On the Theory of Mental Arithmetic

On the Theory of Mental Arithmetic By Image of Zhusuan

The abacus, invented in china, is a valuable scientific and cultural heritage left over by Chinese forbears. As early as Wan Li time, Ming Dynasty, the abacus was spread to overseas and was attached high importance to its value especially in calculating field. Before the 1970s, the reckoning levels by the abacus in Japan and South Korea are higher than that of Jilin Province. But only after ten years, the reckoning level by Zhusuan in Jilin Province has come up to the most advanced world standard. On the 12th competition of Reckoning by Abacus of Jilin Province in October, 1995, each record of reckoning by the abacus was about 10 times higher than that of whole country 15 years ago. This achievement was achieved because the officials, the specialists and the workers of our province engaged in Zhusuan not only realized the importance of the theory but carried it out in practice. They made use of the power of all levels of organizations to propagate it, encouraged the teachers and the students to experiment on it and gave awards to the advanced workers engaged in Zhusuan.

I’d like to talk something about the contents of Mental Arithmetic by Image of Zhusuan, instructing method and beneficial function of the theory:

I. The Teaching Contents of Mental Arithmetic By Image of Zhusuan

Mental Arithmetic by Image of Zhusuan is a new development of the traditional Chinese reckoning by Suanpan.

The addition-subtraction method of Zhusuan means getting rid of the complicated 52-sentences-pithy-formula, making the use of the relationship of complement and assembling number, carrying out the operation of addition-subtraction. This the 52-sentence-pithy-formula of Zhusuan can be summarized as three rules:

1. When it needs adding, look at the beads close to the frame. If the beads are enough to add, do it. When it needs subtracting, look at the beads close to the beam. If the beads are enough to subtract, do it.

2. When adding, if the beads under the beam are not enough, add five and subtract assembling numbers. When subtracting, if the beads are not enough, add assembling numbers and subtract five.

3. When adding, if the stick is full with ten, carry one and subtract the complement. When it needs subtracting and if the beads on the stick are not enough, move back one and add the complement. So, as long as we understand the three rules, combine counting with it and then exercise by two hands the continuous addition and subtraction of the fixed number of 1, 2, 3, 4, 5, 6, 7, 8, 9, we will grasp the addition-subtraction method of mental arithmetic by image of Zhusuan.

The Multiplication-division method is a method with which to teach the Unit’s Place Rule, ‘Carrying Rule’ and to apply the Unit’s Place plus carried number. The method makes children to command the skill of one-digit-number timing poly-number and calculating the product by brain so that people can simplify the multiplication-division method into the addition-subtraction operation of the whole mass and calculate the product and the quotient.

From what discussed above, we can find that the main part of Mental Arithmetic by Image of Zhusuan is develop from the addition and subtraction of abacus. This algorithm is suitable for children to learn and master. I think there are four characteristic of Mental Arithmetic by Image of Zhusuan.

(1) Easy to learn.
The children of 4 to 10 years old can master the skill easily and the instructors only instruct children under the command of textbook, even if the instructors may not be proficient in reckoning by the abacus.

(2) Interesting to learn.
The tool of Mental Arithmetic by Image of Zhusuan is the abacus. As children see it the abacus is both toy and teaching aid. Children can study the reckoning by the abacus in the process of play.

(3) Scientific feature.
The best age for studying the Mental Arithmetic by Image of Zhusuan is at the initial stage of elementary school. So long as pupils exercise everyday, they can master the following contents of reckoning by abacus within to years including the four fundamental operations of arithmetic addition, subtraction, multiplication and division with the method of mental arithmetic by image of Zhusuan. The pupils can receive the result of study but not forget and be benefited all their life.

(4) Practical feature.
When mastering the skill of Mental Arithmetic by Image of Zhusuan at initial stage of elementary school, quite a lot pupils can operate the four fundamental operations of arithmetic without any paper or other tools and can calculate the answers immediately after seeing the questions. This method is very practical in calculation in elementary and middle school.

Mental Arithmetic by Image of Zhusuan is a great contribution to human in calculation filed by Chinese. Especially, in the time of computer, the creation of Mental Arithmetic by Image of Zhusuan can also improve the intelligence of human and international exchange.

II. The Instructing Method of Mental Arithmetic by Image of Zhusuan

From 1983, the Association of Zhusuan of Jilin Province edited Instruction for Mental Arithmetic By Image of Zhusuan by Six Steps according to the practice for many years and combining the theory of psychology and physiology, deciding the contents of the instruction system, including : 1. basic knowledge of Mental Arithmetic by Image of Zhusuan; 2. addition and subtraction of Zhusuan; 3. addition and subtraction of Mental Arithmetic by Image of Zhusuan; 4. multiplication of one figure; 5. quick multiplication of mental and abacus combination; 6. quick division of mental and abacus combination. The emphasis of this instruction system is : “Instructing pupils in accordance with their aptitude; moving beads by double hands; guiding by abacus; memorizing by image”. The basic purpose of this system is to inspire children’s intelligence.

Five steps are explained here in order for pupils to study and for instructors to follow:
(1) “Instructing pupils in accordance with their aptitude”.
This step means the instruction and exercise of basic knowledge of Mental Arithmetic by Image of Zhusuan. The ages of pupils are between 4 and 7 years old. They should be instructed by less theory but more practice. During the play children can master the basic knowledge by interesting guiding. By many times of exercise, to complete the instruction of “moving beads by double hands”.

(2) “Following in order and advancing step by step”.
This step indicates the instruction and exercise of addition and subtraction for quick calculation. There are two meanings : with the growing of children, their psychology activities became more experienced than ever. They are conscious of their activities instructed by their purposes. The transform from thinking in images to abstract thinking occurred in the children’s mind. Children should be instructed from the easy to the difficult and the contents of instruction should be increased with the growing of children. kthe progress should be suitable for children to avoid losing interest due to pupils thinking ability. The instruction way of “proceeding from the easy to the difficult is suitable at every stage of exercise”.

(3) “Guide by abacus”.
This means the exercise of addition and subtraction of brain calculation. That is to move the beads by double hands on the abacus. By this way, the speed of reckoning on the abacus should be limited within the scheduled time. The image of the change of figures and beads should be formed in the brain. At the same time the “image memory” should be reflected. The speed of Mental Arithmetic by Image of Zhusuan, the accuracy and the skill level depend on the continued time of “image memory” in the brains of pupils, the numbers of beads and so on. “Moving the beads by double hands” has special function to cultivate children’s potentials and intelligence, that is already proved by psychologist.

(4) “Uniting four exercises”.
The “four exercises” include moving beads, writing by pen, seeing and reading, which are mainly used in the training of one-figure mental calculating multiplication. The combination of Zhusuan and Mental Arithmetic by Image of Zhusuan means that the learners move the result of Mental Arithmetic by Image of Zhusuan to abacus and then write down on the paper and finally recover the abacus. The purpose of this exercise is to strengthen the skill of Mental Arithmetic by Image of Zhusuan and to improve the frequency of moving beads; The combination of writing and Mental Arithmetic by Image of Zhusuan means that the result of mental reckoning should be written down by pen over and over again to improve the speed of numerals written and to strengthen the skill of mental calculation. The combination of seeing and mental calculating means that the instructors write down the questions, and the learners look at the questions and calculate. This is to train the reaction means that the instructors read out the questions and then the learners reckon by brain and write down the answers. This is to train the reaction of sense of listening and brain of the learners.

(5) “Comprehensive Exercises”.
This is mainly used in the instruction and training of the quick multiplication and division by the combination of Zhusuan and Mental Arithmetic by Image of Zhusuan. The core of multiplication and division of multi-digit by Mental Arithmetic by Image of Zhusuan is quickness and accuracy. Quickness means that the frequency of moving beads at the basic knowledge stage should be quick; Accuracy means that the accurate product of addition and subtraction is needed. The comprehensive exercises are based on the contents above-mentioned. If learners had a foundation at steps above-mentioned, they will have a good result at this step.

Six Steps Instructing Method of Mental Arithmetic by Image of Zhusuan divides the contents of Mental Arithmetic by Image of Zhusuan into six steps. The first grade primary school pupils spend one hour everyday to study the first half of this method, then they will complete it about one year and will be able to get a satisfying achievement. By this method, the teaching of addition and subtraction in the 6th grade of primary school can be finished in advance as well as the “abacus-mental image and image memory” can be formed in children’s brain. High success rate and great achievements have been received in training the best learners through the Six Steps Teaching Method of Mental Arithmetic by Image of Zhusuan and Moving beads by Double Hands in Reckoning by Abacus.

Since 1983, hundreds of top pupils have been trained in Jilin Province. From 1988 to 1993 only five years, large numbers of excellent pupils with world advanced level were trained. These pupils include Li Xin, Li Yinji, Luo Bingbing, Sun Xiaoguang, Zhang Lihong, Zhu Qing Ying, Yuan Honghua and Hou Lijuan. Up to the present, Jilin Delegation of Abacus-mental Quick Calculation has won seven champions in National Competitions and eleven champions in National Minority Competitions. Especially in the selecting competition of National Zhusuan delegation in May 12, 1994, Jilin Delegation got a outstanding achievement. Among the six members of the new National Zhusuan Delegation, there are four from Jilin. Both of the first and second players are from Jilin province. During May 15-18, 1994, The First National Olympic Mentality TV Competition organized by CCTV was held in Hangzhou. Among ten competitors five are from Jilin Province. After a heated competition, the gold medal was won by Li Xin, 12 years old, from Jilin Province, and the silver medal was won by Li Xin’s male companion Sun Xiaoguang, 13 years old.

In order to popularize the Mental Arithmetic by Image of Zhusuan to make learners understand easily and instructors teach conveniently, the Association of Zhusuan of Jilin Province designed the “217 Engineering” for these purposes. The Engineering is favorable to examination, comparison and competition. The study time of each step, training contents and requirements for quality are more specified in the “217 Engineering”. During the first and second grades in primary school, pupils will spend two years to finish the whole contents of Six Steps Teaching Method of Mental Arithmetic by Image of Zhusuan that is called 2; During the third grade, pupils will spend one year to consolidate what they have learned, that is called 1; During the three years, 7 examinations will be held, that is called 7. So the engineering is titled “217 Engineering”. The contents of “217 Engineering” include: during the first half school year of first grade, the first and second steps of the Six Steps Method should be completed, and then the operation of addition and subtraction will be examined in winter vacation. The third step should be completed during the second half school year in the first grade and the addition and subtraction of mental calculation will be examined in the summer vacation. During the first half school year of the second grade, the fourth step should be finished for the examination of the addition and subtraction of mental calculation and one digit multiplication in winter vacation. The last two steps will be finished in the second half school year of the second grade and then to examine the multidigit multiplication of Mental Arithmetic by Image of Zhusuan in the summer vacation. During the third grade in primary school, pupils will spend one year to consolidate and review what they have learned and will take part in three examinations. The way of examination and the standards of acceptance are as follow:

Firstly, basic knowledge of Mental Arithmetic by Image of Zhusuan. It will take three months for basic knowledge amounting to 36 class hours. Requirements : continuous adding and subtracting of fixed number 1 and 5, 150 times per minute; continuous adding and subtracting of 2, 3 and 4, 120 times per minute and continuous adding and subtracting of 6, 7, 8 and 9, 100 times per minute.

Secondly, the addition and subtraction of Mental Arithmetic by Image of Zhusuan taking three months amounting to 36 class hours. The first examination of “217 Engineering” will be carried out to examine the addition and subtraction in ordinary 6 degrees. The passing rate of 30% is required.

Thirdly, addition and subtraction of Mental Arithmetic by Image of Zhusuan will take half a year amounting to 72 class hours. The second examination will be carried out to examine the addition and subtraction in ordinary 5 and 6 degrees. The passing rate in 5 degree of 20% is required and in 6 degree the rate of 80% is required.

Fourthly, the multiplication of mental calculation in one digit, taking half year amounting to 72 class hours. The third examination will be carried out to examine the addition and subtraction of ordinary 5 and 6 degrees. The passing rate of 80% in 5 degree and 100% in 6 degree are required. 60 multiplicative questions should be reckoned up within ten minutes, in which the multiplicands are 4 to 5 digits numbers and the multiplier are from 2 to 9. All these questions belong to mental calculating multiplication.

Fifthly, the multi-digit multiplication and division of “Mental Arithmetic by Image of Zhusuan”. It will take half a year amounting to 72 class hours. The fourth examination will be carried out to examine the addition and subtraction in ordinary 5 degree and the multiplication and division in ordinary 6 degree. The passing rate of 100% is required for addition and subtraction in 5 degree and 30% for multiplication and division in 6 degree.

Sixthly, one year is needed for learners to consolidate and improve what they have learned. One class hour each week amounting to 48 class hours. The fifth examination will be carried out in the first half year. The multiplication and division in ordinary 5 and 6 degrees and the examination of addition and subtraction in ordinary from 1 to 4 degrees can be selected by learners. The passing rate of 20% of multiplication and division for 5 degree is required and 80% for 6 degree is required. The sixth examination will be carried out in the second half year. The multiplication and division in ordinary 5 and 6 degrees and the examination of addition and subtraction in ordinary from 1 to 4 degrees can be selected by learners. The pass rate of 80% for multiplication and division in 5 degree and 100% for 6 degree are required. In the final examination, learners who pass the 5 degree examination may apply for every other degree and learners who fail in 5 degree multiplication and division examination are required to take part in the making-up examination. According to the requirement, all of learners should surpass the standard of ordinary 5 degree. These learners will obtain the certificate of the “Republic of China” issued by the Association of Zhusuan. The certificate is valid lifelong. After the learners grow up and become cashiers or accountants, they needn’t to take part in the Zhusuan examination. Through the practice for more than ten years, it has been proved that Six Steps Teaching Method of Mental Arithmetic by Image of Zhusuan is the best material for children to study Zhusuan. The Mental Arithmetic By Image of Zhusuan Method can be more spread to children. The combination of the Method and “217 Engineering” will give a greater contribution to improve the calculating quality of people and carry forward the Chinese traditional culture.

III. The Cultivating Intelligence Function of Mental Arithmetic By Image of Zhusuan

It has been proved by practice that this education is provided with the function of cultivating intelligence for children. Recently, the Association of Zhusuan in Jilin Province, the Education Commission of Jilin Province and Psychics Testing Group of Northeast Normal University together took a test on children. Using the Wecosler Children’s Intelligence Table, the group tested some children from the First Kindergarten of Jilin Province Government and the Third Primary School of FAW. The results of random sampling indicated that the language intelligence, operation intelligence, general intelligence, memory, attention, will and oral expressing ability of children who have learned Zhusuan are much better than those children who have not learned.

There are five functions which are beneficial to children:
1. Promoting the early development of children’s intelligence, The mode of thinking of children is mainly thinking in images. Children are full of curiosity but lack of attention and are interested in changing to specific things. Abacus is the tool of Mental Arithmetic by image of Zhusuan, but for children, it is also a changeable toy. Psychology indicates that the intelligence developing in the children period is with large potentiality. In the training of Zhusuan, the eyes, mouths, hands, brains, pens and beads are used alternatively, and everybody can operate the abacus by hands. By these actions the nervous system of sense organs of children are often in the active conditions that the children’s interesting in study will be hindered. By the unremitting training of Zhusuan, children’s willpower can be fostered as well as the good studying habit for children can be shaped. Through taking part in the competition. Children can participate in the social activities to mould their temperament.

The education of Mental Arithmetic by Image of Zhusuan, cannot only foster the children’s temperament, to make children more clever, but also enhance the memory of children. Comparing with those children who didn’t study Mental Arithmetic by Image of Zhusuan, children who have studied Mental Arithmetic by image of Zhusuan can save half time in their school subjects and can achieve good records as well. For example, Mao Dongjie, a pupil of Shiyan Primary School of Jingyu Country, has gained a good achievement. Before studying Mental Arithmetic by Image of Zhusuan, she is at the 18th position in her class in the study of general knowledge courses. After studying Mentla Arithmetic by Image of Zhusuan for one year, she became the bet pupil in the whole grade. Her father, a middle school teacher, found his daughter’s memory, attention and the ability to analyses questions be improved remarkably. Mao Dongjie is not individual, most pupils who have learned Mental Arithmetic by Image of Zhusuan like her.

2. Improving children’s memory.
After certain conception of space and things, depth and range of thinking is strengthened, children’s memory can be improved remarkably. Through the process of mastering the bead (audio-visual) and mathematics (abstraction) children can form a transition from thought in Image to abstract thought. So that the ability of children’s abstract thought can be effectively developed and raised. A State Information Competition was held in the primary school in Longshan District, Lisoyuan City. Some pupils with good level of Chinese from Centre School could not recite the questions. The other pupils who have studied Mental Arithmetic by Image of Zhusuan recited the questions with about thousands Chinese characters quickly, and finally won the championship. It id introduced that after entering middle school, the pupils who have studied abacus are not only good at mathematics, but also good at reciting English words.

3. Lightening the calculation burden on pupils.
To cultivate the calculating ability is one of the main tasks of primary school. It influences the cultivation of other abilities and the study of all subjects and is also a difficult point for teachers. But the difficulty can be easily solved by the combinating of reckoning by abacus with mathematics. During the first and second grades of primary school, pupils can finish the four fundamental operations of arithmetic so long as they spend 30 minutes to study reckoning by abacus. Their abilities to calculate exceeded the requirement of Teaching Curriculum greatly. Because of strong abilities to calculate, quite a lot pupils can reckon the questions without using paper as well as can save half times to finish the homework.

The amounts of calculation in arithmetic in primary school and in algebra, trigonometry, geometry, physics and chemistry in middle school are quite a lot, about 40-50% of the whole contents of science subjects. In their studying of Mental Arithmetic by image of Zhusuan, pupils spend one hour to exercise abacus, thus, they can operate the four fundamental operations of arithmetic skillfully. Chang Liang, a middle school student graduated from the 25th Primary School of Changyi District, Jilin City, said : “Comparing with his classmates, he only spends one tenth of time to calculate the science subjects, during the time saved he can study English and play electronic organ”. View the things as a whole, the education of Mental Arithmetic by image of Zhusuan can lighten the calculating burden on pupils greatly although more time is needed everyday in primary school.

4. Improving the school records comprehensively. A united examination was organized by the Education Bureau of Gong Zhuling City. In the examination, there were 14 primary schools from Er Shijia Town. The average marks of mathematics and Chinese are 81.8 out of 100, but 88.6 marks for pupils who have learned abacus, 6.8 marks more than ordinary pupils; The passing rate of whole town was 97.3%, but it is 100% for pupils who have studied Zhusuan, 2.7% higher than the ordinary pupils; the excellent pupil rate in whole town is 42.5%, but 59.7% for pupils who have studied Mental Arithmetic by Image of Zhusuan. At Gongnong Township, Liaoyuan City, there are 12 classes in second grades, in which four classes were experimental units. During the united examination of whole township, the four experimental classes won the first four positions. The excellence and pass rates of experimental classes were respectively 85% (average records of two subjects were more than 95 marks) and 100%; Compared with it, the excellence and passing rates of ordinary classes were respectively 75% and 90%. Thus it can be seen that the education of Mental Arithmetic by Image of Zhusuan can promote the school records indeed.

5. Training Super-Children.
Some researchers from 15 areas of whole country took part in the project about super-children. Researchers consider that super-children are provided with follow characters : (1) strongly craving for knowledge; (2) keen insight; (3) wide-ranging and focused attention and excellent memory; (4) shape thinking and creativity; (5) having self-confidence and strong willpower. In the practice of Mental Arithmetic by Image of Zhusuan training, we realized that most children who have studied the abacus reckoning have the characters of super children to a different extent. They obtained the achievements by studying hard not depending on innate talent. It is proved by practice that to foster super-children, studying Mental Arithmetic by image of Zhusuan is a very effective way.

The above-mentioned are my personal understanding on the instruction, training and research of Mental Arithmetic by image of Zhusuan in kindergarten and primary, from now on, I will continue the practice of education of Mental Arithmetic by Image of Zhusuan. Just like Confucius said : “For study, if you do not catch the opportunity, it will be lost”. I’d like to make my contribution with colleagues to children’s intelligence development.

Source: Senior Accountant Liu Shantang (China)

- Executive Director of Calculation Method / Principle Committee of China Zhusuan Association

- Chief Coach of National Training Team

- General Secretary of Jilin Zhusuan Association