Also, many Japanese children take extracurricular math classes. There are more than 20 thousand private mathematics educational organizations in Japan. Schoolchildren of any age can study there: both first-graders and high school students. Many of them are taught a quick counting system using mental arithmetic.

Additional classes take one to two hours and take place two, and in some cases four times a week. On them, children first learn to solve examples using a counting board - an abacus, and then move to the next level, at which they begin to count in their heads.

At such lessons, children are given sheets with examples printed on them; their task is to spend as little time as possible on solving them. And this is in addition to four school math lessons (45 minutes each) per week.

After a couple of years of training in mental arithmetic, Japanese children can multiply seven- and eight-digit numbers in their heads faster than a child from any other part of the world can answer how much seven is eight.

Thirst for victory

Japanese children really like speed counting. Many consider it a new sport and participate in city, regional competitions and national championships.

This approach differs significantly from the generally accepted one, which calls for protecting children from competition in every possible way. However, many people forget that excessive care for children is no less harmful. Indeed, in this case, children will not know the joy of victories that they won thanks to their own efforts.

By ceasing to evaluate children, you can deprive them of motivation for further development.

Passion and talent

No one is born a math genius. Research shows that it takes 10,000 hours of practice to become an expert in a new field. If you want to succeed in math, be prepared for the fact that it will take a lot of time and effort.

If we look at the math competitions that the Japanese participate in, from speedy retelling of multiplication tables in elementary school to more complex mental arithmetic calculations in high school, it becomes obvious that it is the spirit of competition that keeps the Japanese fond of mathematics.

Language

Trying to understand the reasons for the success of Chinese people in mathematics, Malcolm Gladwell in the book “Geniuses and Outsiders: Why do some have everything and others nothing?” places special emphasis on language. The names of numbers in Chinese are short and can be pronounced very quickly: 4 sounds like “si”, 7 sounds like “ki”. The smaller the words, the faster they can be remembered. The author shows this using the example of the sequence of numbers 4, 8, 5, 3, 9, 7, 6, which an English-speaking person will remember 50% of the time the first time, and a Chinese person will remember it completely. The secret is that our short-term memory, on average, stores numbers in a period of no more than 2 seconds - most likely, you will remember the number of numbers that you can pronounce during this time.

The author notes a more logical name for complex numbers in Chinese than in English. The same can be said about the Russian language. For example, in the word sixteen, we first use the derivative from the name of the number “six”, and then denote one ten - “-eleven”. In the word sixty-one, we act more logically: first we designate the number of tens “sixty”, and then indicate the units - “one”. In Chinese, Japanese and Korean, the number naming system is more logical: sixteen is pronounced like ten and six, sixty-one is pronounced like six tens one. This gives Asian children a number of advantages: they learn to count faster than European children and perform arithmetic operations more easily. To adults, such differences in number naming seem insignificant, but they are significant to children.

Culture

Another factor that helped the Chinese succeed in mathematics is not directly related to science - it is the culture of rice cultivation.

Growing rice required a lot of labor and 3,000 hours of work in the fields (European peasants worked on average 1,200 hours a year). Unlike European peasants, the Chinese grew and harvested crops twice a year and did not have long rest during the winter.

Malcolm Gladwell notes that the work in the rice fields was difficult and painstaking, but resulted in “meaningful work.” Unlike European peasants, the Chinese were not in complete slavery to the nobles and did not give them most of their income. Landowners set a fixed rent, above which each community could take the harvest for itself. The peasants knew: if you work better, you will get more.

The hard work of the Chinese is reflected in many proverbs dedicated to work, the most striking: “The family of a person who gets up before dawn all year round will not be in poverty.” What does mathematics have to do with this? This science, like no other, requires persistence, perseverance and the willingness to sit for a long time on each task.

To summarize: the success of Asian countries in mathematics is associated with the cult of hard work, language, a large number of school lessons and additional classes

Back forward

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“Counting and calculations are the basis of order in the head.”
Pestalozzi

Target:

• Learn ancient multiplication techniques.
• Expand your knowledge of various multiplication techniques.
• Learn to perform operations with natural numbers using ancient methods of multiplication.

1. The old way of multiplying by 9 on your fingers
2. Multiplication by Ferrol's method.
3. Japanese way of multiplication.
4. Italian way of multiplication (“Grid”)
5. Russian method of multiplication.
6. Indian way of multiplication.

Progress of the lesson

The relevance of using fast counting techniques.

In modern life, each person often has to perform a huge number of calculations and calculations. Therefore, the goal of my work is to show easy, fast and accurate methods of counting, which will not only help you during any calculations, but will cause considerable surprise among acquaintances and comrades, because the free performance of counting operations can largely indicate the extraordinary nature of your intellect. A fundamental element of computing culture is conscious and robust computing skills. The problem of developing a computing culture is relevant for the entire school mathematics course, starting from the primary grades, and requires not just mastering computing skills, but using them in various situations. Possession of computational skills is of great importance for mastering the material being studied and allows one to develop valuable work qualities: a responsible attitude towards one’s work, the ability to detect and correct errors made in the work, careful execution of a task, a creative attitude to work. However, recently the level of computational skills and transformations of expressions has a pronounced downward trend, students make a lot of mistakes when calculating, increasingly use a calculator, and do not think rationally, which negatively affects the quality of education and the level of mathematical knowledge of students in general. One of the components of computing culture is verbal counting, which is of great importance. The ability to quickly and correctly make simple calculations “in the head” is necessary for every person.

Ancient ways of multiplying numbers.

1. The old way of multiplying by 9 on your fingers

It's simple. To multiply any number from 1 to 9 by 9, look at your hands. Fold the finger that corresponds to the number being multiplied (for example, 9 x 3 - fold the third finger), count the fingers before the folded finger (in the case of 9 x 3, this is 2), then count after the folded finger (in our case, 7). The answer is 27.

2. Multiplication by the Ferrol method.

To multiply the units of the product of remultiplication, the units of the factors are multiplied; to obtain tens, the tens of one are multiplied by the units of the other and vice versa and the results are added; to obtain hundreds, the tens are multiplied. Using the Ferrol method, it is easy to multiply two-digit numbers from 10 to 20 verbally.

For example: 12x14=168

a) 2x4=8, write 8

b) 1x4+2x1=6, write 6

c) 1x1=1, write 1.

3. Japanese way of multiplication

This technique is reminiscent of multiplication by a column, but it takes quite a long time.

Using the technique. Let's say we need to multiply 13 by 24. Let's draw the following figure:

This drawing consists of 10 lines (the number can be any)

• These lines represent the number 24 (2 lines, indent, 4 lines)
• And these lines represent the number 13 (1 line, indent, 3 lines)

(intersections in the figure are indicated by dots)

Number of crossings:

• Top left edge: 2
• Bottom left edge: 6
• Top right: 4
• Bottom right: 12

1) Intersections in the upper left edge (2) – the first number of the answer

2) The sum of the intersections of the lower left and upper right edges (6+4) – the second number of the answer

3) Intersections in the lower right edge (12) – the third number of the answer.

It turns out: 2; 10; 12.

Because The last two numbers are two-digit and we cannot write them down, so we write down only ones and add tens to the previous one.

4. Italian way of multiplication (“Grid”)

In Italy, as well as in many Eastern countries, this method has gained great popularity.

Using the technique:

For example, let's multiply 6827 by 345.

1. Draw a square grid and write one of the numbers above the columns, and the second in height.

2. Multiply the number of each row sequentially by the numbers of each column.

• 6*3 = 18. Write 1 and 8
• 8*3 = 24. Write 2 and 4

If multiplication results in a single-digit number, write 0 at the top and this number at the bottom.

(As in our example, when multiplying 2 by 3, we got 6. We wrote 0 at the top and 6 at the bottom)

3. Fill in the entire grid and add up the numbers following the diagonal stripes. We start folding from right to left. If the sum of one diagonal contains tens, then add them to the units of the next diagonal.

Answer: 2355315.

5. Russian method of multiplication.

This multiplication technique was used by Russian peasants approximately 2-4 centuries ago, and was developed in ancient times. The essence of this method is: “As much as we divide the first factor, we multiply the second by that much.” Here is an example: We need to multiply 32 by 13. This is how our ancestors would have solved this example 3-4 centuries ago:

• 32 * 13 (32 divided by 2, and 13 multiplied by 2)
• 16 * 26 (16 divided by 2, and 26 multiplied by 2)
• 8 * 52 (etc.)
• 4 * 104
• 2 * 208
• 1 * 416 =416

Dividing in half continues until the quotient reaches 1, while simultaneously doubling the other number. The last doubled number gives the desired result. It is not difficult to understand what this method is based on: the product does not change if one factor is halved and the other is doubled. It is clear, therefore, that as a result of repeated repetition of this operation, the desired product is obtained

However, what should you do if you have to divide an odd number in half? The folk method easily overcomes this difficulty. It is necessary, says the rule, in the case of an odd number, discard one and divide the remainder in half; but then to the last number of the right column you will need to add all those numbers of this column that stand opposite the odd numbers of the left column: the sum will be the required product. In practice, this is done in such a way that all lines with even left numbers are crossed out; Only those that contain an odd number to the left remain. Here's an example (asterisks indicate that this line should be crossed out):

• 19*17
• 4 *68*
• 2 *136*
• 1 *272

Adding the uncrossed numbers, we get a completely correct result:

• 17 + 34 + 272 = 323.

Answer: 323.

6. Indian way of multiplication.

This method of multiplication was used in Ancient India.

To multiply, for example, 793 by 92, we write one number as the multiplicand and below it another as the multiplier. To make it easier to navigate, you can use the grid (A) as a reference.

Now we multiply the left digit of the multiplier by each digit of the multiplicand, that is, 9x7, 9x9 and 9x3. We write the resulting products in grid (B), keeping in mind the following rules:

• Rule 1. The units of the first product should be written in the same column as the multiplier, that is, in this case under 9.
• Rule 2. Subsequent works must be written in such a way that the units are placed in the column immediately to the right of the previous work.

Let's repeat the whole process with other digits of the multiplier, following the same rules (C).

Then we add up the numbers in the columns and get the answer: 72956.

As you can see, we get a large list of works. The Indians, who had extensive practice, wrote each number not in the corresponding column, but on top, as far as possible. Then they added the numbers in the columns and got the result.

Conclusion

We have entered a new millennium! Grand discoveries and achievements of mankind. We know a lot, we can do a lot. It seems something supernatural that with the help of numbers and formulas one can calculate the flight of a spaceship, the “economic situation” in the country, the weather for “tomorrow”, and describe the sound of notes in a melody. We know the statement of the ancient Greek mathematician and philosopher who lived in the 4th century BC - Pythagoras - “Everything is a number!”

According to the philosophical view of this scientist and his followers, numbers govern not only measure and weight, but also all phenomena occurring in nature, and are the essence of harmony reigning in the world, the soul of the cosmos.

Describing ancient methods of calculation and modern methods of quick calculation, I tried to show that both in the past and in the future, one cannot do without mathematics, a science created by the human mind.

“Whoever studies mathematics from childhood develops attention, trains the brain, his will, and cultivates perseverance and perseverance in achieving goals.”(A. Markushevich)

Literature.

1. Encyclopedia for children. "T.23". Universal Encyclopedic Dictionary \ ed. board: M. Aksenova, E. Zhuravleva, D. Lyury and others - M.: World of Encyclopedias Avanta +, Astrel, 2008. - 688 p.
2. Ozhegov S.I. Dictionary of the Russian language: approx. 57,000 words / Ed. member - corr. ANSIR N.YU. Shvedova. – 20th ed. – M.: Education, 2000. – 1012 p.
3. I want to know everything! Large illustrated encyclopedia of intelligence / Transl. from English A. Zykova, K. Malkova, O. Ozerova. – M.: Publishing house ECMO, 2006. – 440 p.
4. Sheinina O.S., Solovyova G.M. Mathematics. School club classes 5-6 grades / O.S. Sheinina, G.M. Solovyova - M.: Publishing house NTsENAS, 2007. - 208 p.
5. Kordemsky B. A., Akhadov A. A. The amazing world of numbers: A book of students, - M. Education, 1986.
6. Minskikh E. M. “From game to knowledge”, M., “Enlightenment” 1982.
7. Svechnikov A. A. Numbers, figures, problems M., Education, 1977.
8. http://matsievsky. newmail. ru/sys-schi/file15.htm
9. http://sch69.narod. ru/mod/1/6506/hystory. html

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In Russia, we are accustomed to multiplying numbers in the traditional way, which we were taught at school, by writing the multiplier numbers in a column (). However, in Asian countries such as Japan and China, it is customary to think differently. For the contemplative Eastern mentality, indispensable visualization is important. Even the Arabic numerals generally recognized in the world are written by the Chinese and Japanese in hieroglyphs. It is with the peculiarity of the Asian graphic system that the Japanese and Chinese method of multiplying numbers is associated.

This video shows how to multiply in Japanese and Chinese:

Many people will think that this method of Japanese or Chinese multiplication is too complicated and confusing, but this is only at first glance. It is visualization, that is, the image of all the points of intersection of lines (factors) on one plane, that gives us visual support, whereas the traditional method of multiplication involves a large number of arithmetic operations only in the mind. Chinese or Japanese multiplication not only helps you quickly and efficiently multiply two-digit and three-digit numbers by each other without a calculator, but also develops erudition. Agree, not everyone can boast that in practice they know the ancient Chinese method of multiplication (*), which is relevant and works great in the modern world.

*) Japanese or Chinese multiplication table? Archaeologists in Japan found a wooden tablet with a fragment of the multiplication table, which was allegedly made in the 8th century. Scientists believe that such tables were used by Japanese imperial officials who needed to master various sciences, including arithmetic.
The discovered tablet is the oldest of all previously found in Japan. It is interesting that the hieroglyphs used to write the numbers are very similar in graphic style to those that were used as official writing during the Chinese Tang Dynasty of the 7th-10th centuries. Based on this, scientists assumed that the table was copied from a Chinese arithmetic textbook of that time, that is, the entire Japanese multiplication table was borrowed from China.

It was to their neighbors in China that high-ranking Japanese went every year to learn from them various sciences, such as arithmetic. The ancient Chinese multiplication table was not a simple one, as it involved multiplying two-digit numbers by each other. It is unlikely that all Japanese officials could learn such a table by heart, which is why they carried something like cheat sheets with them to work, a fragment of one of which is the tablet found by archaeologists in Japan.

So, the Japanese multiplication table was borrowed from the Chinese, who, according to some hypotheses, were one of the creators of the first arithmetic system, as evidenced by archaeological finds containing fragments of the multiplication table, the age of which scientists estimated at 2700-3000 years.

Illustration copyright Getty Images Image caption I wouldn't have a headache...

“Math is so difficult...” You’ve probably heard this phrase more than once, and maybe even said it out loud yourself.

For many, mathematical calculations are not an easy task, but here are three simple ways that will help you perform at least one arithmetic operation - multiplication. No calculator.

It is likely that at school you became acquainted with the most traditional method of multiplication: first, you memorized the multiplication table, and only then began to multiply each of the digits in a column, which are used to write multi-digit numbers.

If you need to multiply multi-digit numbers, you will need a large sheet of paper to find the answer.

But if this long set of lines with numbers running one under the other makes your head spin, then there are other, more visual methods that can help you in this matter.

But this is where some artistic skills come in handy.

Let's draw!

At least three methods of multiplication involve drawing intersecting lines.

1. Mayan way, or Japanese method

There are several versions regarding the origin of this method.

Having trouble multiplying in your head? Try the Mayan and Japanese Method

Some say it was invented by the Mayan Indians, who inhabited areas of Central America before the conquistadors arrived there in the 16th century. It is also known as the Japanese method of multiplication because teachers in Japan use this visual method when teaching multiplication to younger students.

The idea is that parallel and perpendicular lines represent the digits of the numbers that need to be multiplied.

Let's multiply 23 by 41.

To do this, we need to draw two parallel lines representing 2, and, stepping back a little, three more lines representing 3.

Then, perpendicular to these lines, we will draw four parallel lines representing 4 and, stepping back slightly, another line for 1.

Well, is it really difficult?

2. Indian way, or Italian multiplication by "lattice" - "gelosia"

The origin of this method of multiplication is also unclear, but it is well known throughout Asia.

“The Gelosia algorithm was transmitted from India to China, then to Arabia, and from there to Italy in the 14th and 15th centuries, where it was called Gelosia because it was similar in appearance to Venetian lattice shutters,” writes Mario Roberto Canales Villanueva in his book on various methods of multiplication.

Illustration copyright Getty Images Image caption Indian or Italian multiplication system is similar to Venetian blinds

Let's take the example of multiplying 23 by 41 again.

Now we need to draw a table of four cells - one cell per number. Let's sign the corresponding number on top of each cell - 2,3,4,1.

Then you need to divide each cell in half diagonally to make triangles.

Now we will first multiply the first digits of each number, that is, 2 by 4, and write 0 in the first triangle and 8 in the second.

Then multiply 3x4 and write 1 in the first triangle, and 2 in the second.

Let's do the same with the other two numbers.

When all the cells of our table are filled in, we add up the numbers in the same sequence as shown in the video and write down the resulting result.

Media playback is unsupported on your device

Having trouble multiplying in your head? Try the Indian method

The first digit will be 0, the second 9, the third 4, the fourth 3. Thus, the result is: 943.

Do you think this method is easier or not?

Let's try another multiplication method using drawing.

3. "Array", or table method

As in the previous case, this will require drawing a table.

Let's take the same example: 23 x 41.

Here we need to divide our numbers into tens and ones, so we will write 23 as 20 in one column, and 3 in the other.

Vertically, we will write 40 at the top and 1 at the bottom.

Then we will multiply the numbers horizontally and vertically.

Media playback is unsupported on your device

Having trouble multiplying in your head? Draw a table.

But instead of multiplying 20 by 40, we'll drop the zeros and just multiply 2 x 4 to get 8.

We will do the same thing by multiplying 3 by 40. We keep 0 in parentheses and multiply 3 by 4 and get 12.

Let's do the same with the bottom row.

Now let’s add zeros: in the upper left cell we got 8, but we discarded two zeros - now we’ll add them and we’ll get 800.

In the top right cell, when we multiplied 3 by 4(0), we got 12; now we add zero and get 120.

Let's do the same with all other retained zeros.

Finally, we add all four numbers obtained by multiplying in the table.

Result? 943. Well, did it help?

Variety is important

Illustration copyright Getty Images Image caption All methods are good, the main thing is that the answer agrees

What we can say for sure is that all these different methods gave us the same result!

We did have to multiply a few things along the way, but each step was easier than traditional multiplication and much more visual.

So why are few places in the world teaching these methods of calculation in regular schools?

One reason may be the emphasis on teaching “mental arithmetic” to develop mental abilities.

However, David Weese, a Canadian math teacher who works in public schools in New York, explains it differently.

"I recently read that the reason the traditional multiplication method is used is to save paper and ink. This method was not designed to be the easiest to use, but the most economical in terms of resources, since ink and paper were in short supply." , explains Wiz.

Illustration copyright Getty Images Image caption For some calculation methods, just a head is not enough; you also need felt-tip pens

Despite this, he believes that alternative multiplication methods are very useful.

"I don't think it's helpful to teach schoolchildren multiplication right away, by making them learn the multiplication table without telling them where it comes from. Because if they forget one number, how can they make any progress in solving the problem? Mayan method or The Japanese method is necessary because with it you can understand the general structure of multiplication, and that is a good start,” says Weese.

There are a number of other methods of multiplication, for example, Russian or Egyptian, they do not require additional drawing skills.

According to the experts we spoke with, all of these methods help to better understand the multiplication process.

"It's clear that everything is good. Mathematics in today's world is open both inside and outside the classroom," sums up Andrea Vazquez, a mathematics teacher from Argentina.