Blog Post #13: Mathematics of medieval European universities

 Three things that surprised me from the article:

1) "Numbers were identified with the various gods. He considered the odd numbers to be male and the even ones to be female. He made a strange distinction between the "divine number," a sort of general concept of number which existed only in the mind of the creator-god, and scientific numbers, which were the common numbers known to men on earth." - pg. 267 I find it surprising that numbers were once identified with gods, as if nobody could really know where they came from. I am also curious about how they determined odd numbers to be male and even numbers to be female.

2) "The arithmetic of the schools did not receive the wholehearted approbation of all the people." - pg. 271 Considering how mathematics is used as a basis in many fields (such as science, engineering, architecture, etc...), I find it mind-boggling that it was once received as something rather trivial and not of importance. 

3) "Dialectic or logic became so important that it tended to obscure all the other arts." - pg. 270 

Blog Post #12: Dancing Euclidean proofs

 Two things that surprised me or made me want to stop and think about the article:

1) I found it interesting that the "exploration of multiple representations and modes of cognition has been shown to enhance and deepen mathematical understanding building more holistic mathematical understanding through a knowledge of the equivalence of varied modes of representation." When I learned math, I mainly did practice problems to deepen my understanding and never really thought about physical activities as another mode of cognitive learning for this subject.

2) I also found it intriguing that there is a need to "spark an appreciation of the beauty of mathematics". I have always viewed math as a tool that was used to solve problems as I never interpreted it as an art form. To see that mathematics has beauty in it because it has an artistic interpretation is really eye-opening. 

How might this kind of activity be helpful for math learning and understanding math history in a high school mathematics class? What kinds of constraints or obstacles might you encounter?

This kind of activity could be helpful for math learning and understanding math history in a high school mathematics class especially when learning about geometry units. This is because it can be a little bit difficult to visualize how angles are proportional to each other on paper and a physical interpretation could be easier to grasp. Constraints could be picking examples that are simple enough to embody and space to do physical interpretations with the whole class. 

Blog Post #11: 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?

Euclid and Euclidean geometry is still studied today because he wrote "The Elements". "The Elements" provides the basics of geometry. Within the 13 books, the books 1-6 are about plane geometry, books 7-9 deals with number theory and books 11-13 deal with third dimensional geometry. "The Elements" also begin with definitions and five postulates that set the ground rules for the following books. This book has been really important over centuries because it not only provides the basics of geometry, but the "standard of [proof] rigour was to become a goal for the inventors of the calculus centuries later." 

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

I am not sure if there is beauty in the Euclidean postulates, common notions and principles for proofs, but I can say that it is extremely useful for setting the ground rules for writing geometric proofs. For example, it specifies that you can draw a line between 2 points and assumes the existence of points, lines and circles. The second definition of beautiful in the dictionary is that something is "of a very high standard". In the supplementary article, it states that "Euclid's Elements is remarkable for the clarity with which the theorems are stated and proved. The standard of rigour was to become a goal for the inventors of the calculus centuries later." So, his proofs were of a high standard that became a role model for later mathematicians, which is beautiful.

Blog Post #10: Explication/ commentary on poem about Euclid

 Explication and Commentary on the poem:

"Euclid Alone Has Looked on Beauty Bare"

A reference from here: https://lifeorange.com/writing/writing2.htm

In the first stanza, immediately, I question: Who is Beauty? Knowing that Euclid is a mathematician that has written many proofs, Beauty is probably a reference to some mathematical truths that have yet to be discovered by other people. This is because the Beauty is described as "bare", like something that has been uncovered from the unknown. The following lines in the stanza seem like it is criticizing those who "prate" for foolishly speak of the "Beauty" without understanding what it is. In the second stanza, it seems like Euclid is the only one who is able to analyze the "Beauty", or mathematical truth, as he was able to "anatomize" its light. And because of this realization, he was able to make the "Beauty" real (proving her as a truth) as she can now set her massive sandal on stone. 

"The Euclidean Domain"

This poem seems to be ridiculing the original poem. In my opinion, the first stanza asks if Euclid was the only person who ever saw Beauty bare, or other variations of her. I think that this is basically acknowledging that there are other mathematicians out there that have probably seen multiple sides to the same mathematical truth and proven it one way or another. The second stanza asks how the poet knows that the Beauty exists and whether or not she can justify that by having seen the Beauty clothed. The third and fourth stanza states that the poet is calling herself one of the praters that have misunderstood the Beauty because she states that only Euclid has ever analyzed the Beauty correctly.

Parody:

Euclid alone has looked on Beauty bare. 

Let all who prate of Beauty speak their mind, 

And lay them beside Euclid upon Earth and continue 

To ponder themselves, while they stare 

At the Beauty, intricately drawn nowhere

In shapes of shifting lineage; let geese 

squawk and fly off into the distance.

O blinding hour, O holy day, terrible day,

When first the shaft into their vision shone

Of light anatomized! Together

have looked on Beauty bare. Fortunate they

Who, though once only and the but far away, 

Have heard her massive sandal set on stone.

Blog Post #9: Was Pythagoras Chinese? article reflection

Does it make a difference to our students' learning if we acknowledge (or don't acknowledge) non-European sources of mathematics? Why, or how?

I think that it makes a difference to our students' learning if we acknowledge non-European sources of mathematics if the information can provide support for students' understanding of why we use math in a relevant way. For instance, "The Jiu Zhang suahshu gave the reader the tools to solve problems and did not concern itself with proving the authenticity of the tools", whereas "The elements was nearly the opposite, as its chief concern was giving rigorous proof of ideas presented.". In this way, a students' learning is further supported because this gives the perspective that math is not only about proving what is true, but it is about applying it in problem solving.  Sometimes, to teach basic concepts to school students, a rigorous proof might be confusing and too complicated. Some perspectives from different cultures might give students and easier grasp on the concept than others. For example, when looking at the Greek vs. Chinese mathematicians ways of determining pi, we understand that the "Chinese proof requires little background knowledge due to its visual nature" using a method of circle division instead of squaring the circle as the Greeks did. 

What are your thoughts about naming of the Pythagorean Theorem, and other named mathematical theorems and concepts (for example, Pascal's Triangle... check out its history.)

I think that the naming of Pythagorean Theorem and other named mathematical theorems should be named based on which person brought a greater influence to its application across the world. In my opinion, I don't believe that it should be based on who necessarily discovered it first. I would argue this because the theorem was not only discovered and proven by a particular mathematician, but that mathematician recorded the theorem and made sure to spread influence so that people of all classes can benefit from this knowledge in their work and daily lives. It was discovered that Pythagoras had proven his theorem and became well known for it by introducing it to society in his cult; also 'European jesuit missionaries began entering China in early 17th century" to spread their scientific influence while being influenced by Chinese perspectives, in which the Europeans spread math knowledge around the world to implement science into improving the lives of many. Although I think that this is the way that theorems should be named, I believe that others who have contributed towards the proving of the theorem should be credited for their perspective as well because it is equally as valuable. 

Blog Post #8: Personal reflection on the process of Assignment 1

 I worked on Problem #10 which was the Ahmes' Loaves Sharing problem. I think that it's mind-boggling that the Egyptians used a method of false position to come up with numbers to this hypothetical problem of dividing up 100 loaves of bread between 5 people with 2 complex criteria: the loaf division has to have arithmetic progression and 1/7 of the 3 largest shares must be equal to the 2 smallest shares. 

The process of finding the solution to the problem was to reverse engineer the problem by multiplying each share by 100/60 and then adding all the entries together to ensure that they add up to 100. And then taking 1/7 of the largest 3 entry sum and ensuring that they do in fact equal the smallest 2 sum. 

As a reflection, this is a pretty interesting exercise that I would not be able to quickly accomplish by doing guess and check. I would definitely need to use algebra to solve the problem efficiently. 

Blog Post #7: Assignment 1: Solving ancient problems in ancient and modern ways

 

The Powerpoint Presentation from class is posted here: https://docs.google.com/presentation/d/1SFpx71zq2HDgFA1etofldv6vAquwBHJS/edit?usp=sharing&ouid=106697084074657608536&rtpof=true&sd=true