Arithmetic Thoughts and Uses
“Grandpa, we have two very interesting visitors in class today.” Nancy greets Grandpa cheerfully the minute she gets home.
“You are really lucky.” Grandpa responds, “Tell me who they were and what happened.”
“They are Mrs. Shaw and her university professor. He is Dr. Chien who wrote about arithmetic teaching methods and Chinese culture. Mr. Shaw told us the professor was a very learned scholar in Hong Kong.”
“Mrs. Shaw must be very nice?”
“Yes, Grandpa. She said she used to be a teacher too. But, she now stays home to take care of her family.”
“So what did Dr. Chien tell you?”
“He told us that the ancient Chinese had a lot of inventions in mathematics used in many areas, such as calculating changes in the universe, making predictions of all kinds, oceanic navigation and games. For example, there is the Lo Shu Square with 9 numbers arranged in a matrix of 3 rows. The numbers add up to 15 horizontally and vertically. Here, I have copied them down. The numbers are: 4 9 2, 3 5 7, and 8 1 6. It is very neat.”
“That is fascinating! I wonder how the ancients discovered it.” Grandpa shows his curiosity.
“Dr. Chien said it was the legendary Emperor Yu who discovered it while trying to tame the big flood. He saw it on the shell of a huge tortoise.”
“I wonder how it is useful.”
“Dr. Chien said that many scholars explored what could be derived from those numbers. Eventually, they developed the concept of Yin and Yang, two basic forces that determined the stability and change of cosmic bodies. Grandpa, what is yin and yang?”
“I don’t really know, Nancy. But, I remember my Mum told me that everything in life has two sides, like negative and positive, good and bad, helpful and obstructive, and so on. We should learn to accept the negative and seek for positive ways of countering it. My Mum had experienced a number of wars and famines in her life. But, she survived them and led a useful and peaceful life. I remember she used to have five coins, each having two different sides. Whenever she met with a difficult situation, she would ask me to hold the coins in my hands, shake them and throw them on the table. She then read the way the coins showed, like how many showing the yin side and how many showing the yang side, as well as how the coins spread out in the four directions. Then, she would meditate for a while and make her decision on what to do. I wish I had learned her way of calculation. I regret I did not.”
“Grandpa, that should be a lot of fun. It is too bad that you did not learn it. I would love to shake the coins for you.”
“What else did Dr. Chien show you?” Grandpa asks.
“He told us that mathematics is more than regulated calculations. They are ways of thinking and imagining, the ways we human beings understand and organize the world. He played a game with us. First, he asked Mr. Shaw to show us how to divide 360 by 5. When we got the result of 72, he told us that in Chinese culture, people used to believe that there were 72 trades or occupations. But, it was just a general belief to suggest that there were MANY trades. It had little to do with the actual number of 72.”
“That’s indeed interesting.” Grandpa responds quickly. “I’ve always thought the same. I had believed that the type-setting skill I learned from my uncle was one of the 72 trades.”
“Dr. Chien played a game with us by asking our fathers’ occupations, and what trades we knew. We did it with lots of fun.”
“What did you name?”
“I said type-setting and computer programming.”
“What about the others?”
“My friends had named some old trades, such as blacksmith, barbering, basket weaving, shoe-making, brick-making, tailoring, and farming. The new trades included bus-driving, piloting, dental assistants, hairdressing, container management and cosmetic surgery. But, all we got amounted to 51 only.”
“What did Dr. Chien say?”
“He told us that numbers are not definitive things. They are often used to express quantity, such as large and small, plenty and few, suggestive or emphatic, and so on. He said that the 72 number originally came from using the 5 cosmic elements of metal, wood, water, fire, earth as a divider for the 360 days of a year. Then, people used it habitually to denote the concept of many. Even the Monkey King Story described the extraordinary power of the magical money of doing 72 transformations. When people refer to the students of Confucius, they say there were 3000 students of whom 72 had become
outstanding scholars. But, there was no record in this regards. There was mention of about 20 of his distinguished students only.”
“What about the sayings of Confucius?” Grandpa asks.
“Dr. Chien told us that when Confucius said: ‘In the company of three persons, one could be my teacher’ he meant in the company of many persons. Likewise, when he said: ‘act after you have deliberated three times’ he meant deliberated thoroughly and not three times. In the Chinese culture, we consider 1 to be basic or few, 2 is more than 1 but not too much more. So we use the number 3 to express the concept of many or plenty. To say a lot we use the number 9, which is 3 times 3. In daily life, when we agree with an idea or someone, we say: ‘I agree 10 times strong’. When we wanted to express an emphatic agreement, we would blow up our expressed feelings by saying: ‘I agree with 120,000 times, a real exaggeration.”
“You guys really had a 12 times interesting lesson. How did Dr. Chien conclude the lesson?” Grandpa asks with humor.
“He told us that mathematics is not just divided into two branches, like pure and applied mathematics. It is not limited to the usual operations of arithmetic, algebra, geometry, or other forms of higher mathematics. It is both the human exploration of abstract concepts and the application of materials for utility, such as engineering or economics. He urged us to open our minds to human ingenuity and fantasy, always to put aside our restrictive opinions in order to reach far into time, space and the heart of fellow human beings. In the end, he told us the story of the frog living at the bottom of the well who insists that the sky is just two strokes from one end to the other. It accused the frog living on open ground that it was lying when it said the sky was so big that even one million strokes could not reach the border boundary of the sky. Mr. Shaw thanked Dr. Chien for an inspiring lesson and invited him to visit again.”
“That is a very good conclusion. I know the Frog in Well fable. Dr. Chien is really wise to use a simple story to teach a very difficult topic. Nancy, would you be interested in studying mathematics when you go to university?”
“Dr. Chien asked us that question too. Matthew , David and Elaine raised their hands to say they would. I did not join them because I am not sure if I will. Grandpa, I am taking Dr. Chien’s advice to keep myself open to other options.”