Babylonian ‘algebra’ from Crest of the Peacock

 I can see that a lot of the Babylonian word problems have to do with agriculture and farming (at least from the examples we have seen so far), that the "variables" they used were quantities they use in farming. I think it has to do with the kind of social background at the time. The Babylonians were using something they see a lot in their daily lives. So to state a general mathematical principle, they had to resort to using terms like "ush", "sag" and "lagab". These were the "variables" resorted to using when there was no proper algebraic notations.

I do not think that mathematics are all generalization and abstraction. The Babylonians probably developed these problem solving skills (including algebra) when they faced real problems in life ( like food distribution, area of farms etc...). They had to come up with systems and ways that allowed them to solve these problems. So the origins of all these algebraic expressions and equation solving systems might well be some really specific life problems they encountered. And from these very specific problems, they developed systems to solve them and later on generalized them.

If I were to work on different areas of mathematics without proper algebra it would be very tough. I was taught algebra when I was really young and a lot of the mathematical concepts I learnt later on are heavily dependent on algebra. Without algebra, we may have to resort to words and phrases of the English language. An example of this is Calculus: Instead of using y'', we may have to say "the rate of change of the rate of change of the displacement of the car". As you can see for something tangible like length, we can just use a word for it, but when it comes to abstract concepts like the second derivative, there is no easy one-word description to represent the quantity. That's why I am really impressed that the Babylonians did play a part in the development of Calculus. 

Create your own Babylonian-style base 60 multiplication table for the number forty-five

Here are the 5 pairs of numbers which multiply to 45 (base 60)

4        11,15

40      1,7,30

12      3,45

16      2,48,45

32      1,24,22,30

The pair 50        54 also seems interesting because the place value is missing so when you did the division, the quotient is actually a bigger number.

Crest of the Peacock -- an introduction to non-Eurocentric math history

 The first thing that stood out to me is "history is written by the winners". What this means is history was written by winners of wars and conflicts. You can see that our current mandatory education system was established around 18th and 19th centuries and this was brought about by the industrial revolution in the western world. So textbooks were mainly written by Europeans. Therefore, you see that the material were very eurocentric. I wonder what would have happened if textbooks were written maybe 1000 or even 2000 years ago? Would we see more contributions from the rest of the world like Arabs, Indians and Chinese?

I was also mesmerized by figure 1.4. It is a very interesting map which shows the development and transmission of mathematics around the world. Interactions among the different civilizations like wars, pilgrimages and trades helped the transmissions. Before I read this article, I knew there were inter-civilizations interactions but I never knew that it was this extensive as shown in figure 1.4.

Another thought I had after reading the article was that concepts like Pythagorean theorem, algebra, equations solving are so common place in today's schools. However, these were discoveries made by our forefathers, after putting countless hours or even their whole lives into them. Reading histories of mathematics helps us appreciate more what people thousands years ago did for us (does not matter what civilization they were from) and that we do not take these for granted.

Base 60

 Speculative phase: I think base 60 was used due to time: 60 seconds in a minute, 60 minutes in an hour.

- 60 is also divisible by more numbers than 10? 

- There are 12 zodiacs in the Chinese calendar. I am not sure whether that is related to base 60 or not

- I have also seen angle expressed in the form of something like 20°48'. So to convert it to degrees or the modern way of it, we take 48 divided by 60 = 0.8 and that is the decimal places. So it becomes 20.8°

Research phase:

- 60 is a the smallest number that is divisible by all numbers from 1 to 6. I guess this is a very convenient number to use especially when we are talking about time.

- In fact, the Chinese also uses a sexagenary (60 years) cycle to name their years. There are 10 heavenly stems and 12 earthly branches. Of course the names are all in Chinese but for simplicity sake you can think of the 10 heavenly stems as A, B, C, D, E, F, G, H, I, J (I actually learnt them in Hong Kong when I was still in elementary school but it was so long ago I totally forgot about them) and the 12 earthly branches as 1,2,3,4,5,6,7,8,9,10,11 and 12. The 12 earthly branches are also the 12 times of the day in the Chinese time, each earthly branch represents 2 hours, 12 of them make up the 24 hours in one day The way the Chinese name the years looks like this: A1,B2,C3,D4,E5,F6,G7,H8,I9,J10,A11,B12,C1,D2.... and with this pattern we are back to A1 after 60 years. Even up until today, there are still lots of Chinese using this system. Chinese also like to name significant events after the name of the year. A prime example is the the Xin hai Revolution in 1911 that overthrew the Qing dynasty (the last imperial dynasty of China) and established the Republic of China. Here Xin is one of the heavenly stems and Hai is one of the earthly branches.

Why Teach Math History?

 Our first reading: Why teach math history?

Pre Reading thoughts: I think we should teach Math history to a certain extent. I would do it by topic. For example, when we are learning area/perimeter, I may go over the history of pi with my students. My initial thought is if we are spending too much time on the history of pi, we will have less time for the actual Math material.

The first thing that catches my attention is that Math is perceived as part of the western culture. However, the history of Math allows us to become more aware of less known cultures. This not only allows students to learn Math, the teaching extends to other subjects like history, geography and social studies.

The second thing is “shortcut” hand method of doing multiplication of two numbers between 6 and 10. It is through these interesting little tricks that trigger students’ interest in Math. As a teacher candidate, I will definitely be implementing these in my lesson plans. 

Finally, I agree with the idea of implementing the history of Mathematics in school to a certain extent. The history of Math is so rich that it gives us a lot of Math problems to work with during classes. This can also show students that history is not just names and dates. It can be very fun too!

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