Re: Class reflection

    Before starting this class, I thought the history of math would be bland and dry to learn. However, as the class went on, I really enjoyed learning about the history of math as it's not just the events that happened years ago in math, the history of math is really about how human beings evolved over the years. It's fascinating to learn that math plays a very important role in the history of cultures and science. Math started as a human's basic needs to live, but it's interesting to see math itself involved in music, arts, literacy, and even astronomy. It's curial for new teachers to learn about the history of math and how math has always influenced our lives from many perspectives because knowing that gives us a better understanding of why we teach math and what's the purpose of teaching math. I also enjoyed learning math in an "old math" way to solve current questions. It's unbelievable sometimes to see how people back then solve some current math questions! 

    I really liked the layout of the lessons, it's very logically make-sense. I also enjoyed knowing those little math stories along the way that I may talk to my students about! 

Re: Reflection on Assignment 3

    When I was doing some research on Ada Loveless, I felt a sense of proudness for her as a female back then and still had high education and dedicated work. She had a very unfortunate childhood abandoned by her father, but that doesn't stop her from being herself and pursuing what interested her. 

    I also thought about the timing of all milestones in science are not coincident. A lot of her work was produced during the time of the industrial revolution. A certain revolution will bring out a lot of related inventions, and the difference machine is one of the products of the industrial revolution. It's fascinating to me to think about the history of math has come a long way!

    I also enjoyed working with Matt and Ivan! 

Re: Assignment 3 - Ada Loveless

 Our group consists of Matt, Ivan and myself. Below is the link to our "draw my life" video of Ada Lovelace:

https://drive.google.com/file/d/11r9fq8ZYN3fp1VBWtcU23RsmgoG1JArr/view

Re: An introduction to the mathematics of the Golden Age of medieval Islam

After reading the article, I realized how geographic locations have always influenced the history of math. Also, astronomy has an enormous influence on the history of math. From Medieval to China, there have been many records showing the close relationship between astronomy and math. It's interesting to see how humans have always been curious about space and other planets in space, and math is often used in calculations of related matters. Many new inventions in the field are from science and space, which further approved that math is beneficial and important in other science subjects. I believe it's important for teachers to introduce math history to students in different aspects to understand other related subjects to make math more interesting. I wonder if people already knew the existence of other beings in space. 

It's fascinating to see how many mathematicians are also experts in other fields like astronomy, architecture, and literature. Most of them are well-educated with curiosity. They are often very observant. They all have the common characteristics of curiosity and are eager to learn, which are the crucial traits to success. 

Lastly, math history seems always to travel to different regions in its own course, and they all have similar common ground or theory behind them. It's fascinating and strange how the history of human beings and knowledge evolves. When you think about it, it seems like there has alwasy been an invisible hand controlling the history of science along the way and it's crazy to think about how our technology is so advanced nowadays, which makes me think what will the technology look like in another 100 year? 

Re: trivium & quadrivium

 "Quintillian (a.d. 35-95) suggested that a boy study grammar, music, geometry, and astronomy until he was sixteen, after which age he might advance to higher studies." I think it has some reflects on today's educational system. People need to study themselves before getting an advanced education. I read somewhere saying that people read enjoy reading books because books help them understand themselves. In one man's life, no one will ever get tired of getting to know themselves. All the learnings are more fun and sustainable if you enjoy it, "grit" is the most important factor for success. The pure happiness a man can have when doing something he enjoys cannot be described in words. That's why it's important to learn generally before specializing in certain areas. It is also why having a healthy mindset of learning is the first step of getting a higher education. People have to have the good fundamental basic knowledge to receive and make the best use of higher educations. 

Knowledge starts with human beings' basic needs; that's why learning from experiences is the best way of learning. That's why the new curriculum emphasizes the importance of applying knowledge in real-life situations. I think different subjects should be integrated altogether to provide a meaningful connection with each other. It also creates a multi-dimension way to think and learn!

Re: Alice Major on Mayan and other numbers

     It was fascinating to read this article on numbers and personality. It offers such a unique perspective way to see the meanings behind all the numbers. I think teaching should be put into context, giving stories to it. Psychologically, it is proven that students learn better when they involve their own life experiences. Infuse your own personality into math numbers can be creative and fun for students. 

    I also really liked the concept of how language has an influence on how people understand numbers in different cultures. For example, number 4 means stability in some cultures but means death in Chinese culture because it has a similar pronunciation of "death" in Chinese. A lot of buildings would skip floor level 4 in the elevator in China. Obviously, there is no scientific approval on that, but it just adds so many characteristics to different numbers.

    Using such a concept in lower-grade math classrooms will be very helpful and enjoyable for students. It gives students the freedom to make up their own personal connections to numbers and remember some boring math fundamental facts more fun! As someone who doesn't have great memorization, making up related stories helped me remember boring academic concepts. 

Re: Reflection on Assignment 1 (Sorry, it's so late)

 I really enjoyed the process of doing the assignment. It was such a different experience to have a group project online. I never really thought about how ancient math works in our modern math, especially from a math education perspective. Having the opportunity to learn more about how the pyramid was created and how people calculated its volume. My part was introducing how the pyramid was created in the old days, which gives me an in-depth understanding of why it's important to teach the history of math in a class. "whys" gives students the curiosity and motivation to study the "hows" in a math classroom setting. It also offers the most mathematical proofs with actual contexts, which gives a story for teachers to tell the class, making the context more interesting and attractive for students. For students who enjoy math, it gives them a sense of fulfillment of learning math and knowing the importance of math in tour lives. 

My teammates are both delightful to work with. Marius can always make jokes out of many tough situations and find positive aspects of everything I lack. Jacob is also very much knowledgeable of many things; he always offers a unique perspective on the topic, a lot I can learn from both of them! 

Re: Dancing Euclidean Proofs!

I have always thought of proofs as something very logical and routined and I like that aspect of it a lot. The structure of proofing something has its beauty in the form of clean lines and clear step-by-step procedure. However, dance to me is very soft and indirect. Proofs to me are like the school of Bauhaus in design, it's very logical and clear-minded with very little warmth to it. 

After reading the article, especially after watching the dance itself. I felt the warmth and logical beauty of proof through the dance itself. More accurately, I think the movement of the human body is a piece of art itself. 

I find it's fascinating that we can use our legs and arms to "prove" such complicated proof. The rhythm of the dance mirrors the logical side of the proof, it's planned with thoughts but flows so seemingly. 

For non-math people, it becomes so much easier to understand the basic concept of the proof through the use of our own bodies. 

I was also surprised to realize the natural symmetries of our body which is very mathematical in some ways. In conclusion, I appreciated the fact that they used the form of dance to show the beauty/logical thoughts behind proofs!

Re:Explication and commentary on a poem about Euclid (and a parody of it)!

 Euclid is often referred as "the father of Geometry". Geometry exists everywhere in nature, it is the most bare version of beauty. Euclid uses math to show the beauty of nature through the eye of geometry, he also used nature to explore the beauty of mathematics. These two things are naturally inseparable. Euclid's vision on geometry opened all of ours eyes, he created mathematical language to express the beauty of geometry around us. He had a pair of eyes to see through things and noticed the bare bone of lots of things that are related to math. 

Thanks to Euclid, we were able to use math to express/sketch the things we see in this world. We were able to document a lot of historical events using math! 

From the first poem, it seems like It's Euclid who lifted the veil of mystery of the mathematical behind all the natural beauty. The harmony of nature and math was all along there waiting for humans to discover it, and Euclid noticed it. The gate of math then revealed at that moment. Time never stopped, nature never waited for anyone, nature takes its course and it seemed like Euclid was the savior of the dark time of math and opened the gate of pandora's box of mathematics. 

The second poem seems like is criticizing the prasie most people had towards Eulid. It aruges that beauty is in everyone's eyes, math belongs to all of us. No one could be the maker of nature. In other words, no one was the maker of mathematics. Math has been always existed among us, in our day-to-day life. Math is like the air, we breath in and out every day. 

Maybe it's a bit off the topic, it reminded me of a movie called Prometheus. Who is the God, who made us? and, who made them? Who made this world and who created the nature which is incredibly beautiful and mathematical? 

Why do We Teach Math History

When talking about history, I think most of us would have a sense of curiosity and admiration. Reading about the history of math would give us a claerer view of "whys" and "hows". Teaching the history of math will help students to explore the "whys" behind the math we are talking about today, letting them to appreciate the ins and outs behind all the formulas.

I would think students who don't like math wouldn’t take any interest in learning the history of it, but if educators could introduce math via the medium of teaching history, students might take another look at what is math in a different perspective. Math was created from our daily lives which is closely tied to human history, students should be able to notice the beauty of math from learning its history first. I think one of the most common reasons that math was not the most liked subjects in school is because it seems so far away from our daily life, students think there is no use of math. However, it's actually the opposite as math was created from people for practical purpose in their daily lives. Teaching students about it would hopefully help them to see the importance of math and math education. 

Re: Constructing a Magic Square

 This question reminds me of Sudoku I used to enjoy playing. 

To start with the first number, I decided to start with the number in the center. I picked 5 as my first tray-on number for the center number since it's the middle number among 1-9. Since now the middle number is determined to be 5, we know that A,D & B,C should be from the following numbers, 

15-5=10

1,9

2,8

3,7

4,6

Then I tried to put 9 on the upper right corner then realized that it won't work since the number is too big that there won't be so many options left for other spots in the square. Then I tried to put 9 in the middle spot on the top row, then I solved the whole squre. 

See below, 

Re: Research online the significance of the Eye of Horus and unit fractions in ancient Egypt.

    The Eye of Horus reminded me of the pineal gland in our brain which in some regions was called the "third eye". The pineal gland looks like the Eye of Horus and it's the part of the human's brain, the primary function of the pineal gland is to produce melatonin. Melatonin has various functions in the central nervous system, the most important of which is to help modulate sleep patterns. It's interesting how I personally take melatonin sometimes when having trouble sleeping. 
   
     The human pineal gland grows in size until about 1–2 years of age, remaining stable thereafter, although its weight increases gradually from puberty onwards. The abundant melatonin levels in children are believed to inhibit sexual development, and pineal tumors have been linked with precocious puberty. When puberty arrives, melatonin production is reduced. That's why in some cultures, "the third eye" is considered as the symbol of purity and soul. 

    Interestingly enough, there is some math behind this as well. the Eye of Horus was thought to be used by the ancient Egyptians to represent one divided by the first six powers of two

The right side of the eye = ​¹⁄₂

The pupil = ​¹⁄₄

The eyebrow = ​¹⁄₈

The left side of the eye = ​¹⁄₁₆

The curved tail = ​¹⁄₃₂

The teardrop = ​¹⁄₆₄

This reminds me of the golden ratio as well just with different base numbers, which is essentially everywhere in our lives. 

Re: Reading Article Was Pythagoras Chinese - Revisiting an Old Debate

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

Yes, most of math theorems/theories are similar across different cultures. However, they are not totally the same/duplicate of each other. Every culture has its own perspective in terms of understanding the theorems and the purpose of developing the theorems are also different. I think it's important to let students view/understand the same theorem from different perspectives will help them to foster the ideas into their more practical learning. As Canada is such a multicultural country, It's important to introduce non-European sources of mathematics to the class so that students will learn to appreciate the versatility of mathematics around the world. Mathematics is truly a great lens to see the world not only mathematically but also historically! 

•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.) 

I personally don't feel it's wrong to use someone's name for the theorem, but It can mislead sometimes as that person might not be the first one who discovered the math behind it. However, as we are teaching/learning in a western country, it's understandable to first introduce the western version of the theorem which makes it's even more important to share the non-European sources of mathematics in class. Sadly, schools most likely only introduce westernized mathematics in schools in Asian countries and forgot about they have their own history of mathematics! We should at least introduce them to the students so that they are reminded of their roots which should be embraced by people all around the world. Math is not just some formulas that were recorded in books, it's rather a part of the history that witnessed the development of cultures and human beings. Math doesn't belong to "someone", it belongs to all of us!

Re:Ancient Egyptian 'algebra': The method of 'false position' (estimate, check, adjust)

 

Amy just got paid from her part-time job working as a sales assistant in a clothing store. Her paycheque for two weeks was $1500. Her friend Sara asked her how much does she spend on her outfits, Amy answered "I don't know how much exactly but the money I spent on clothes is half of the money I spent on my food and rent, and I already spent all of the money I got from my current paycheque." How much did Amy spend on her outfits? 

Modern Solution: Let x be how much Amy spent on her outfits. 

x+2x=1500

3x=1500

x=500

False Position: Let x be how much Amy spent on her outfits. 

Try x=100, x+2x=100+200=300

We need the answer 1500, which is 5 times as big as the current answer, so x must be 5 times as big as the trial number. (100*5=500)

Check: If x=500, x+2x=500+1000=1500

Re: History of Babylonian Word Problems

I really enjoyed reading the chapter! When I finished reading the first two pages of all the word questions. I made some notes aside, I saw there was a pattern to them! Every word question has a subject (you, I, she, he, and they) and a verb (doing something), and there's always a time and a place for the word problem. While reading the problems, I begin to have a picture of in my head, illustrating the event/problem is being described. That's so interesting because I would not have the same "picture" of the problems if they are simply a theory/concept like greek math was mainly about! Here we see the importance of having word problems that are familiar in our life as it's easy to read our own experiences rather than thinking in someone else's shoes.

I would think Babylonian math would be the "first-class math" if I didn't keep reading since it's relatively real-life related and in order to have the ability to solve those problems, they will have to know the reasoning behind it, right? However, after finishing reading the whole chapter, I then realized that in order to come up with some sort of theory/theorem like how greek did, it's unquestionable that they will have to know the ins and outs of it first. On the other hand, knowing how to solve problems doesn't necessarily require any deeper understanding of the concept. In another word, solving word problems is like training yourself to imitate Claude Monet's paintings, you can have your tweaks here and there but it's not your creation. However, the authentic meaning of education is to deeply understand the fundamentals behind all the "whys", or at least the encouragement to achieve that! 

In my own experience, it's very challenging for both students and teachers to translate mathematical theories/theorems into a language that most students can understand without doing any word problems, or we can say without imitating a sample solution. Relating back to the article, I don't think there is such thing as the first/second class of math. Pure, applied or practical math are related, it's like building a skyscraper, every piece of concrete counts. As educators, we should encourage students to understand math in their own ways. There is a right/wrong answer but not in terms of how we learn. 

Re: Babylonian ‘algebra’ from Crest of the Peacock

 I think using algebra or single letter notation in math helpes solving problems in a much easier fashion and also easier to present and understand. However, I don't think it's necessarily a must in terms of solving math problems, a lot of math questions can be represented in words. So why do we have all the notations that we are using now? From the book Crest of the Peacock, we know that the fundamentals of mathematics were gathered from different countries/regions, in order for them to communicate notations had to be invented/created. It made more sense if the communication of math only happens in the same region, but communicating with others from outside the region might bring confusing. I would rather think of notations as the "regulations" for the math world. It's a universal tools for people to communicate in mathematical language at a ease. 

As for whether math is all about abstractions and generalization, I don't agree with that. As the concept of math is so broad and everyone learns/understands math differently. However, I do think abstraction is a great tool for students to ease into math as it gives the students the freedom to potray math as they like. Generilzation is also important as it gives the students some foundations/facts to imply math in a abstract way. Both are curial in math learning. 

I think using pictures/illustration would be a good way to state some math problems like geometry, or easy number theories. Stating calculus without algebra would be very hard. I still think notation's excitence has its purpose there. 

Re:Create your own Babylonian-style base 60 multiplication table for the number forty-five

knowing the example 2   22,30 with base 60, I then worked backward. setting the column 1 number first and I can then find combinations numbers for column 2. 

For example, I picked 4 as my column 1 number, so I know that column 2 must be (a+b/60)=45/4, since it's a base 60 question, it will be easier to change the base 45 to 60 which gives us 45/4=675/60. Simplify 675/60, gives us (11+15/60), therefore we have the answer of [4.  11 15]. Please see the below answer by using the same method. 

Re: The Crest of the Peacock by George Joseph

 

Three things that surprised me: 

1. The most significant developed mathematical activities was flourished around the twin rivers. It's so interesting to see how geography had a huge impact on how people's migration which leads to the rise of the economy in a certain area. Advanced mathematical development encourages the growth of the local economy and a growing economy also has a positive influence on the evolution of mathematics. 

2. I'm surprised that "by the second half of the first millennium AD, the most important contacts for the future development of mathematics were those between India and the Arab world." It's interesting to see that a lot of ancient math developments started in Asia and the Middle East. Maybe it's because of the silk road. Every culture has its mathematical history, but surprisingly they all have some common understanding/knowledge of numbers. 

3. I noticed that a lot of philosophers were involved in the revolution of math in history. It's interesting to see how math is not only related to numbers back then. Math is the product of the economy, philosophy, and astrometry, etc. That made me think that math is the foundation of a lot of things/subjects. I also find it's fascinating how Asian countries consider math is the most important subject in schools! Pure math might be boring, but reading about history and doing some researches might encourage students to take a like in this most-disliked course among high school students. 

Re: Homework to be posted by Monday, Sept. 21 at 9 AM on your individual blog: Base 60

I think the reason why 60 might be a convenient, significant, or especially useful number to use as the base for a number notational system is related to how we count time. 1 hour consists of 60 mins and 1-minute consists of 60 seconds and also 6 o'clock divides the clock/circle perfectly into 2 halves. In many ancient countries, 6 and 9 two magic numbers, especially 6 is considered as a number with peace and balance. When we think about time, a year has 12 months which is 2 times 6 as well. 

Surprisedly, we still have 60 seconds in 1 minute and 60 minutes in 1 hour! Seems like how we record time never really changed. In many cultures, there are still a lot of traditions that have close relationships with number 6, or 60. The Chinese zodiac has 12 animal symbols, and so as western horoscopes! When you think about the reasoning behind it, time was pretty much the only common knowledge/understanding of people across the world know despite different languages were spoken. 

After doing some research, the reason why Babylonian mathematics had a base 60 rather than 10 is that 60 has more divisors than any smaller positive integer which is so fascinating. Scientists even found out that they knew the formula that's today known as the Pythagorean theorem. Talking about the history behind 6, there was this "most commonly accepted theory holds that two earlier peoples merged and formed the Sumerians," USA Today reported. "Supposedly, one group based their number system on 5 and the other on 12. When the two groups traded together, they evolved a system based on 60 so both could understand it." Knowing the history behind all this really made me think about how math was created among people all around the world and it was serving such important purposes in people's lives.