Dancing Euclidean Proofs

 

It is weird re-looking at this video while on practicum, because I start asking myself how would I be able to incorporate this into the classroom. Especially because I don't think Euclidean geometry is included in the curriculum anywhere. Compared to when I first watched the video I thought: "wow, this is beautiful and really brings another layer of life and beauty to the proofs I love". I suppose that the main message I'm going to take from this video, with consideration of my practicum, is that teachers can make Math more exciting for students that are drawn to other fields, in this case dance and art. 

Actually, just yesterday I had given my Math 8 students a puzzle that had a story element to it. While walking around the room and checking on their progress, I asked the first student that I had talked to who had solved the puzzle, if she would be willing to present the solution to the class and she exclaimed "Of course, I'm a theatre kid". It was exciting to see her be so excited about solving and presenting her solution. Later when we were working on linear equations in the class, you could see her enthusiasm diminished. 

The article also pointed out that these multi modal understandings or different modes of cognition deepen mathematical understanding and makes it more holistic. I find that this finding is true in our mathematical practice. One of my school advisors also teaches a Workplace Math 10 class where students sometimes still struggle with identifying the radius of a circle. I have seen an Education Assistant go to student and make a big circle with both her arms to demonstrate the different between diameter and radius. Although students just watch her do it, I think I might start asking students to actually do it themselves in my classes. Even a simple activity, that is similar to the dance may help students understand these ideas more. 

I am kind of excited about trying to maybe incorporate some these smaller math movements into my classroom. I think they also double up as a brain break, as well. 

One last thing I wanted to note from the article is that made me stop was the considerations and affordances of the environment. I think that this also teaches students something that is applicable even in a typical math class. We have to work with what is given to us, and sometimes this requires a little bit of imagination and flexibility. I think it is important to empathize these skills in a classroom. 

Euclid and Beauty

Why is Euclid and Euclidean geometry still studied to this day? Why do you think this book has been so important (and incredibly popular) over centuries?

I think that Euclid is still studied, in part, because of his contributions to mathematics. He really came up with these solid, foundational ideas and proofs that are important to how the world fundamentally works. When I think about why Euclidean geometry is still studied to this day, I think that the same question could be asked for a variety of other areas of mathematics. I suppose that at least the first couple books of Euclid's Elements do not have clear practical applications, but I think that they could. Especially the later books regarding number theory could be very useful in a variety of computer algorithms. 

I think that this book has been so popular because of its elegance in proving and constructing these ideas, such as constructing an equilateral triangle. When I took a Euclidean geometry course at UBC, I remember being mesmerized by the ideas that the constructions and proofs used. It wasn't like math I have ever seen before, but it still held the logic and reasoning that I am fond of. I want to believe that similar ideas contribute to the general popularity of these books.

Is there beauty in the Euclidean postulates, common notions and principles for proofs? How can we define beauty if these are considered beautiful? 

I think there is definitely beauty in these things. I think that watching someone draw the proofs by hand is quite beautiful. You start with simple things like a point and line and end up with complex drawings resulting in an interesting proof. 

If these are consider beautiful, I suppose beauty is defined closely to elegance, and possibly rigour. I actually think the completeness is what makes these things untraditionally beautiful. The Euclidean work takes ideas that may seem obvious and break them down so small to show how they hold up and are constructed. It is comparable to creating brilliant and beautiful ideas with virtually nothing. 

Euclidean Poetry

It appears that these poets are basically comparing Euclid to some sort of a math messiah. The capitalization of Beauty implies that it may be a metaphor for God, which Euclid has somehow brought to earth. Also the references to Greek mythology, as well as biblical languages, and references to paeans (which to my knowledge are like church hymns) suggest that Euclid's work is being highly appreciated. 

I argue Millay was trying to convey in her poem that these fundamental principles being built into logical arguments is "beauty" in its purest form. There is no debate about how these principles work and they will always work. When Euclid first introduced this type of reasoning to the world, he introduced the world to pure "beauty". 

On initial impression, it appears to me that Kramer is suggesting that even with Euclid's contributions, we still do not truly see "Beauty" in all its glory. In my personal view, I think that he is suggesting how few people, and maybe even Millay, herself, truly understand the workings of Euclid. Also how much easier it is to "see Beauty clothed" which I believe implies that we see and understand the aftermath of these discoveries/principles, but we don't really know them truly. 

Just some of my five cents.  

Response to: Was Pythagoras Chinese

Let's Acknowledge Non-European History 

I think that if we are to credit European math history it is also important to credit non-European math history. When I was first debating this stance in my head, I thought that it didn't really matter to students, especially if history is not the focus of the math lesson. But then I thought, even the names of these Greek mathematicians are incredibly dominant in how we teach math and these European names really do matter (even if it is easy to discount their importance to students). I look back to when I was in school and I was fascinated with Ancient Greece, and I would be so excited to learn about how the math was present in this history as well. But if I had known about the contributions from other cultures, would I have also taken fascination with them? I think it is kind of unfair that these European mathematicians get this seen more often. I think that giving credit to all the various contributors is important, as well because it recognizes various cultures and demonstrates the commonality of human civilizations. 

Naming Theorems 

At this moment, I feel annoyed and maybe frustrated with all the theorems that give credit to a person who contributed to them. I think these feelings stem from the fact that it is so difficult to just give one person credit, especially when it is not clear if they were necessarily the true discoverer of these ideas. Then, I think what else would we name them? I suppose practical names would be beneficial. I think in class, a peer told me that in China the equivalent of the Pythagorean Theorem is called the Gou-go Theorem which means the Short-Long Theorem. I think that names like these are more practical, fair, and universal. But then what do you do for theorems where there isn't a simple descriptor name, I'm not totally sure -- maybe just make something up, vote on it?  

Reflection on Assignment 1

Like other concepts in this class I think that it is so amazing that these mathematical ideas have been around for so long and in areas where it is unclear and mysterious how they came to these conclusions. I suppose I enjoy the idea that student that do not necessarily enjoy the typical math classroom, may really take interest in these types of lessons. There is more of a curiosity element to it. I also felt some frustration when trying to solve the ancient way because it was very unclear, which is similar to the feeling that I have when faced with a difficult proof. At the same time I think I would really enjoy having student do a similar activity and teaching similar concepts that we are learning in this class. These historical techniques seem to also appeal to"practical" math, while also having a more personal touch, which many students seem to crave. I hope to help my students appreciate math more with these concepts. 

Egyptian Fractions

I suppose the benefits of using unit fractions in ancient Egypt were revealed in class. It was an easy way to spilt meals amongst workers. I think it would also be possibly used for things like dividing property or inheritances, as the word problem suggests. 

The use of unit fractions today is less obvious to me. I suppose it could be used a tool for doing mental calculations. I know I sometimes figure out what is 10% or (1/10) of something is as a reference point for things like tipping or calculating discounts. I also looked up Egyptian fractions on wikipedia and they
suggested that today they are useful for rope burning puzzles. I think they could also be used like the Egyptians used them, when my family is dividing food, sometimes like a lasagna or a cake or a kit kat bar (when there are three of us), we will sometimes do something similar, where everyone get approximately one size of the first portion and then whoever wants seconds gets approximately a second portion of a different size.   

Here is some of my work for the horse inheritance puzzle: 

Solving Pythagorean Triples in Modern and Ancient Ways

 Here is the link to Heesung's and I's assignment.