Article response: Trivium & Quadrivium

" Plato, …, conceived of such education as the sole occupation of the first thirty-five years of a man's life. He would have the first twenty years spent on gymnastics, music, and grammar, the next ten on arithmetic, geometry, astronomy, and harmony, and the next five on philosophy.1*"(p.264) 

I am interested in the subjects involved in the Greek educational system. These subjects are gymnastics, music, drawing, reading, writing, grammar, rhetoric, arithmetic, geometry, astronomy, natural sciences, harmony, philosophy, and dialectic. I have been thinking the similar thing. I suggest psychology and logic as essential studies for the modern world. We need to understand how people's brain is working, and how to think logically. These two points are important and useful.

I agree with Plato that ideally, people should have their first 30 or 35 years focusing on learning different types of subjects. However, not everyone can have such treatment. In ancient Greece, women were generally not given a formal education. Nowadays, obtaining education is a luxury for many people in the world. I am sad about this. I hope everyone could get the chance to go to school, go to university.

"Among the Greeks computation or reckoning, the arithmetic of business was called logistic and was considered to be entirely different from the study of number as such, which philosophical study was called arithmetic."(p.266)

The philosophical study was called arithmetic. This reminds me of Zeno's Dichotomy paradox and Achilles and the tortoise paradox. These paradoxes are a set of philosophical problems, but they are also mathematical mysteries. In ancient Greece, mathematics, astronomy, and philosophy are three subjects bundled together. I just wonder why we don't learn astronomy anymore. I still can remember how excited I was when my high school organized an event at an astronomical observatory. I think that learning astronomy could make the students curious and become more open. Ancient people like to gaze at the night sky, and modern people like to gaze at the cell phone screen. Although our brain can reach everything through this little screen, our heart is placed in a box, I mean we cannot feel those feelings.

P.273

It makes sense that the symbol of multiplication X is derived from above method. I think the solution is : 100 - 20 - 30 + 6 = 56. I wonder what if 3 x 7?   Using Recorde's method, (10-3 = 7 ) and (10-7=3 ), 3 x 7 =? It will go back to the original question. I think solution is : 10-3 = 7, then 100 - 70 - (10 - 7)x3 = 21. 

4 x 9 's solution: 10 - 4 = 6 , then 100 - 60 - (10-9)x4=36

Article response (Nov 24th) : Numbers with personality

How does our brain work? How cognitive systems are structured or functioning? One day they could be found out. While reading Major's article we can learn that “Imagining personalities for numbers involves cognitive systems that are linguistic as well as mathematical.”

The story of Taxicab Numbers and how Ramanujan found the uniqueness of number “1729” is as intriguing as how “Goldbach's conjecture “comes into being. After reading Major's article, I couldn’t help wonder if positive integers are impersonalized in Ramanujan’s mind when he studies their patterns and property. And that might give him intuition to realize the relationship between cubes and sums when G. H. Hardy told him the story of this number on the taxicab. We can even assume if he is given other numbers, Ramanujan could also discover their uniqueness mathematically. Since all integers are talking to him.

I would like to introduce these stories to secondary math students. Since this makes mathematics much more fun and student can feel more related. In my class, a new talent like Ramanujan might just need a story to be inspired for lifelong interest or a great math discovery. I will try adding these fun elements naturally when teaching integers and cubes.

Some digit numbers do have personalities for me. Influenced by Chinese culture I like number 6, 8, and 9 and feel they are lucky numbers. In Chinese tradition, each digit number has a meaning. For example, the number three means “life”. It is considered a good number. The number 4 is considered an unlucky number because it sounds like the Chinese character “death”. Normally, I don’t think these symbols have personalities. They are useful and important. I respect their usage and rules.

Response to "Dancing Euclidean Proofs!"

Once again, I am impressed and moved by the perfect combination of Math and Art in this course. I really enjoy the beauty this video brought to me, especially when I know exactly what the dancers are trying to convey. The background music with a little sound of waves is soothing. The dance on the beach sand is expressing some classic propositions from an ancient mathematical and geometric treatise. What a wonderful feeling! 

During the decision-making processes in creating the dances, three co-authors (Milner, Duque, and Gerofsky) came up with many good ideas. There are two of them stood out to me. The first one is that the authors decided to use both arms to spin in circles in Dance 1. For proposition 1, we know there is no diameter in Euclid's original diagram, however, they creatively found a beautiful solution to the imbalance problem. Moreover, with "extra" arms they naturally made an equilateral triangle and a fluid dance. The second one is that the authors decided to dance on the beach in order to draw in the sand to record the movements. Like they said, the new element-- Land, added beauty and more possibilities. When dancers disappeared after the dance, the trace left in the sand reminded me of a famous painting "The School of Athens."  In the painting, Euclid is drawing a theorem for his students with a compass. I somehow entered that painting as I was looking at the trace in the sand and hearing the background music. Just like the authors say in the article that "as we dance the proofs, and as a live audience might view them, we somehow enter the page." (p.243)

" The body does not move in a vacuum, but in response to stimuli from the land and place."(p. 245) Using the environment as part of the proof is a great idea. Old civilizations found wisdom from the natural environment. I think people naturally like to feel connected to nature. When I was young, I always drew trees and mountains in my pictures. Even though I couldn't explain why at that time, I felt trees and mountains are indispensable. 

To demonstrate the propositions of Euclid's Elements through movement and dance was not easy. Many things and details related to proofs and choreography were needed to be considered. But at the same time, the entire process must be full of interesting ideas and beautiful moments.

Explication and commentary on a poem about Euclid!

Ancient Greek mathematician Euclid was born around 365 B.C. in Alexandria, Egypt. We know almost nothing about his personal life, but we all know he was the father of geometry. That's because he wrote The Elements, the most widely used mathematics and geometry textbook in history. Euclid's Elements is the earliest example to discuss geometry in a systematic approach. It includes 23 definitions, 5 postulates, 5 common notions, and 48 propositions. Although many of these results had been stated by earlier mathematicians, none of them like Euclid showed these propositions as a comprehensive system. Euclid was the first one to see the whole picture of geometry. 

Two poems are a tribute to Euclid's work. Because of his Elements, we changed the way of how we look at the world. To my understanding, Beauty in the poem represents the world, and Euclid was the only one who "looked on Beauty bare". Mathematicians before him have seen just a part of Beauty, and people after him have only heard a distant echo of Beauty's step. Euclid was the only one who knew how to see the world. With Euclid's Elements, can we see Beauty one day?