Word Problems

 To me, word problems have always been the "uh-oh, here it is" part of the test. I can appreciate them for their ability to measure the application portion of math in a way that other questions cannot so I understand their importance. I know that I wouldn't use them in the absurd ways that they are used some times. I am referring to those questions with 560 watermelons and an eleven year old classmate opening up a movie theatre and selling 35 tickets. Simply including your students' names does not fill the void of lack of relatability. Word problems have to be posed in situations that are actually relevant to your students' lives. For example, posing a modelling question around social media brand deals and rates of pay in relation to follower/subscriber count is something I think today's teenagers could relate to. I do find this a personal hurdle in teaching because how can I draw the line between personal feelings towards social media and what kids are into. That's a topic for another day, and another course perhaps.  

Surveying by Ancient Egyptians

Two questions I have regarding the wall painting from the tomb of Menna are:

1. What is the thing in the middle of the bottom painting (the thing that we thought may have been an oven)?

2. What is the role of the two people standing on the right side of the bottom painting?

I want to note that when we first looked at this picture I was completely clueless. I could tell there were people holding a rope on the top but that was as far as I could grasp. I also noted that whoever painted this also made sure to draw in what seems to be grey hair – older people? I bring this up to appreciate my classmates. How quickly were people noticing that there were knots on the rope signifying units of measurement. And that they were harvesting some sort of grain. That there were scribes, some people doing labour with others recording. I was a little lost for words listening to my colleagues talk about their findings and interpretations. I thought it was remarkable that they were able to deduce such things so quickly. 

One that that stood out to me was how the author of the essay explained cubits, and that cubits were also broken down into “palms” which were measured using fingers. Including everything we’ve learned in this class so far, it got me to thinking how much limbs, and fingers were used to count, do arithmetic and generate mathematics from the ground up. Now we have tools to do all the measuring for us but I tried to think, “if I had absolutely no tools and was asked to measure something how would I do it?” Using my arm almost seems innate but the safety net is that I can confirm my findings by falling back on the tools we have. Early civilizations did not have these tools, and yet the constructed such complex and comprehensive concepts of mathematics and the most important part may be that it was crucial to every day life. 

Number Systems and Time

I'm not sure that I ever thought about the geometries of time until Jacob asked us last week in the orchard garden. I definitely associated certain months with temperatures due to the varying extremes we had in the weather but I'm not sure that I ever asked the question why things are the way they are. I think humans intrinsically have difficulty letting things be as they are; there is an extreme discomfort in not understanding or making sense out of. This leads me to think, there aren't really twenty-four hours in a day, is there? Just because we say it is so, doesn't make it so - it's just our interpretation of it. We took daylight, tracked it using a sun dial, split the intervals into twelve because it was a duodecimal system at the time, then decided the night also needed to be split into twelve, then realized the inconsistencies between the lengths of the intervals, and then at some point squished and molded everything to fit a nice and perfect equal twenty-four parts and called it a day. I understand the importance of the twenty-four hour day in the current state of the world but I don't consider it's importance in general. In the day time we should handle our business, and at night time we should rest - that seems innate. 

Also, for such a stringent species measuring seconds now by the atomic time it seems odd that we are fine with two months in a row having thirty-one days and then throwing in an extra day to the month with the least amount of days every four years so that we stay accurate to the year actually being 365.25 days. I don't know, seems like we pick and choose when to be straight and narrow (perhaps a larger tell of society as well *cough* *cough*).

*edit*

Technically there are no inconsistencies between the two articles however the article written in Scientific American was much more definitive in its justification for why Babylonian's used base-sixty. The article in University of St. Andrew's Math History is less clear. O'Connor and Robertson list several reasons (including the large number of factors and the ability to count to 60 using knuckles and joints) of why Babylonian's used base-sixty however they do not definitely say which of the reasons it is. It is important to note they list these theories as they have been proposed by others. There are no surprises to me when it comes to the calculation of time but I do wish it wasn't so unnatural. I do find it surprising that the Egyptians decided to split the daylight hours into twelve. I understand why twelve (due to its significance in terms of lunar months, knuckles/joints in one hand, etc.) however I don't understand why they felt the need to split the day at all. 

The Crest of the Peacock - A Response

This is a little off topic but one thing that surprised me was the nonchalance with which a Babylonian clay tablet being in a Berlin museum was mentioned. Earlier on in this class we were shown an ancient Indian artifact with the earliest known record of “zero” being held in a library in the UK. The irony of this article, to be exposing our Eurocentric views while casually mentioning that a priceless artifact from an ancient civilization is still being held somewhere in Europe is something I can’t wrap my head around. In recent times we have been visiting the concept of orientalism and rightfully so, it’s important for us to understand that current world views are dominantly Eurocentric. There’s something about physical, and tangible examples like stolen artifacts not being returned to this day always makes me stop in my tracks – the example in this article was no different.

The clarifications of the progress of mathematics during the dark ages was something that was especially surprising. Although I was somewhat privy to the history of math and science during the earliest Islamic Caliphates, I did not know they were so crucial to modern day mathematics. It is a shame that such rich history from then, and other places including India and China have been reduced to the term “the dark ages”. It would be great and I may with some more knowledge argue that it is a necessity that the history of math and science be taught through true world views and not a Eurocentric one. 

Babylonians and Base-60

Honestly, I have no idea why they would choose a base number of 60 without doing any other research. Perhaps it had something to do with seasons? Cyclical changes for their crops or a way they measured time through the Sun. Sixty is also the first number with many factors (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60) and maybe that contributed. This is requiring too many brain cells, I’m going to go find out why now. 

Wow, would you look at that. It seems as though base 60 was used for its large number of factors and divisors. There also seems to be enough evidence to suggest that hand and finger counting methods gave five and twelve (fingers and knuckles respectively) which combined to create a system of sixty. 

In our current world, base 60 is what we use to measure time. 

*edit*

When I was initially thinking about why Babylonian's would use a number system with the only reason I could come up with in my head was the large number of factors. I was able to confirm my guess through the following link: https://www.thoughtco.com/why-we-still-use-babylonian-mathematics-116679

We also discussed in class how Babylonian's used to count to sixty on their hands but I wanted to confirm; through a quick Google search I found the following explanation on Reddit (I know it's unreliable but it went in line with what I heard in class): https://www.reddit.com/r/todayilearned/comments/e3i0j/til_that_sumerians_and_babylonians_used_a_base60/.

 

Teaching History of Math

In my opinion, the secondary math curriculum already contains too much. My pre-reading ideas is not to add more to the curriculum, including math history, but instead to remove or restructure the existing curriculum. Adding math history could take up time that can be better spent on allowing for teachers to teach basic, fundamental concepts at a more deep rooted level so that students can gain a relational understanding of mathematics. An issue that stands out for me is the topic of math history as well. In my opinion, you have to be interested in a subject to want to learn about its history. Students already dislike going into their math classes – to pile on top of that history that seems unrelatable would be too much for the teacher. I think there are better ways to induce curiosity and creativity in the math classroom. 

There were a couple of points in the reading that stuck out to me. The notion that mathematics is presented in a polished, organized structure when it was never found like that. This is something I agree with, and never thought about before. Teaching kids how mathematicians of the past came to discover a basic concept we teach today could be a means of inspiring a more exploratory method to approach math with. Although the author presented an entire section on how to integrate math history in the classroom I think he may have added to an already existing issue of an overburdened curriculum. In one of the examples, he used a project (that was described as taking one to two semesters) that was done by graduate level students in University. The issue with this as a supporting point is that students at the graduate level still pursuing mathematics are an anomaly – their interest in math far exceeds that of most students. His other examples as well, including worksheets and readings only add to the curriculum. 

I don’t think I read this article at a good time. My bias lies too heavily against it. Although I do agree that the exploratory nature of studying the history of mathematics and mathematicians can inspire a different form of learning, I don’t believe it can be applied at a public school level. There were two things in the article that I thought were good ideas – a whole course dedicated to the history of mathematics and the notion of experiential mathematical activities. I think the latter would also have to be presented to students wanting to pursue further mathematics as well. I think teaching the history of mathematics to students already interested in math is a great way to induce further curiosity and learning but with the current curriculum and varying levels of understanding amongst secondary students, I believe more resources need to be put into how we teach math. 

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