Wednesday, August 1, 2007

Chinese Philosophy and Math

I have been reading Fung Yu-Lan's book 'A Short History of Chinese Philosophy'. In the introductory chapter the author is talking about the spirit of Chinese philosophy. In the latter portions of the chapter he addresses why it is that Chinese philosophy is often written in brief and disconnected ways when compared with the Western philosophy and its more articulate style. In Chinese philosophy aphorisms, allusions, and illustrations are used to create this brief and disconnected style. Their use leads to suggestiveness, and Fung points out that this suggestiveness is less limiting than a more articulate approach. There are ideas that are suggested and understood that could not be articulated well without limiting the scope of thought.

It seems to me that the traditional methods of teaching math, namely memorization, drill, and practice, are more articulate (there is a certain way to solve this type of problem, and that method should be used to finish some set of exercises). Once a student has completed the exercises they should have a grasp on the mathematics.

A problem that is to be used to teach mathematics cannot be this articulate. There needs to be more suggestiveness, thus allowing the students to use whatever mathematically accurate approaches they see fit. Having a classroom discussion on the ideas that come from the various solutions will help the students define certain mathematical concepts in ways that will be beneficial to themselves.

Fung quotes a passage from the Chuang-tzu, a significant book of philosophy, which may help distinguish between ideas suggested and methods articulated, and how ideas suggested can be the more powerful of the two.

"A basket-trap is for catching fish, but when one has got the fish one need think no more about the basket. A foot-trap is for catching hares; but when one has got the hare, one need think no more about the trap. Words are for holding ideas, but when one has got the idea, one need no longer think about the words." (pg 12)

Later on Fung says that 'words are something that should be forgotten when they have achieved their purpose. Why should we trouble ourselves with them any more than is necessary?' (pg 13) It seems to me that the traditional methods of teaching math are having our students deal with the words. The formulas they memorize and the specifically formatted equations they learn to solve are the words used to describe the ideas of mathematics. Our students, however, don't get to experience the ideas of the mathematics because we are spending so much time focusing on the words.

The last paragraph of the first chapter of Fung's book says the following.

"Kumarajiva, of the fifth century A.D., one of the greatest translators of the Buddhist texts into Chinese, said that the work of translation is just like chewing food that is to be fed to others. If one cannot chew the food oneself, one has to be given food that has already been chewed. After such an operation, however, the food is bound to be poorer in taste and flavor than the original."

Do we assume our students are not capable of chewing the food of mathematical ideas? Are we continually chewing this food for them? I think we are. It is no wonder that our students are not enjoying math classes as much as we would like them to, or even as much as they would if we would let them.

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