I like to make some comments about various remarks made in recent postings about computational tools such as abacus, slide rule, graphing calculators, as well as other less familiar tools such as those used by professional surveyors: planimeter (a device that yields the area enclosed inside a simple closed curve as one traces around the boundary).
Before going further, a colleague sent me a quotation attributed to a former president of NCTM. I am not sure about the name. If anyone happens to have first hand information, please post it. I think it is a wonderful statement in the context of "reform":
"If the past has nothing to say about the present, then the present has nothing to say about the future."
Almost anything we do in the classroom may be viewed as *reform* when it is not an exactly replica of what has been done by someone else. In this context, there are very few things that could really be viewed as *totally new*.
Before going on with comments about abacus etc., I like to say a few words about the planimeter. It began when an e-mail colleague asked me if I knew the principle behind the working of a planimeter. My response was: What is a planimeter and what does it do? I was then given a rough description. It sounded very interesting so I went to a few local colleagues and ask each of them if they had seen one. Several different descriptions were given to me by math., by physics, and by engineering colleagues. None of them very accurate. For each, I had a "theory". None of them seemed very satisfactory. I then recalled a key piece of information: it was used by surveyors and invented in the 19-th century. I then remembered that a casual tennis friend whom I had not seen in over 15 years was a partner in a local surveying firm which was first established around 1870's. I called him up and asked him outright: Do you have a planimeter? That hit paydirt. I asked him if a colleague and I could come over at some convenient time to see how the gadget worked. By this time, I had already dispatched e-mail to colleagues in Denmark and was given references where I could see a picture of this gadget. However, it did not reveal enough details. To make a long story shorter, two of us were invited over and one of the engineers gave us a demonstration. He was delighted to accommodate two "crazy" mathematicians where one (my colleague) carried a video camera to tape his demonstration--this was late in Friday afternoon when they were packing up for the weekend. They also had a PC with a software package that will also give them the area enclosed by a simple closed curve. The engineer told us that when he is in a hurry, he uses the planimeter. It takes about 10 seconds for him to get the area. By using it two or three times, he could get the accuracy down to less than 5%. For his clients, this is totally satisfactory. In contrast, it takes much longer to load up the software, set the scale, scan the chart and get the reading. Thus, many of his colleagues still use the planimeter. None of them were able to explain the mathematical theory behind the planimeter. After seeing how it worked, it was not too difficult for me to decipher the correct theory. Later, I found the same explanation in a German calculus text first published in the 19-th century. There were pictures and descriptions of other devices. The principle behind the planimeter is the area integral in polar coordinates. It is ingeneiously implemented by very simple mechanical gears. Yet, I have heard from colleagues in reforming calculus asserting that it is pointless to discuss polar coordinates and techniques of integration. Just do Riemann sums on the graphing calculator. The old fashioned planimeter is not only more convenient to use (when one does not need extreme accuracy) but encompasses advanced ideas in mathematics (as well as mechanical principles)--Green's theorem or fundamental theorem of calculus in 2-dimensions that is usually taught in a third semester calculus course. The generalizations to higher dimensions are used in understanding electricity, magnetism, and many more recent advances in physics as well as other areas of sciences and engineering.
Let me now return to the abacus.
For those familiar with the history of mathematics in China, abacus became universal around the 13-th through 15-th century and is still widely used in many countries.
Previous to that, counting rod calculation was the way computations were made in China (for at least 1,500 years). According to Sir Joseph Needham (1900-1995), Chinese mathematics reached its zenith around the 13-th century. At that point, the Chinese mathematicians were judged to be the best algebraists in the entire world. It was a mystery to historians what caused the precipitous drop in the subsequent 6-7 centuries. No doubt there were many factors related to its society, its educational system, etc. Any "theory" would likely lead to an interminable debate. This is the nature of the beast called "history". My reading suggests quite strongly that one of the culprits may very well be the
It was much faster to use than the counting rods. For example, there was a competiton in 1946 held in Tokyo between a person using the soroban (Japanese version of the abacus) and a U.S. army clerk using an electric calculator. In performing + and -, the abacus won by a large margin. In multiplication and division, it lost by a small margin.
The efficient use of the abacus required the user to remember a huge number of mnemonics. In effect, the mind did the one and two digit calculations while the abacus kept track of the record. For the skilled user, it eventually became an exercise of eye-hand coordination that bypassed thinking--think of professional typists converting a hand written document to typed pages. [One of the first class I took when I came to the U.S. as a 10-th grader was a typing course. My English vocabulary was about 200 words at the kindergarten level. By midsemster, I was typing at about 50 words per minute. The teacher was totally mystified. What he did not realize was that I followed his instruction faitufully:
Do not type one character at a time, type each word as a combination of finger movement and read ahead to the end of each line so that you can type a sentence at a time.
Since the task was to produce a copy, one bypasses any attempt to understand the content of the materials to be typed. Since my English vocabulary was quite limited, I was not distracted by the 'meaning' of the material to be typed. Much later, when my English improved, I was able to use this acquired skill to compose directly on the typewriter. Converting to word processor took much more effort because I was not able to see the process and had to remember many, many rules. The more advanced technical word processor is even more difficult. Instead of seeing the two dimensional formulas, one has to master a huge number of rules that convert these to coded symbols in lines. It is fine if one does not make any mistake. To proofread a math. manuscript is extremely painful. Even if one knows the mathematics, this is a highly tedious task. This is especially true when each user decides to customize. [There is a vague analogy with the use of graphing calculators and the use of paper-pencil calculations as well as with the planimeter.]
In contrast, the counting rods were much more complicated in that they actually were used to do "two-dimensional computations". For example, the Gauss-Jordan elimination algorithm to solve a system of linear equations was already in use via the counting rods around 100 AD. The square and cube root algorithms were already in use to solve practical problems related to area and volume computations. These were related to land surveys and commerce, division of properties, fair taxations, etc. etc.
In terms of mathematics education throughout the history of China, the basic problem seems to be that the Chinese adopted the problem-solving approach. The text consisted of a number of "real world problems" related to fair taxation, division of properties, carrying out measurements, etc. etc. While the master mathematicians were able to figure out from the accompaning variety of solutions what was going on, the common people could not. When the government used these as official texts to train the civil servants to perform their duties, the introduction of abacus simplified things tremendously. In many of their duties, it was not necessary to figure out the reasons, one could in many cases just tabulate what is needed (think in terms of the sales tax table next to many of the old fashioned cash registers). As the economy advanced, what the common people learned from the civil servants were the skills of using the abacus without the understanding of the more advanced mathematics that existed in the days when people used the cumbersome counting rods. This represented a loss of knowledge.
This together with the repressive dynasties and the internal turmoils as well as other factors spelled doom for mathematics--there was a huge military-industrial complex at a scale that was not exceeded until WWII. There were also huge construction projects, etc.
Geometry did exist in Chinese mathematics. These was plenty of practical geometry problems solved via algebraic techniques. The entire analytic or axiomatic approach as exemplified by Euclid's elements was unknown to Chinese mathematicians until around 1600 when Matteo Ricci introduced it to a court appointed master mathematician Xu Guangqi. In the book (Chinese Mathematics, a concise history, by Li Yan and Du Shiran), beginning on p. 190, one can read about the first entry of western mathematics into China.) Matteo and Xu translated the first 6 books of the Elements (on geometry), the project took place in the period: 1600-1607 (Ricci first arrived in Macao in 1582 and entered China in 1583 and he was the most influential of the Jesuit priests).
Xu's comments in the preface of the translation of the Elements contained the following statement:
[The elements] proceeds from the evident to the particular details, from doubt to certainty. What appears useless is very useful, in fact, it is the foundation of everything [postulates and common notions appear useless but in fact they are the 'foundation of everything'], it is true to say that it is the basic form of the myriad forms, the medium for a hundred schools of learning.
In the essay 'Discourse on the *Elements of Geometry*', Xu went on:
This book has four 'no needs': no need to doubt, no need to guess, no need to test, no need to change. It has four 'cannots': cannot elude it, cannot argue against it; cannot simplify it; cannot change its order. There are three 'supremes' and three 'cans': it looks unclear, in fact it is supremely clear, so we can use its clarity to understand other unclear things; it looks complicated, in fact it is supremely simple, so we can use its simplicity to simplify the complexity of other things; it looks difficult, in fact, it is supremely easy, so we can use its ease to ease other difficulties. Ease arises from simplicity and simplicity from clarity; finally, its ingenuity lies in its clarity.
In the same prefacde, Ricci wrote:
The Grand scholar [that is, Xu Guangqi] is very enthusiastic, he wanted to complete the translation but I said: No, please first distribute it. Help those interested to study it and if it proves useful, then we shall translate the rest. The Grand scholar said: Very well. If this book is to useful it should be used. It does not have to be completed by us. So we shall stop the translation and publish it.....
[Note: Book 7 through 10 are on number theory and 11-13 on solid geometry. 14 is an addendum to 13, and 15 was added on in the 3rd century AD. 7 through 10 would have been closer to the Chinese since algebra was highly developed by this time in China. The ideas expressed by Xu might then be clearer to other mathematicians in China. As I see it, the translation did not make much of an impression because there were not enough mathematicians of the caliber of Xu that really appreciated what Xu had understood. Namely, Xu was not able to prepare the teachers--who were likely to be the civil servants at that time. The elements were finally translated in its entirety to Chinese in 1857 by A. Wylie and Li Shanlan. I would like to note that Xu and Ricci had different agendas.]
The other math. texts translated into Chinese about the same time was:
The Epitome of Practical Arithmetic, (1637 AD by Matteo Ricci and Li Zhizao). Also known as the Treatise on European Arithmetic.
This text apparently treated some methods of calculation with reference to traditional Chinese math. texts which were unknown to the west. The most important part of this book is the systematic introduction of Western methods of calculation using pen and paper. In the introduction to the first chapter, Li said:
In using pen and paper to represent abacus calculations start with 1 up to 9 and according to what the number is write it down on paper. Getting to ten, do not write down ten [the Chinese character that resembles the plus sign and is pronounced roughly as 'shi'] but write down 1 advanced one position to the left and put 0 in the original position .... from ten advance to a hundred and from a hundren to a thousand, from a thousand to ten thousand and so on.
[Note: base ten system was used in China since about 14-th century BC and preceded the Arabic system by about 2300 years. The Chinese sandwiched characters for 10, 100, 1000, 10000 between the digits and omitted any character for zero. Thus, 103 would be recorded as 1 hundred 3 while 130 would be recorded as 1 hundred 3 ten. This, I believe is called the multiplicative system as opposed to the current place value system. In the abacus and counting rods, there is no necessity for a special symbol for zero since it is just represented by an empty space. Thus, there were two aspects to calculation: the part related to the technology and the part related to record keeping and the related corollaries of understanding each part.]
The 4 basic arithmetic operations of fractions were described in this treatise [Decimal fractions were already in use in China by 100 BC. The counting rod procedure naturally led to fractions and further division led to decimals.] For reasons that were not clear to me, for fractions in the Treatise on European Arithmetic the denominator is put on top while the numerator is put on the bottom. This sort of method to record fractions was not only different from the traditional practice in China, it was also different from the method used in the West at that time. This way of recording fractions gives no advantage in convenience and it also made the Chinese people reading it feel uneasy with it. However, this method of recording fractions continued till towards the end of the Qing Dynasty (around 1900) when it was corrected.
[Note: This upside down notation probably contributed to the fact that the text was not very effective. This was no doubt very unfortunate because Li included in the text many methods of calculations from ancient math. texts in China that were not present in Western mathematics. For example:
Method of rectangular arrays (Gauss-Jordan elimination). including method of positive and negative numbers. Method of finding a root of a quadratic equation (a corollary of the method of extracting roots. Li included an example of extracting an 8-th root. This presumably embodied the Horner method and the Pascal triangles, both appeared earlier in China. Arithmetic and geometric series and the formula for their sums (these already existed in earlier Chinese texts and certainly predates the fabulous story about Gauss's feat of adding 1 + 2 + .... + 100 = 5050 as a schoolboy. The point of that story was that Gauss did it without knowing that it had been done earlier, and no doubt the same applies to Gauss-Jordan elimination. Yes, one may re-discover things, but there are very few Gauss'.
Thus, although mathematics was stagnating in China during this period, it was still ahead of the West. However, this was the end of the Ming Dynasty (1368-1644) which introduced the civil service examination system that stamped out all ideas of original thinking and also had the extreme repressive policy where criticisms of official policies could lead to the death of the entire family (up to 'degree 9'). The subsequent Qing Dynasty (1644-1911) was "foreign" while the preceding Yuan Dynasty (1271-1368) was also "foreign" (the Mongol Dynasty). The two "foreign dynasties" ran the country largely through the vast bureacracy that were in existence. The only emperor of note was Kang Xi near the beginning of the Qing Dynasty who was very much interested in mathematics and other sciences. However, the pre-occupation of the time was calendar reform. The missionaries clearly understood the priority and Ricci asked and got the church to send many more missionaries versed in computing the calendar. The intention was to gain the confidence of the emperor and spread the teachings of the church. This backfired when the church declared that the Ancestral worship was to be banned. Not only did the population but the court were outraged. When a court intrigue supported by the missionary failed, all the missionaries were expelled from China. In the meantime, the country was ran by the bureaucrats following the system instituted at the beginning of the Ming dynasty. Aside from these, there were constant internal battles to gain control of power.