Final Blog Post

As I look back on my blogposts for 442 there are two themes that stick out to me - practicality and simplicity. Maybe the simplicity of the lives of ancient civilizations allowed for practical mathematics. I in no way express any negative connotation by saying such a thing; in fact I think the simplicity is something lost in today's world. Regardless, I find the importance in such a fundamental approach to exploring mathematics. This course has taught me to not neglect the origins of math concepts and perhaps starting at these places where mathematics originated can make for a more fulfilling lesson. If we take something such as quadratic equations and just move forward with applications and how to solve such equations without first exploring how Al-Khwarizmi came to discover it, we would be eliminating a potentially interesting historical lesson for students. We also get to do math through investigation and exploration. 

I'm realizing now that this is because of the realizations that we had as students in 442. We also learned about the origins of so many math concepts we knew no history about. We got to learn about ancient Chinese mathematics and about the many contributions that were made through these ancient civilizations. We were able to learn math history from a lens outside the perspective of one that is Eurocentric and that is so, so important. 

I'm realizing as well the importance of math history in terms of critical mathematics education. It takes a deconstructing approach and an important step in deconstructing current worldviews is to offer accurate representations of history that haven't been distorted by colonization. 

It was wonderful to see all these things of whose history I knew nothing about and furthermore thought that they were only created by Europeans in ancient Greece. I got to see that there is rich history in other places in the world and more importantly, places that actually relate to my own identity.

Multiplication Tables of 45

I recall sitting in (I think) the garden with Nanxi trying to solve through what these numbers could mean. Intelligently, she figured out how the Babylonians were doing this table of multiplication on base 60 - I'm kicking myself now that I didn't take a picture of her work because she also discovered how the decimals worked as well. As we go down the table from the beginning we see that 2 times 30 equals 60, 3 times 20 equals 60 and so on. When we get to 8, we find our first decimal which is interpreted as 8 times 7 equals 56 and 8 times 30/60 is 4; 56 plus 4 is equal to sixty. I believe there are certain numbers missing because they are prime or have these prime numbers as their factors. For example, 11 is missing and 22, 33, 44 and 55 are also missing. Although, I am not sure why they are missing but I think I just thought of an idea while writing this sentence out. 

If we take the number 11, we could say that whole number wise, it fits into 60, 5 times (which would give 55). Now we need 5 more, the closest to which we could get by multiplying 11 by 27/60 (which gives 4.95). We are still left with 0.05 to make up and our current answer would look like 5, 27. We can already see that to get to 0.05 we would be doing another 27/60 at the third (or fourth?) place value which would yield another decimal value with 5 - we end up having a repeating decimal similar to the fraction 1/3. Babylonians I'm thinking, must have identified that there would be numbers with repeating decimals and thus excluded them from the table. What I think is fascinating is that they also identified that other numbers that are multiples of these numbers would also end up with repeating decimals and chose to exclude them as well. 

Their fraction notation is similar to ours with the exception that they only simplify as far as base-60. Our 3/4 is their 45/60, our 1/2 is their 30/60 and so on. Wait, then there is a difference because our 3/4 is represented as 0.75 while theirs is represented as ", 45". Because we use a base-10 system, our decimals are over powers of 10 while Babylonians construct their decimals over powers of 60. 

A table of 45 would look (I think) like this:

Interestingly enough, I'm not sure if I'm correct but I was noticing that with this odd-number base, the answers in the right side could only contain odd numbers. As soon as you encountered an even number, the decimal would most likely start repeating itself. 

Art History Project Reflection

For our art history project, Brandon and I decided to present on the history of the Monty Hall Problem. When we were first tasked with this assignment and told it would be done in partners, with no hesitation I knew that I wanted to be partners with Brandon. For two reasons, one being that he wasn't in class that day and I didn't want him to be stressed about not having a partner but more importantly, Brandon and I get along really well. There's many ways that we relate to each other and I think the common ground of both working full-time is a channel of empathy that cannot be overlooked. Going through this program while committing such hours to work out of necessity is not something many can understand but Brandon and I are on the same boat with that. This showed in the way that we collaborated with each other. We split up the work, offered each other help and and took autonomy over our work. We didn't hassle each other and we trusted one another. The funnest part about this project was honestly standing at the white board with Brandon working through all the probabilities, explanations and extensions. 

As much as I've heard the Monty Hall Problem, I never fully understood the proof for it until I did this project with Brandon. It was also extremely interesting to learn about the Bertrand's Box Paradox from which this problem originated. 

I'm also really glad the class liked our little presentation bit in the beginning. I was thinking it might be a little corny but hearing everyone laugh and compliment us was a wonderful feeling. A big shout out to Jacob as well - what a guy right? Ask him to speak any prize and his wholesome self said, "a hug". 

Dancing Euclidean Proofs

I will be perfectly candid here - dancing is not my thing nor interest. I was not especially captivated by the Dancing Euclidean Proofs Susan showed us. The article however, was interesting to read. I think my indifference towards the dance fueled the interest with which I read the article; I did not expect there to have been so much thought put into the routine. This is not one of the things that I will write about in regards to what stood out to me but I think it worth mentioning and that is when the authors describe using their second arm in the first proposition. I was a member of the audience watching the performance; I did not consider the work, attention to detail or thought that went into it. Jason Ellis taught our EDST 401 class about the apprenticeship of observation and it seems I did not fully ingest that lesson.

The two things that made me stop and think both came at the end of the article. The first was when the authors mention how there is as much math within our bodies as there is in nature and this made me stop for selfish reasons. Recently I have been paying close attention to the UBC fountain which was such a spectacular allocation of funds on my walks to and from different classes. There is something extremely captivating about the wave patterns; it is like they are moving yet still. I was thinking that if the water comes out of the spout in a consistent manner, will the waves always be the same? Consistent as in the amount of water, the angle at which it comes out, the height of the water, the way it drops back down to the basin, the weather (wind, etc.) and other things - basically if the fountain was always the exact same. I don't know, I think it would. It's like the little waves (they're almost like pyramids of water sitting atop the surface of the water) are just constantly replacing each other. One of these days I'm going to bump into somebody or something while looking at the fountain. That being said, the author's are right when they say there is as much math in/on our bodies as there is in nature. The patterns and symmetry in which my arm hair grows is a testament to that.

The second thing is when the authors were discussing the land and their dance within it. When describing the circumstances they faced with the environment, interestingly enough, they said "...our particular geographical setting became an active limiting agent in our representation." I think, their geographical setting liberated them. They used the term limiting agent to describe the consistency of the sand which "ruined" their initial plans however I argue that whatever the land dictates you do is truly liberating and what is supposed to occur on that land. If we want to pay homage to the land and embrace it, you do so by taking in stride what the land gives you - which the authors/dancers did. 

It could be helpful but I think you are more than likely to encounter students like me that don't want to take part in activities too outside their comfort zone. Although there may be potential benefits I believe in keeping students comfortable within reason. You can't cater to all their wants or else you might get to a point where students do no math at all but you also don't want to push them to a place where they aren't comfortable at all. Experiential learning itself is always going to be enlightening and each type will offer its own benefits. Experiential learning does not always have to be around body movement. Kids can learn about measurements through a baking activity and this might be way more impactful than some exercises on a worksheet. Food for thought, literally.

Individual Presentation Reflection

Pure mathematics has always been the branch of math that I have found most intriguing. I don't think I discovered the name for it until my proofs class in my first year of undergraduate studies. It's like, math from scratch. You have nothing, now create. You have this concept, now prove it. Although I'm awful at it, I find that it is the only thing that I have no problems being stuck at. Which is interesting now that I think about it. 

I feel like I rediscovered this while researching on the history of geometric constructions. It may be the purest of the purest forms of mathematics. When I realized that these mathematicians were creating theorems, proofs and conjectures based solely on concepts and not facts, I was amazed. They were creating factual experiments solely through proofs and conjectures. Euclid does not define a right angle as an angle that is ninety-degrees. He defines it as the two angles created by one line intersecting another line being equal - or "right". I had written a question for potential future students of mine in how I would incorporate this into my classroom where I asked "draw me a square without using a ruler or compass and prove that it is a square". I sat on this question for a long time thinking about how I would answer it and I still have not come up with an answer. But there's beauty in that, right? Why do I need to have the answer to this question - it's meant as an exploratory introduction to geometric construction and it would be informative regarding student thinking. 

Lui Hui and Zu Chongzhi

I think depending on the topic or unit, it can greatly benefit our students to acknowledge and discuss sources of mathematics that aren't Eurocentric. I say this for two reasons. The first, it offers mathematical diversity. We are often plagued by tunnel vision in mathematics and once we find one way to do something or one way on how something is done, it is difficult to step out of that thinking and allow for more possibilities. Exploratory or investigative mathematics does just the opposite; it allows individuals to ask and discover mathematical reasoning. When one comes to these realizations themselves without explicit aid, they have an ability to comprehend other possible solutions. Other routes to solutions are what I mean by mathematical diversity. Teaching concepts like the relationship between the sides of a right triangle, pi and base-sixty number systems gives students the ability to see things differently. Another reason it is good is to combat the obvious bias of Eurocentric views. It is interesting, in the article "Was Pythagoras Chinese- Revisiting an Old Debate" by Ross Gustafson my "spidey-senses" tell me that if history was reversed and the Greeks wrote the Jiu Zhang Suahshu and the Chinese wrote The Elements, Gustafson would have praised the time (earlier discovery) instead of the rigorous proofs. We live in an Eurocentric world after all, and those are just my two cents. I think we can find beauty in both discoveries. 

In regards to the naming, this requires a long discussion. Unfortunately, we live in a post-colonial world. This affects many factors of our day to day lives that should not go overlooked. For several years, world maps used exaggerated the size of North America and inaccurately depicted the sizes of South America and Africa. Cultures that do not align with Western traditions are spoke of with negative connotations. Something as simple as eating with your hands is considered barbaric. Along these lines we can also find naming conventions. Naming is often a reframing of history so that it starts somewhere in Europe. So on this point, I don't like it at all. We should know the actual earliest discovery of math concepts and celebrate all who stumbled upon something related (so long as it was not stolen). 

Euclid

Euclid's Elements is widely regarded as one of the most important works in mathematics. It has been in use for over 2000 years and remains important until this very day. Although it is not unknown that the content found in the book are not uniquely Euclid's, Elements was groundbreaking for its organization, clarity and exposition. How Euclid proved his theorem's and proofs have become a standard. Aside from his remarkable communication in mathematics, I think the topic itself is why Elements has stood the test of time. The concepts touch on both basic and advanced, and geometry is ever present in our lives and in education/academia. His proofs can be learned as foundational understanding for kids in the public school system. Elements can also be a core part of those pursuing higher education in mathematics - in fact, one might argue that it is imperative to learn about Euclid's work for anyone pursuing such a field of study. 

If there is beauty in Euclid's postulates is subjective - it depends on who you ask. I think I can find beauty in the simplicity of it and certainly, I can appreciate Euclid's thought process in using these simple (but proved rigorously) facts to prove much more complicated theorems. I think where I have difficulty finding the beauty in it is its specificity to mathematics. A line segment isn't such a common aspect. 

What do they say, "beauty is in the eye of the beholder"? That's my best answer to the last question, "how can we define beauty if these are considered beautiful?" It's always about perspective, and one's opinion on what's beautiful cannot be held with higher importance or validity than someone else's opinion. We may bring in the aspect of popularity or the opinion of the masses. For example: if one hundred people think something is beautiful and one person does not, then maybe we have an argument but what is important to note is that it would still be something to be argued - it is not a fact. 

Something I find beautiful is a small aspect of relativity. Stephen Hawking in his book "The Universe in a Nutshell" speaks on Einstein's theory of relativity. He brings up an example of two planes starting at opposite ends of the earth. Both airplanes are set with extremely accurate clocks and they set off at the same time - one flies east and one flies west. Both planes arrive at their initial point having recorded slightly different times! The rotation and speed of the earth contributes to the plane flying east. I find that the beauty in this lies in the fact that it goes against basic human understanding. If you describe this scenario to someone and ask them what the clocks will show, they will most likely say the same time. I don't know, my sister gave me this book many years ago (I think I was in middle school) and I never got past that page with the airplane because I couldn't comprehend it. At my big age, I still don't think I do. 

Maybe that's where the beauty lies; beauty in the sometimes amazing yet incomprehensible nature of science and mathematics.