Reflection on EDCP 442

It is a little bit hard for me to admit that since high school I have grown to have a much greater appreciation of history. It is hard for me to admit because history was always my dad's favourite area and I wanted to be different from him, and it never really stuck to me in high school. I mean I have a general sense of history, but recalling details or a timeline within a decade is often difficult for me. Maybe that is partially why I enjoyed this class more than an average history course. We touched on so many things -- so I think we really learnt some of the highlights, the things that make math unique in each culture and the complexities of how math was actually "formed". I think I have learnt a about a lot of different things that I wish to look more into later on in my career, as well as topics that I think are incredibly important for a high school math teacher to hopefully pass onto their students. 

I think the top concept that I wish to carry on into my classroom is getting away from this Eurocentric view of mathematics. When I think back to some of the stories or methods that we learnt, some of the Eurocentric ones were so much more well documented, but they were also vaguely known or kind of familiar to me. Many of the ideas I have learnt about from other cultures often do not have the same level of document (in some cases for some incredibly devastating reasons), but it so amazing to put yourself into minds and thought patterns of ancient times that are less familiar. In some cases, we talked about how these different methods can help students better understand or have a different method to try out. I think these varying methods are a good way for students to inquire and find patterns, and I think it is something that is generally given less credit, or maybe just less time devoted to, as students age. As a teacher I hope to keep this mind and incorporate some of these lessons and strategies into my lessons. Not only does it open the mind to new patterns, connections, and methods, but it also humanizes math a little bit and shows that it isn't as black and white as it is often thought of. 

Assignment 3 - The Kamal and Astrolabe

Kamal

Among some of the oldest tools used in navigational history is the arabian tool, kamal. The kamal was in its simplest form just a piece of wood and a line of string; however, it was used by Arabian sailors to determine their latitudes. As such, sailors would be able to use knots on the string of the kamal to ‘remember’ specific latitudes. For example, if a specific knot was known to be the latitude of a town, sailors would use the kamal to return to that latitude and then sail directly east/west back to the town. What was particularly useful about the kamal at the time was the ease of use and the fact that it could be used by the illiterate. 

To use a kamal, sailors would use their arm to extend the kamal with it being parallel to the ground and with the string also being perpendicular to the string. Often to ensure the string remains taut during measurement, the user may use their teeth to hold onto the knot. If the user was on the latitude that matches the latitude of the knot, the user would find that the Polaris star would fall exactly atop the edge of the kamal. The usage of the kamal is dependent on the mathematics of trigonometry and the relative position of the Polaris star. Fundamentally, the kamal allows for the measurement of latitudes through trigonometry with reference to Polaris and this latitude is stored physically on the rope through the position of the knot. 

Astrolabe 

The astrolabe is another ancient navigational tool that we decided to research. We found that the mathematics behind this instrument include many more applications beyond navigation. 

In ancient Greece, it was a combination of a planisphere (a map of the stars) and a dioptra (an ancient version of a protractor). At this time it was used to calculate time and date using the position of the stars and planets. Hypatica of Alexandra (~400 AD) is sometimes incorrectly credited to their invention, although they were invented at least 3 centuries prior to her birth. She did however teach others how to construct them. 

In the 8th century mathematician, Muhammad al-Fazari, created the first astrolabe in the Islamic world. This time period is where astrolabes really took off as a navigation device, as well as an important tool in Islamic religion. In the 10th century astronomer, al-Safi, listed about 1000 different uses of the astrolabe with most applications related to astronomy, astrology, navigation, surveying, when to plant crops, when to raise taxes, rising and falling of tides, timekeeping, prayer, Salat (type of prayer), and Qibla (direction of Mecca).

It has a ring on the top to keep it held directly upright. The largest circle part, called the Mater, in Latin, has all kinds of information, such as zodiacs and times, and keeps the astrolabe together. Within the Mater, are multiple changeable plates that are made for particular latitudes which are often double sided. In the Islamic world plates would often be made for Mecca, Cairo, and Cordoba. On the top would be a map of the stars. Then on the back there would be a straightedge that you would align with the sun, which would then translate into various information. These different parts are highlighted in our rendition of the astrolabe.

References

astrolabe (2018). In James Evans (Ed.), (1st ed.). Oxford University Press.

Bennett, Jim, Navigation: A Very Short Introduction (Oxford, 2017; online edn, Very Short Introductions online, Feb. 2017), http://dx.doi.org/10.1093/actrade/9780198733713.001.0001, accessed 29 Nov. 2021. 

Booth, C. (2017). Hypatia: Mathematician, Philosopher, Myth. Fonthill Media.

Ives, R. L. (1961). Navigating with the kamal. Journal of Geography, 60(6), 268. Retrieved from https://www.proquest.com/scholarly-journals/navigating-with-kamal/docview/1290486548/se-2?accountid=14656 

Kolev, R. (2013). The babylonian astrolabe: The calendar of creation. Neo-Assyrian Text Corpus Project.

Martin, L. C. (1923). Surveying and navigational instruments from the historical standpoint. Transactions of the Optical Society, 24(5), 289.

Nielbock, M. (2017). Navigating with the Kamal. arXiv preprint arXiv:1710.04482

SciShow. [SciShow]. (2017, February 27). 6 Creative Ways People Used to Navigate the Oceans [Video]. YouTube. https://www.youtube.com/watch?v=2Bv8dc19HQ0&t=145s&ab_channel=SciShow

Staples, E. Indian Ocean navigation in Islamic sources 850–1560 CE. History Compass. 2018; 16:e12485. https://doi.org/10.1111/hic3.12485

The British Museum. [The British Museum]. (2018, June 28). How to use an astrolabe I Curator's Corner S3 Ep1 #CuratorsCorner [Video]. YouTube. https://www.youtube.com/watch?v=N8oWGwcdFmA&ab_channel=TheBritishMuseum

Also the slides we used. 
 

Mayan Math and Myron Medina

 

This article was a nice one to end off with for the year! It makes me reminisce about when I was in elementary school and trying to find some sort of prediction between our names and "cubby number" we were given. I think it is the root of my fondness of the number 5 -- since my last name starts with a "D", I also tend to be given either number 5 or 6 in a group of 30, and I never really liked 6, for some reason. I think I have also always had this likeness for 5 because in a elementary music class we would sing a song about multiplication table of 5. It would start along the lines of "5 times 5 is 25...." and end with a little poem about barley and Charlie. I still love it and remember it satisfying me as a child, with all the rhythm and patterns of the 5's times table. 

Its no surprise that it follows that I also really enjoyed Medina's poems at the end of her article. I loved reading them and seeing exactly how she brought those numbers to life in a way that I relate to, although I am not sure if they were intentioned to be inspired by the Mayan characters, as in my brief reading I only noticed a couple subtleties. 

Its strange to me that as I was reading the final prompt for this blog post that I really do not have as much of a connection to months/letters/days of the week as I do to numbers. Or maybe I am just more drawn to numbers more than the average person, so this peculiarity to numbers sticks out to me.

I would consider showing this concept to my secondary math students, if we have time, or maybe foe enrichment, or maybe for a "get to know you" activity. I love personalizing numbers and it almost feels like a bit of a controversy to me as well. For example, most justifications for numbers I can try to relate to, but Pythagoras' definition of male being 2 and female being 3 is just so wrong to me. I think of females being stereotypically symmetrical (ie. two halves) and males are just more odd. I see here that I'm taking the word odd to mean two different things, but the same applies to females. I consider females to be more "even-headed" so I think of even numbers as more feminine. I suppose it is not good that I am bring gender stereotyping into numerals, but maybe it is a bias worth considering.   

In regards to Taxicab Numbers, I was unable to access the Wolfram Alpha article, but after a wikipedia search on Taxicab Numbers and Ramanujan, I suppose I am unsurprised that Ramanujan was the person to have such a sentiment towards numbers. It is quite remarkable that he would consider such detailed facts/patters for four-digit numbers. It kind of makes my pattern search feel a little bit less special, but I do appreciate that such a famous mathematician shares a similar thought pattern to me.