30 Sept 30: Babylonian ‘algebra’ from Crest of the Peacock

How could one state a general mathematical principle in a time before the development of algebra and algebraic notation?

I think it's very possible to do this. As we've seen in the reading, the Babylonian explanation simply entails more wording. All the symbols and algebraic notation that we use today can be translated into English. Therefore, we could state a general mathematical principle by simply writing out the words. Mathematical symbolism, at least in my opinion, is just compact wording, so we don't have to write essays every time we do math. Kind of like how we make up a word for an animal instead of describing every time what we're talking about.

Is mathematics all about generalization and abstraction?

Modern mathematics, definitely is. All of the branches of mathematics have taken the Hilbert school of thought in terms of reaching new levels of generalization and abstraction. Before, mathematics was closely linked to physics, and we would make up math for the physics we didn't know. Then, using this reasoning, the math would be developed further, using generalizations. I think there's a quote by Vladimir Arnold, who was opposed to the idea of axioms in mathematics, that says that mathematics today is "too much Hilbert and not enough Poincaré".

how could you imagine stating general or abstract relationships without algebra?  

Lots and lots of essay-like pages, as explained in the first paragraph :).

28 Sept - Babylonian-style base 60 multiplication table for the number forty-five

Note that the ";" represents the "sexagesimal point". That is, anything after this symbol represents the 60ths, 3600ths, etc. 

Five pairs of numbers that multiply to forty-five:

4           11;15    (4 * 11.25)

3,0         0;15    (180 * 0.25)

50,0         0;54    (50 * 0.9)

7;30       6         (7.5 * 6)

0;36      1,15     (0.6* 75)

Sept 23 - Crest of the Peacock Response

The first thing that surprised me was that "the history of the Africans or the indigenous peoples of the Americas often appears to begin only after their encounter with Europe." (p.1) When we think of a mathematician, it's often that we picture a bearded old man (at least I do). We forget that mathematics has had quite a long history, and that at some point in time, Baghdad was the center of mathematics in the world. It is perhaps human nature to make ourselves important and diminish the value of other cultures. I also found it interesting how the author dug deeper into the trajectories of how mathematics was developed. I was surprised how the development of mathematics is not as Eurocentric as I had thought and that all these other cultures had an impact on its development, especially Greece. The figure on page 10 signifies how history is in general. It's never a straight line of influence, but a mishmash of a multitude and interconnected pieces influencing each other.

The "plurality of mathematical development" (p.12) was a very interesting point that sparked my attention. I never knew that Eudoxus was influenced by Egyptians. Everything makes sense now, as in this world it's hard to be one hundred percent independent. Something, somewhere, will influence us, whether it will be consciously or unconsciously. When it comes to the history of mathematics, it seems obvious now that it was influenced by many bodies of knowledge and that nothing is linear. The points the author makes, also help in teaching mathematics. If students see mathematics as a broad subject where many cultures had a part in, then it's easier for them to relate to it, especially if they are from different backgrounds. It's hard for everyone to relate to an old bearded white male. But knowing that many cultures have contributed to mathematics, and presenting some history of mathematics in the classroom, can break that stereotype.

Sept 21 - Base 60 response

(1) My speculation is that the number 60 has some kind of religious or cultural significance. For example how the number 9 is significant in many tribes, as it's the approximate time the baby is in the womb. It doesn't seem to be more convenient than 10 at a first glance. The only difference, mathematically, between 60 and 10 is that 60 can be divided by 3.

(2) We use 60 when measuring time. There are 60 seconds in a minute, 60 minutes in an hour, etc. To number of angles at the center of the circle is 360 degrees, which is 60x60. I am unfamiliar with the Chinese zodiac, and I am not sure where else we'd use 60.

(3) From just reading on the web (I couldn't find a legitimate source), I've found that the number sixty is a "superior highly_composite number" (https://en.wikipedia.org/wiki/Superior_highly_composite_number). This means that sixty is divisible by 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. Therefore, the reason why the Babylonians might have used it is because it's easy to work with fractions, if we make the common denominator 60 or 3600.

16 September - response - Integrating history of mathematics

I believe math history should be incorporated into my teachings because there are certainly some students who simply like history. Apart from that, teaching history of math and showing how math is developed is one of the key aspects to understanding where math comes from. Since the axiomatic approach has taken hold of all mathematics, students get the polished subject in the end, without seeing what it took to get there. For example, it might take fifty years to reach a good definition. When we see end result, the main question that comes to mind is "why would you have it this way", which can only be answered by studying how it was made.
A concrete example of this would be the idea of a "limit", which came from the problem of rigorizing infinitesimals. Calculus is presented to students in a polished way, with the introduction of limits. I believe that this is very misleading to the students and is very disingenuous towards the enterprise and journey of mathematics in our society.

A quote that struck me was: "history can provide role models of human activity." Most high school textbooks, seem to be focused too much on making math "useful". This might be an over-correction to the known problem of students asking "why would I need to know this". Perhaps it would be best to inspire instead. Showing small biographies of mathematicians and their accomplishments can be a way of achieving this. A lot of mathematicians grew up in poverty or had to deal with many societal pressures. Some of them who made discoveries at a young age (i.e. Abel), can definitely inspire the young generation. When leaving out the makers and most of the story of mathematics, it becomes dehumanized. As the author puts it, "mathematics is often regarded as a discipline which is largely disconnected from social and cultural concerns and influences." Therefore, I believe, just like the author, that the teaching of history is indispensable in teaching mathematics.

I wholeheartedly agree with the author. After reading the article I've already decided to take on the section "7.4 Ideas and examples for classroom implementation" for my own future classes. I think it would be great for students to "participate in a situation in which they have to decipher and understand a
known piece of correct mathematics but whose treatment is not modern. "(p.206) Not only would they grow as students, but this undertaking would help me grow as a teacher as well.

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