Trivium & Quadrivium

"Logistic was practical and utilitarian, a study for children and slaves; logic was a liberal art, a study for free men" I stopped at this point it really showed the progression we've made in terms of who we allow education for. Let alone the prejudice against women and people of color in recent times, going back even further we education was even more exclusive and elitist, not allowing others to learn and having slaves. I know that the educational system isn't perfect, but we have made some good steps forwards in being more empathetic and inclusive. Even with post-secondary education, I think Canada has the right idea with subsidizing for their citizens, I compare it to the US where some students can't afford post-secondary school and cannot afford to get a degree. I hope that we can move into a world where education isn't discouraged because of pricing, but everyone can have an equitable shot of success, no matter their socio-economic background. We shouldn't discourage smart people from succeeding because of money.

 "towards the end of this period, the Hindu-Arabic number system was beginning to be known in Europe"  This caught my attention because it reminded me of my assignment I just recently completed. I hadn't learnt about the impact that Hindu-Arabic math had, especially on our decimal place-value system. It originated in the east and moved over to the west, yet in school we aren't taught that. We are just told what it is, and because so much of the curricula is taught from the perspective of ancient Greek philosophers and mathematicians, I just assumed it originated from there. This highlights the importance of teaching mathematical history as having an accurate account of history could inspire people of different race/ethnicities to feel properly represented.

"The arithmetic of the schools did not receive the wholehearted approbation of all the people" This quote made me stop because we were talking about medieval mathematics and yet this problem still persists today. Mathematics to some is seen just as computation, and doesn't have much depth to be explored into. I believe this starts from a place of math anxiety in common days, where people are turned away from mathematics at a young age and do not come back around to appreciate its beauty. This could also be because students haven't been properly taught math in a way that inspires creativity or inquiry. As we move forward with developing how we teach, we can hopefully inspire young math students to see how interesting and applicable it can be.

Numbers with Personality

 

 

To think that each of the 'positive integers was one of his personal friends' seems a little outlandish to me. My first thought was then 'are the negative numbers his enemies then?', which could lay into the personification of numbers moreover, but was strange to me nonetheless. I believe that the quote is seen as playful and more of a connection that the speaker has to the world, connecting him through various means such as numbers. The world is often surrounded by numbers and as we walk through the world we encounter numbers every day, to have them as your friend would be to see the beauty and positivity of numbers, having a positive relationship with them and appreciating them for all they do. 

I think that it would be a cool idea to bring into the classroom, perhaps towards a concept with exponents and large numbers or scales, as they're often hard to conceptualize. As someone with a wild and colorful imagination, perhaps we could use the sheer scale of a large number like a monster or golem and show creativity through that lens. For scale factors students can draw themselves and a monster overlain on them, and find the scale that would proportionally make their monster correct. For representing large numbers, we could perhaps give a large number a name or represent it by a drawing, and refer to it as that in future classes. I'm not entirely sure how I would bring in personifying numbers strictly as I don't completely know the benefit in it.

Personally I don't think numbers have personalities, although like the article mentioned, I have seen instances where people see numbers as lucky or unlucky. Specifically in Chinese culture the numbers 4 and 8 have significant meaning. In sports I've also seen people thinking numbers are lucky, or associating a trait to a certain number. In hockey a famous defenceman named Bobby Orr wore the #4, so naturally many young kids would wear that number to imitate him. In terms of months of the year, I think we could associate weather to personalities, with Winter being cold and a little dreary, but Summer being light, happy, and excited. There are also ties to astrological signs that many people believe in.

Assignment 1 Reflection (sorry it's late!)

 

As far as first assignments go, I was really impressed with how our group gelled together. I had the pleasure of working with Karishma and Ivan on the ancient Babylonian problem regarding Sagitta and Chord. What worked really well for us was delegating work evenly, but also providing critical feedback to each other so we could continuously build upon ideas. This reminds me of many of the things we talk about in class such as collaboration with other teachers to have better lesson plans/classes. I feel because of this we were able to have a better flowing presentation, but also had our own ways of explaining the problem. My part of the presentation was to think of the extension question/problem, and I tried my best to have fun with it. I've found that practice math questions can be boring for students, so I try to find ways to make them exciting. In class during our brief post-discussion Susan mentioned how kids may relate better to dragons than squares and circles, something that resonated with me as part of the fun in math is creating problems that extend from the imagination! 

In terms of the actual material, it was different putting myself in a Babylonians' 'shoes' and figuring out the applications of the mathematics. One of the cool pieces I learnt from Karishma was how the Sagitta and Chord could possibly be used for building archways like the Ishtar Gate. The mathematical advancements that the Babylonians had was almost indescribable, they were so much further ahead of their time. 

Dancing Euclidean Proofs

The first thing that stopped me when reading this article was how I could find a way to physically implement mathematics into something I'm passionate about: hockey. It didn't have to be about Euclidean proofs necessarily, but more geared to embodied learning as a whole. I think that sports is a great way to show embodied learning and can have really great physical representations, especially with something like skating that allows gliding and movement of the puck, or even the use of the stick. If I were to translate ice to the dancing Euclidean proofs, perhaps the use of figure skating could be used. With hockey however I could create drills or skating patterns that would represent mathematical proofs, or even mimic the first Euclidean proof dance as a drill that could work on edgework and skating. While reading the article all I could think of was how I could implement these ideas into something I love.

A quote that made me stop was "If you sit down to study the Elements from a book, you are in a sense completely detached from its representation on the page" . What I found so interesting about this was of course that I have experienced this detachment in other subjects or even with math. I feel this is a reason why math can be seen as 'boring' for some students, as rigorous proofs through text may just be words on a page that they don't care for. Over the short practicum what I've learnt is kids do not learn well when all they do is sit in their seats and take notes or read -- the more they move and become part of the work, the more they become engaged and focused. I think that math is notorious for having proofs that are boring and hard to understand for most people, so having embodied learning really gives the student a better sense of how proofs may work.

A second quote that stems from the first is "You become the active agents responsible for the making and understanding the representation" , and although this has to do with dancing in the text, in reality this is what we want all of our students to be able to do. We want to give them the necessary tools and knowledge to understand the material, but we want them to actively engage with it and form relational understandings through whatever form they're most comfortable with. We should encourage students to learn through analogies or physical representations, as it can give the context a sense of realism and practicality. I remember in physics we used the right hand rule, this physical representation really made sense of how forces may work on an object and therefore made the content easier.

Euclidian Poems

Euclid is credited as being one of the founders of geometry, having influenced and proved many theorems that have set the basis for what is commonly referred to as Euclidean Geometry! In Euclid's Elements, rigorous mathematical proofs are used to lay the foundation for geometry. 

The poem written by Edna St. Vincent Millay is particularly interesting in the sense that it seemed like an ode to Euclid, praising that he "alone has looked on Beauty bare". What's interesting is the capitalization the word Beauty, almost personifying it and giving the word meaning. I believe what the author is trying to say is that by working rigorously through mathematical proof, Euclid has seen beauty in an existential form and perceives the world differently than 'most people'. She is putting Euclid on a pedestal and saying he is an intellectual far beyond anyone else as he "alone" has seen Beauty. She then reiterates this at the end of the poem by saying "fortunate they who, though once only and then but far away, have heard her massive sandal set on stone" she is referring to people who have come close have only seen 'her' (referring to Beauty) 'massive sandal set on stone' once again personifying and giving life to the term; but also showing the distance between Euclid and others. While he has witnessed Beauty bare, others have only heard the sandal set on stone, and that is as close as they've gotten. I believe that this poem is made to make Euclid seem like an intellectual like no other; no one comes close to his levels of intellect, and it seems like no one ever will.

The second poem by David Kramer however seems more like a direct opposition to the ideals of the first poem. From the first line "Euclid alone has looked on Beauty bare?" the mere idea that Euclid is better than all is put into question immediately. I believe the poem is criticizing those who put Euclid on a pedestal in the lines "As you sang praise, Orpheus, of Eurydice, Your mouth became the orifice of your idiocy!". The story of Orpheus and Eurydice is a tragic one. To summarize: Orpheus lost his wife and visited the underworld to convince Hades to let her return; he said he would grant this wish, but Orpheus would have to walk back from the underworld and trust that Hades had let her follow behind him. Orpheus was thus instructed not to turn around and trust Hades that she will be following him. Orpheus almost made it to the end, but his anxiety got the best of him, thinking the Gods had tricked him, so he turned around only to see his wife Eurydice's shadow wisp away. I think the poem really targets on the turning around aspect and to continue looking forward. Yes, it is great to praise artists and scientists for their works, but to idolize and say they are the most intellectual people to ever exist is a disservice to other greats. "Has no one else of her seen hide or hair? Nor heard her massive sandal set on stone? Nor spoken with her on the telephone?" I believe this line is expressing that other people have come just as close to Euclid on seeing true Beauty and should be respected for that reason.