Response to "The Arithmetic of The Medieval Universities"

In the section "Christianity and Liberal Arts", there is a distinction between logic and logistic defined by Christianity. "'Logistic is the theory which deals with numerable objects and not with numbers. It does not consider number in the proper sense of the term...' but rather the counting of flocks, addition, subtraction, multiplication and division, always dealing with sensible objects." This definition is very much the same as "useful math" defined by modern day people who question the education of mathematics beyond those mentioned operations. It is interesting to see how some people retain the same view of mathematics when this field has drastically changed and improved our life for hundreds of years. 

In correspondence with the above definition: "as for the quadrivium, as the science are called, since they have little to attract in themselves and produce only a meager profit, most of the students neglect them or else omit them entirely". Thanks to the development of technologies, mathematics nowadays is required more and more by other disciplines, rather than being a stand-alone subject. While there is still perceptions of math being "useless" once you graduate from college, this mindset is being altered gradually. 

Lastly, I really like what Augustine said about mathematics, where "the principles of logic as the inviolable foundations of knowledge ... Side by side with the logic we find the truths of mathematics .. all there truths are necessarily and unconditionally true; they cannot be contested." I am surprised that at A.D. 386, someone already had such confidence in the nature of mathematics, when there was no computer or any efficient ways to communicate with the rest of the world. The fact that some mathematics can be proven true for thousands of years (and stay true for possible eternity) fascinates me a lot. I really like the feeling of proving something to be true with 100% confidence, and I hope my future students could feel the same satisfaction when they are also able to do so with mathematics. 

 

Response to "Numbers with personality"

In Major's paper, it says "our brain creates a space where personal associations and broader social experience can mingle." This seems to be true in Ramanujan's case since he shows affections towards numbers by calling them friends. It's easy for mathematicians to talk in this way because they work with numbers all the time, however, I am not sure if it's a good idea to introduce this OLP to secondary math students. I would probably mention that numbers have personalities for some people, and students are welcomed to find numbers that's special to them by themselves. This is not a process that requires too much teacher involvement, in my opinion. 

There are some numbers that I don't feel comfortable with due to my cultural background, such as the number "four" as mentioned in the article. Although I do not believe in superstition ideas, I still try to avoid the number "four" when it is possible. I don't have particular affection towards other numbers, except numbers 2 and 5. The reason is simple, 2 and 5 are easier to work with in mathematics compared to numbers such as 7. It is usually easier to find patterns from 2 and 5. 

Reflection on Assignment 1: solve an ancient puzzle

In this assignment, we did a presentation on Babylonian multiplication and division. While the provided information on Babylonian arithmetic seemed to be quite understandable and straightforward, the process of bringing everything together to make a presentation was more challenging then what we had imagined.

The first challenge was to figure out the non-existing decimal place in Babylonian mathematics. We started the project before the class where Babylonian reciprocal table was introduced, so I had to figure out why the pairs in the reciprocal table do not always multiply to 1. Despite many documentations on the Babylonian reciprocal table, few directly pointed out the missing of ways to represent different decimal places. Through researching and trying to do multiplication from the pairs, this concept eventually became clear. It is interesting how some resource makes assumptions about facts that are not necessarily evident to the audience. When reading such documents, the best way is to try it out with paper and pencil by ourselves. 

Through this presentation, I had a deeper understanding of the operations in Babylonian times, as well as other ancient math concepts from the presentation of other groups. The group work part made it more interesting because we can get different perspective from our group members. It is a fun and rewarding assignment! 

Response to "Dancing Euclidean Proofs"

One of the goals described in the article is to "make the beauty of Euclid's proofs accessible to mainstream audiences and students of mathematics through the physical beauty of dance" (page 240). I think such practice is very important in pedagogy of mathematics, especially in a time when math-phobia is the norm among general public. The regular mathematics in a textbook is intimidating, even to a math major university student. Using a form that is more popular and more appreciated by the public can reduce the fear towards mathematics, and people can make connection from what they already know about dancing. 

The audience is not the only ones who benefit from such a embodied, performative form of mathematical proofs. The dancers themselves, through "the process of choreographing the dance proofs -- making decisions, practicing, memorizing"(page 244) internalize and mentally recreate the proofs. Recreation using one's own approach/language/tools is the best way to study a complex concept. 

All of the above is possible due to the amazing compliance between Euclidean Geometry and performing arts. It would be almost impossible if anyone try the same approach on topics such as partial differential equations... We don't know if Euclidean had thought about the symmetry of our human bodies when completing his book "The Elements", but geometry really comes from the part of life that is obvious to most of us. And this is the great starting point for math education for the young. 

Response to Poems About Euclid

In "Euclid Alone Has Looked on Beauty Bare" by Edna St. Vincent Millay, it is clear from the title that this poem talks about something that was witnessed or discovered by Euclid and no one else, which is very likely his work on geometry. Here "Beauty" is used to personify mathematics, and "bare" is used to describe the mysterious "true math". Everyone has encountered math in their life, but most people only "prate Beauty" and "cease". Euclid was the only one at that time who had real insight in geometry, and demonstrated his understanding to the general population through the book "The Elements". Although ordinary folks did not "see the bare Beauty", they can now at least "hear her massive sandal set on stone".

In "The Euclidean Domain", the author seems to be replying to the first poem. 

Euclid alone has looked on Beauty bare?
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?

These lines questioned the statement made in the first poem that Euclid was the only one who had seen Beauty bare. I find it funny that "spoken with her on the telephone" was also included here. It is absolutely impossible to "speak to" Beauty bare in early ages. But the inclusion of this line broadened reader's perception of Beauty bare. 

This poem asked Beauty bare to "put away your sandal", which is also a reference to the first poem, telling the first author to give up this biased analogy.