Exploration with Egyptian Division

Although I suppose I could think longer about it, I am doubtful that this Egyptian could work with all whole number divisors, unless there are additional techniques that could be added. I tried exploring with 1/3 and 15 divided by 7, and it didn't seem to work out for me. I'm looking forward to class tomorrow to see if I'm missing something. I also don't think it would work out for all even number divisors, because if you take 6 and divide it by 2, it becomes 3 pretty quickly and I believe that the same problem happens. 

Here's some of my math thinking regarding this topic: 

Babylonian Word Problems

 

I do not think that concepts, such as practicality, generality, abstraction, pure, or applied math, rely on contemporary algebra. I think that the Babylonians show us that these concepts are independent of contemporary algebra. As I wrote that last sentence, I thought 'what about pure math'. I am actually not sure whether the Babylonians showed that algebra is not necessary for pure math, but my gut is telling me that it probably isn't. However the Babylonians seem to demonstrate that these qualities of math have been around for a long time. It is surprising to think that 'impractical' word problems have been around so long. I am still kind of in disbelief that math problems seem to be introduced shortly after the concept of math itself. Furthermore, it is even more surprising that this tradition of learning math has been continued on for millenniums. This historical knowledge speaks to how practicality and abstraction seem to naturally come out of math, even for Babylonians. In comparison, to other historical groups, such as the Greeks, it is interesting to think why Babylonians seemed to prefer this more generalizable and applied math, whereas the Greeks seemed to prefer more pure or art math. My speculation is that this difference may have to do with the values of the culture at the time, as my preconceptions of Babylonians is that they were rather industrial and the Greeks were famous for their art and philosophy. 

Into to Crest of the Peacock

I think the first thing that surprised me with this reading is that I had never really consider how Eurocentric math history tends to be. I suppose that I never explicitly associated math history with being Eurocentric, but a closer analysis shows that it is often biased towards Europe and does not give fair credit to other origins. As I was reading, I kept on debating this idea in my head. Is there still a Eurocentric bias? I suppose there is, but as I'm reading this paper I am going against this bias and nothing in this paper was truly shocking, so I'm unsure. I suppose that there are definitely more Eurocentric characters in math history, although I am bad with remembering historical figures so maybe I am accounting for this bias. 

Relatedly, the second thing that really stood out to me was the comparison of Figure 1.1 and Figure 1.3. Figure 1.3 was an expansion of a small part of Figure 1.1 and it did partially convince me that at one point math history was definitely Eurocentric. It also made me think that I suppose I would likely never question this lack of information in Figure 1.1, if the author had not pointed it out. I think I am also drawn to Figure 1.3 as it quite visual and shows the relationship of math in different areas. It kind of reminds me of graph theory. I also really enjoyed Figure 1.4, but Figure 1.3 was the first to surprise me. 

The third thing that surprised me was the (unverified) claim that calculus may have been discovered 250 years before Newton and Leibniz discovered it. I suppose this claim support the idea that math history has been quite Eurocentric. I am quite curious as to why so much historical information is missing. I wonder if it is just the way it has been passed down between generations. It kind of makes me question historical knowledge in general.  

I also wanted to add that I was also somewhat pleasantly surprised to think about what practical lessons I could take from this history? I suppose the world is so much more interconnected but it is interesting to read about how math would travel and grow independently regardless of a lack of internet. 

Why Base 60?

 

As I was thinking about this the first thing that I thought of was how you might count to 60 on your hands. What else would be a convenient reason for a base? Maybe they split each finger (excluding the thumb) into thirds and counted by 5's. In this case you could count 60 on one hand. It is different from 10 in that is quite a bit larger and there are less patterns that can be found later on. From 60 to 3600 is a large gap compared to 10 to 100. 

Maybe the choice of 60 has something to do with astronomy or physics. I'm not familiar with the history but 60 tends to be a significant number. It is the number of seconds in a minute, the number of minutes in an hour, as well as the number of minutes in a degree. Maybe there is some connection there. I'm pretty sure I remember taking a psychology course where your range of vision is only so many degrees so maybe that is also connected. Maybe they measured how many fingers they could see in front of themselves at an arms length away. I kind of enjoy this guessing game. 

...

After a google search about base 60, I am not sure how accurate I was in terms of what I was guessing. The Babylonians counted in six groups 10, not five groups of 12, so it is unlikely that it was based on hands. Although it seems like other cultures used this method of counting. It was also interesting to find out its applicability to fractions as 60 is divisible by a large number of prime numbers. The example that stood out to me was 1/3 in base 10 is 0.3333... but in base 60 it 20/60 = 0.2. It seems like the use with fractions is a strong contender for why they used base 60. Other theories include the use in physics. But my guess would be the fractions argument as it is just more aesthetically satisfying.

Response to Our first reading: Why teach math History

I think about how Math History was taught to me in high school and university. In high school, I remember once doing a project on comparing Newton and Leibniz in Calculus 12. Beyond this project, nothing really stuck out to me in terms of math history. That being said, I wouldn't be surprised if I had forgotten historical knowledge mentioned in my high school math classes. In university, I feel more fond of the math history I had been exposed to. I remember the main purpose that it served to me was almost a break in mathematical thinking. I knew we were not going to be tested on it, so I just sat and enjoyed the history stories. There had been one written assignment, as well, on the founders of Linear Programming and it was interesting, but I think preferred my professor's in-class stories. 

The section 7.4.7 - Historical Problems caught my attention because these were the examples of historical problems that I enjoyed in my university classes. These problems were part of what did bring math to life for me. 

I think another general idea in the reading was the skills that reviewing math history could allow students to practice or understand. I practically liked the section about errors and alternatives that have come up in math history. I think it really speaks to the idea that at one point even the experts made mistakes and that these concepts are not necessarily intuitive.

Honestly, I am pretty hesitant to take any totally new ideas from this reading, as I am still skeptical about how much class time should be devoted to math history. On the other hand, I would say that this article just made the idea of making math affective more real to me. I always think of students who struggle with mathematical concepts, and how to make it more enjoyable for them or how to help them understand the reasoning for math classes. I think that history is an area that can help explain the reasoning of "why to learn math" to students.