Wednesday, December 16, 2020

course reflection

What fascinates me the most about this course is the range and depth of topics that we have covered. When I first saw the title of the course "Mathematics History for teachers",  I thought to myself, "history and math, how interesting can these two topics be". I thought we were going to read some long articles to find out how important mathematical theories were discovered in history. 

Surprisingly, the history component of this course turned out to be quite interesting. I like how we didn’t just learn about the history behind the developement of mathematics, but also the history of the nations and culturesthat contributed to mathematics. I also enjoyed the artistic aspects that were added to learning about the history of mathematics.

Throughout the course, we have worked on math problems that could be used as starter activities for high school students. For example, the locker problem and the magic square. I have asked my grade 8s to work on the magic square problem during my short practicum. Again, I like the range of topics and questions we have covered in this course. The way that this course is organized is also very interesting. It is my first time taking a course that asks students to write blogs. Anyhow, I think it is very creative and fun.

Since we are learning online, the level of interaction between classmates is limited. One suggestion I would like to make for this course is to have more in-class and blogging interaction and between classmates, so we can get to know each other better.

Thank you Susan and Amanda for a great term! 


Tuesday, December 15, 2020

self-reflction on assignment 3

 I really enjoyed working on this final assignment for the course with my group mates Chloe and Yiwen. Although we have not met each other live in person, but it seems like we have known each other for a long time. We also worked together on the first assignment, and we had no problem with communication and collaboration. 

We didn't know what topic to choose for this art - history project at first. Since we are all from China, we thought we would research an important Chinese figure that has made contribution to mathematics. However, we could not come up with creative artwork to do with such a figure. We then saw the email from our instruction Susuan suggesting on history of sundials. We thought it could be an interesting topic to work on. We also decided to compose a drawing/picture of sundials so we could each be in charge of one part of history and one part of the artwork. 


In all, the collaborative component went well and I can't wait to present our work to the class. 

Sunday, December 13, 2020

Assignment 3

 

Our group chose to represent our topic, the history of sundial, through this piece of drawing. This is because sundials are usually artistic in their designs, and visual representation can easily differentiate the various types of sundials. In history, many nations have individually developed and used sundials to keep track of time. Since there was no direct linkage between all the nations in using sundials in history, we have decided to combine all our findings together in one drawing.


In this drawing, we have put the large sundial in the center with cardinal directions pointing at the geographic location of different regions. Although being a sundial, the large sundial tells a different story from the time. We are focusing on the history of sundial in ancient China, ancient Greek, Renaissance Europe, and Medieval Islam. For each region, we put down the most typical representative sundial used in the era by the mentioned nations.  On top of that, we are representing our findings with drawings that we think are symbolic of the history and development of corresponding sundials. We decided to place the sun at the east where it rises, and the shadow of the gnome separates the three regions that we are going to introduce in detail. 


In teaching, we can show this drawing to the class, and ask students to discuss the history and relations to the given topics. The topic of sundials can be used to explore how trigonometry was used to tell time and improve the accuracy and precision of sundials from different periods of time. This artwork can also be combined with geography or physics classes where it is relevant. We can also include a hands-on activity in class to engage students in making sundials. 


References 

Berggren, J. L. (2007). Sundials: An Introduction to Their History, Design, and Construction. Hands on History: A Resource for Teaching Mathematics, (72), 19.

Berggren, J. (1999). Sundials in medieval Islamic science and civilization. Coordinates, 1(9), 6.

European association for astronomy education. Short history of sundials. Retrieved 07 Dec 2020, from https://www.eaae-astronomy.org/find-a-sundial/short-history-of-sundials

Sabanski, C. (n.d.). Equatorial Ring Sundial. Retrieved December 14, 2020, from https://www.mysundial.ca/tsp/equatorial_ring_sundial.html

Shell-Gellasch, A. (Ed.). (2007). Hands on history: A resource for teaching mathematics (No. 72). MAA.

Sundial Histrory - First Time Keeping Device. (n.d.). Retrieved December 14, 2020, from http://www.historyofwatch.com/clock-history/history-of-sundials/

Vincent, J. (2008). The Mathematics of Sundials. Australian Senior Mathematics Journal, 22(1), 13-23.

Tuesday, December 8, 2020

Assignment 3 draft

 


Art format: One painting and one hand-made sundial 


Reference list:

[1] 2,000-year-old sundial unearthed in southern Turkey's Denizli, Daily Sabah, 20 March 2020

[2]: Archaeologists find Bronze Age sundial dating back more than 3,000 years Ancient Origins, 07 Oct 2013

[3]: Sundials: An Introduction to Their History, Design, and Construction From Hands on history, a resource for teaching mathematics,  2007 J. L. Berggren, Simon Fraser University

[4]: Ancient Chinese Sundials Kehui Deng, 2015

[5]: A brief history of time measurement Feb 2011, University of Cambridge, By Leo Rogers

[6]: Short history of sundials European association for astronomy education

[7]: The mathematics of sundials Australian senior mathematics journal 22(1) Jill Vincent University of Melbourne

[8] http://cultureandcommunication.org/deadmedia/index.php/Sundial  (sundial timeline)

[9] https://equation-of-time.info/sundials-with-shaped-styles

Saturday, December 5, 2020

Blogging on Mathematics of the Golden Age of medieval Islam

 In reading about Al-Khwarizmi's contributions, I am surprised by the fact that he also managed to contribute to geography and astronomy aside from mathematics. It is fascinating to see that many great mathematicians have also made contributions to astronomy. It makes me relate to how math may seem to be a little bit dull when it stands by itself. But when math is tied to other areas such as astronomy and sciences, it usually facilitates in finding and discovery of new concepts and new inventions. We can say that math is contributing to many areas of study implicitly. When teaching mathematics, I can tell my students that math can be applied to many other areas of study so it is important for us to learn math. 


Again, Al-Biruni's interests were wide and deep. He had contributed to astronomy, astrology, pharmacology, and of course mathematics. I am surprised by how some people are born to be geniuses and they are meant to be born to make contributions to the world. Without these smart people, the world wouldn't be moving forward. We need to thank and look up to these people for their contributions. 


Lastly, what surprised me is the fact that these great mathematicians/educators would dedicate their entire life to discovering and finding new things. Many of the works they have accomplished took them years to compile together before they were released. I truly respect their perseverance in all the hard work they have contributed to mathematic and scientific developments. 

Saturday, November 28, 2020

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. 

Saturday, November 21, 2020

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.

Tuesday, November 17, 2020

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. 

Friday, November 13, 2020

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. 

Sunday, November 8, 2020

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. 


Saturday, October 17, 2020

Blogging for Eye of Horus and Unit Fractions in Ancient Egypt

 


The most interesting finding is that people still use the Eye of Horus today. In Mediterranean countries, fishermen would paint this symbol on boats and vessels for protection. People would also make this symbol into jewelry to wear daily for protection against illness. It is also a symbol of power, knowledge and illusion. 



Chinese people dislike the number "4" because its articulation sounds very similar to "death" in Mandarin and Cantonese. In most buildings, floor and unit numbers do not have the number 4, so there are no 4th,14th,24th ..(so on) floors, and no unit numbers such as 104, 114, 204.. etc.  Contrarily, Chinese people favor the number 8 because it sounds very similar to "make a fortune" and "become rich". 

Constructing a magic square

 

Rule: each number from 1 to 9 is used once, and where all the rows, columns and diagonals add to 15 
I also tried to make a 3x3 magic square with different numbers, but I failed. 



Saturday, October 10, 2020

Oct13 Response to Was Pythagoras Chinese?

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


If we do or do not acknowledge the non-European sources of mathematics, for most of our students, I don’t think it will make a difference as long as they learn what they are suppose to learn as prescribed on the curriculum. I think it will only make a difference to those who are interested in knowing more about mathematic history. For example, if we acknowledge the Greek and Chinese sources of mathematics, students who are interested in the Chinese or Greek culture may be more attracted to the information presented. However, as teachers, it is worth noting other non-European sources of matematics so their students recognize that there are other people in the worlds who also contributed to the knowledge we are learning today. Nobody's contribution should not be ignored. 


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


It is interesting to see that most mathematical theorems and concepts are named after the main contributor who discovered or proved the theorems in the Western culture. Whereas the Chiense had named their theorems and concepts differently.  For example, the Pythagorean Theorem is named Gougu Theorem in Chinese because ancient Chinese called the right triangle "gougu". I like the idea to have theorems and conpcets named after the contributors/founders because it show apprecitaion and acknowledgement to their hard work.

The method of False Position

 




Sunday, October 4, 2020

Oct 7 Blogging on History of Babylonian math word problems

Based on the examples of word problems in Babylonian mathematics, it seems like they are created from real life examples. The contents are based on everyday life situations such as calculating grain-pile, bequest, and how much you can buy with a given amount of money. They seem to be practical and general. However, the article also addresses some Babylonian mathematics words problems were beyond real life applications. Old Babylonians might have purposed designed those abstract word problems in a way to test how much they could do with the mathematical knowledge they had at that time. 


In contemporary algebra, we tend to follow the two levels from Babylonian mathematics. First level being practical and general mathematics that can be applied in real life. Second level being extended word problems that are more abstract, and aim to practice the mathematical concepts learned at a higher level. These second level word problems are often the ones we hear students complaining about, "When do we ever get to use this in real life?". I think it is important for students to be able to solve word problems at the first level, but also understand word problems at the second level. It is true that we may not encounter these higher level abstract problems in real life, but this pureness in mathematics is what we need for developments in learning about mathematics. If mathematics were only at the practical and general level, how do we move forward to learn more about mathematics?  If mathematics had remain at the everyday life examples level in the Babylonian era, would we still have the mathematical knowledge in today's world?

Saturday, September 26, 2020

Sept 30 Blogging for Babylonian ‘algebra’ from Crest of the Peacock

 

"In particular, we own to the Arabs in the field of mathematics the bringing together of the technique of measurement, evolved from its Egyptian roots to its final form in the hands of the Alexandrians, and the remarkable instrument of computation (our number system) which originated in India; and the supplementing of these strands with a systematic and consistent language of calculation which came to be known by its Arabic name, algebra. "

                                                                                      -  from Crest of the peacock, page 7


I like this quote because it show how so many nations gathered together their knowlege to reach an outcome on what we are learning today.  

Before the development of algebra and algebraic notation, Old Babylonians used symbolic notations to state general mathematical principles. They used words in their language to describe the mathematical terms. For example, ush means length, sag means breadth, asha means area. So  ush multiply by sag result asha. The Babylonians were also able to solve quadratic equations in its symbolic variant very similar to our modern day notation. 

Math is certainly not all about generalization and abstraction, it is about the thinking behind these generalizations. Without algebra, we could try stating general or abstract relationships verbally with words (jsut like the Old Babylonians). However, this adds more complication to understanding the gerneralizations or abstractions. 

Monday, September 21, 2020

The Crest of the Peacock by George Joseph

 


Three things that surprised me were:

  1. European trajectory 

The ideology of European superiority has surprised me. The article mentions "the contributions of the colonized peoples were ignored or devalues as part of the rationale for subjugation and dominance" (p.4) . In other words, Europeans were taking credits for the discoveries done by other nations, and claimed those were their findings. I think we would call this plagiarism in our modern day society. Other nations such as China, India, and the Hellenistic world have contributed to the development of mathematics, and I agree with the main message that "it is dangerous to characterize mathematical development solely in terms on European developments" (p.12).

2. Figure 1.4 The spread of mathematical ideas down the ages

I am amazed by the intelligence of our ancestors. Regardless of time and place, ancient people in Egypt, Greece, Mesopotamia, India, Arab world, China, and Mayan empire were all under construction for mathematical concepts. Each nation had their own way of approaching and understanding mathematics, and they would transmit their ideas to each other, all working towards the development in mathematics. 


3. Cross-cultural contact between China and India

Besides mathematics, works on astronomy and medicine were also exchanged between these two nations as early as 500 BC (which over 2500 years ago). Once again, I am amazed by the intelligence of our ancestors! 

Friday, September 18, 2020

Base 60 !

 

What special meaning did base 60 have to the Babylonians at that time?  Were 60 tribes nearby? Was 60 is the amount in their money system? 





In our daily life, we use the number 60 as a unit in the time-telling system. (60 seconds = 1 minute, 60 minutes = 1 hour). We also use 60 to make reference to the degrees of a circle (360). 


By doing some resarch online, I found some interesting facts about the Babylonian Mathematics and the base 60 system.

- Babylonians adopted the number 60 as base possibly because it has more divisors than any smaller  positive integers.

-The Babylonian math system did not have zero. 

-The Pythagorean theorem we use today is based on Babylonian math multiplication formula.

                                                                                                                                        (Gill, 2020)


In another article by Sweeney, he notes that "the Chinese or Egyptian culture have developed elaborate and highly accurate divination systems to predict the future, all based on Base 60". People in ancient China would predict the best times for planting and harvesting crops based on the traditional Chinese calendar which has some connection to the number 60. Sweeney also mentions the Pisano Periodicity, which indicates that "nature has a law that living things develop to the fullest to the number 60. " 

For those who are intersted, here are the links to the two articles/

https://www.thoughtco.com/why-we-still-use-babylonian-mathematics-116679

https://vixra.org/pdf/1407.0062v1.pdf

course reflection

What fascinates me the most about this course is the range and depth of topics that we have covered. When I first saw the title of the cours...