Blogging on trivium & quadrivium

 

“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”

It is interesting to see that Nicomachus of Gerasa (c. A.D. 200) personified numbers into male and female, but he also identified numbers with various gods. I am wondering if certain numbers would be identified to represent a specific male god or a female god. He also made some "divine numbers" in relation to god. How did he determine which numbers were divine and what did those divine numbers mean to people who were studying mathematics back then?

"Throughout the Middle Ages, university instruction was based on a lecture disputation method…there were no examinations in the modern sense of the term. The student had simply to swear that he had read the books prescribed and attended the lectures. To qualify for a degree, he was required to participate in public disputations, either defending a proposition or opposing one defended by another student."

After reading this quote, the first thought that came to my mind was "Wow, how lucky were these people for they didn’t have to do tons of exams to get the university degree." Their only final "exam" was through disputation. I find this very similar to our education program because we also don’t have to write any exams, but we are required to do lots of reading and discussions. This type of lecture-disputation for granting a degree would be feasible in most majors in the Arts faculty, but most science degrees would probably require some examinations before granting the degree. I wonder if this lecture-disputation method was also adopted for students in medical schools back then.

"Although it is true that much of what was, in the medieval university, course material for a master's degree is today common knowledge for third-grade school children, and although some of the more profound medieval processes of ratio and proportion are today taught in eighth-grade arithmetic classes, medieval arithmetic must not be regarded as superficial or merely elementary.  Many of the concepts are as challenging to modern graduate students of number theory as they were to medieval students of arithmetica."

This quote here has truly amazed me. It shows how human knowledge has advanced from Medieval times to the current days. A university degree in the Medieval Times is somewhat equivalent to our modern-day elementary school level, but there are still some difficult concepts that remain challenged for our modern-day graduate students. Anyhow, it illustrates how intelligent people have always been. From this, we can tell that people are learning to improve, and have improved from learning. 

Blogging on Mayan and other numbers

" Each of the positive integers was one of his personal friends" 

#1729 = the smallest number representable in two ways as a sum of the two cubes

If we think deeply about any numbers we see in life, we are likely able to link them to something personal. The Hardy-Ramanujan number 1729 was only a taxicab number which would probably have no meaning to anyone other than the driver. However, Hardy and Ramanujan managed to think deep and came up a special property for this number. Through their finding, more people would know that 1729 is the smallest number that can be represented in two ways as a sum of the two cubes. Linking to Major's concept on how human make association to numbers and their personal experience, numbers can be more than just numbers, they can have their special meanings to those who would actually take time to learn about them.

• Is this something that you might want to introduce to your secondary math students? Why or why not? If you would use these ideas in your math class, how might you do so?

Yes, I find this topic quite interesting to get students thinking about math and how math is always around us. Math teachers often hear their students ask "When do I ever get to use this in life". This would be an interesting topic to discuss with students. I can conduct a conversation on how numbers play role in different parts in arts, cultures, and our in everyday lives. I can ask students to think about any number and explain to the class how it is special to them. This activity would really get students thinking about how math is everywhere. 

• Do numbers have particular personalities for you? Why, how, or why not? What about letters of the alphabet, days of the week, months of the year, etc.?

For those who have seen me in real life, no one would have guessed that I am a math major and math teacher. To be honest, I don’t consider myself as a math person because I don't have the math brain that is able to connect everything to math right away. Sometimes I would ask myself, "why did I study math?  How did I up end up with a math degree and I am going to math teacher soon?" Everything seems so unreal. 

 I have told the class in the beginning of the school year that my birthday happens to coincide with the well-known mathematical constant pi. Sometimes I wonder if this is some kind of hint or fate that in my life, I will have to deal with math. (*laugh*) It is probably meant to be that I will have to deal with math for my life (at least part of my life) because I was born on pi day? I really don’t have an answer to that, but for sure the number 314 has a meaning to me. But people don’t see it in me because they don’t see me as a mathematical person. Even I don’t see myself as a mathematical person, but one thing I know is that I was willing to learn and I have worked hard for that math degree.  I will definitely keep up that spirit for future challenges.

Reflection on Assignment 1

For this assignment, our group chose to work on discovering rules in multiplying and dividing by 6 in the sexagesimal system. We have talked about the sexagesimal system quite a lot in class, and our classmates also have some decent knowledge on how to work with numbers in the sexagesimal system with the multiplication tables. 

Clearly, talking about the multiplication part was not challenging in this assignment. The more difficult part was understanding and explaining division in the Babylonian system. Our group explained the concept of Babylonian exmaple with an example. We believed our explanation was clear and everyone understood how it worked. The most challenging part in this assignment was the modern way approach portion. It was not as simple as we have seen with the Babylonian system. The modern way approach actually involves more calculation and requires deeper knowledge in math to fully undesrtand the rules. Overall, I think this is a meaningful assigment for math teachers and I enjoyed working with my partners,Chloe and Yiwen. 

Blogging on Euclidean Proofs Dance

 The idea that mathematical proofs can be represented through dance has truly amazed me. I have to admit that I would not have came up with such a creative idea if I was asked to. The beauty of dancing surprisingly worked well with the Euclidean proofs. Using body and arms as the compass is very thoughtful. I also think that the choreographers chose to make the dance moves on sand is very ingenious. They are not only dancing, but they are also drawing out the Euclidean proofs with their bodies using sand as the paper. The fact that all their work "would eventually be erased by tide and wave" makes the whole scene very poetic. They are able to show the proofs through dance, but they don’t last forever. It makes me link to the idea that there is a time limit on beauty. The whole dance is in fact very aesthetic and enjoyable. 

Blogging on Euclid Poems

 Background on Euclid of Alexandria 

Euclid is a Greek mathematicican who lived in Alexandria in Egypt around 300 BCE. 

He is also known as the Fathter of Geometry. 

He wrote "Stoicheion" or "Elements", which is the most important and successful mathematical textbook of all times. His work also include division of geometrical figures into into parts in given ratios, catoptrics (the mathematical theory of mirrors and reflection), and spherical astronomy (the determination of the location of objects on the “celestial sphere”), as well as important texts on optics and music.

information retrived from https://www.storyofmathematics.com/hellenistic_euclid.html

Euclid Alone Has Looked on Beauty Bare

by Edna St. Vincent Millay

Euclid alone has looked on Beauty bare.

Let all who prate of Beauty hold their peace,

And lay them prone upon the earth and cease

To ponder on themselves, the while they stare

At nothing, intricately drawn nowhere

In shapes of shifting lineage; let geese

Gabble and hiss, but heroes seek release

From dusty bondage into luminous air.

O blinding hour, O holy, terrible day,

When first the shaft into his vision shone

Of light anatomized! Euclid alone

Has looked on Beauty bare. Fortunate they

Who, though once only and then but far away,

Have heard her massive sandal set on stone

The Euclidean Domain

by David Kramer

…Euclid alone

Has looked on beauty bare. Fortunate they

Who, though once only and then but far away,

Have heard her massive sandal set on stone.

—Edna St. Vincent Millay, Sonnet

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?

Proud poets, as you penned your paeans to Beauty,

Did you not think it was your bounden duty

(Though it were one that any might have loathed)

To tell that you have only seen her clothed?

And as you sang praise, Orpheus, of Eurydice,

Your mouth became the orifice of your idiocy!

For Beauty bare you never yet had seen,

’Twixt Hades’ depths and lofty Hippocrene.

O Beauty! Would you, for this mathematician,

Remove (if it would cause to give permission

To look on Beauty bare too great a scandal),

Once only, and then but far away, your sandal?

It appears that Millay has portrayed Euclid as the mathematical god who is above all mankind. Through this poem, Millay has applauded Euclid for his work and findings in mathematics.

In Millay' poem, Euclid is able to see the Beauty Bare in mathematics.

 Now, what exactly is Beauty Bare? 

Since Euclid has looked on Beauty Bare alone, it must be something or someone that could be observed and examined by. In relation to Euclid's work on mathematics, Beauty Bare could possibly be some mathematical related concepts or objects that might have given inspirations to Euclid while he looked on it alone. It is possible that Millay is trying to convey the message that Euclid was able to figure something out from looking at the Beauty Bare alone. 

The parody poem written by Kramer has challenged Millay's illustration on Euclid looking at Beauty bare alone. Kramer began by questioning the fact that Euclid was viewing the Beauty bare alone, and no one else has seen or heard about it. I begin to wonder if Kramer was questioning whether Euclid has done all the findings himself? Or he was just criticitizing the fact that Millay has blindly praised Euclid for doing the findings alone without knowing the legitimacy of his work. 

Anyhow, Euclid has contributed remarkbly to mathematics without doubts.