Course Reflection

Although I was learning mathematics at University for 4 years, I never took any course about mathematics history there. I am so lucky to have Professor Gerofsky as my teacher so that I could learn such abundant content. 

This course explores my curiosity about building connections between mathematics history and teaching mathematics in high school. It helps develop my abilities to stimulate students' interest in mathematics history. I would like to get students to understand more about the way that mathematics developed historically just like how our Professor did to us. Humanity, human endeavor, cultures, they are all could be taught in form of the storytelling, playing traditional games, solving ancient puzzles, etc. 

I enjoyed reading " The crest of the peacock." It shows me a big picture of  mathematics history. The charts in the book helps me understand the development of mathematics in all cultures and places. I also enjoyed working on two projects. One project is about "Ancient Egyptian Fraction", and another one is about "the compass and straightedge construction." I learned a lot by researching on these two projects, and more importantly the whole process was joyful. I believe that high school students would like to know and work on these topics too. They should know who Euclid and Euler are, what stories related to Fermat's last theorem and Gauss's formula are talking about. All in all, mathematics should not be taught in a vacuum. I am so lucky that I can study this program and learn from Professor Gerofsky so that I know why and how to keep working and researching on developing ways of teaching mathematics that incorporate history and culture to make learning mathematics full of fun and fruitful. I know I still need to work hard to enrich my knowledge in this area for that purpose. I appreciate the door this course opens for me, and now I entered and will move forward.  

Assignment 3-Reflection:

Through working on this project, I gained more knowledge on this topic: Compass and Straightedge Construction. And I found that drawing a picture using only compass and straightedge could be a very great activity for high school students. And it is necessary to involve this topic when teaching students geometry. 

The stories related to this topic are interesting. Students could understand how the problem originated. For example, there is a story about the doubling cube. 

[Eratosthenes, in his work entitled Platonicus relates that, when the god proclaimed to the Delians through the oracle that, in order to get rid of a plague, they should construct an alter double that of the existing one, their craftsmen fell into great perplexity in their efforts to discover how a solid could be made the double of a similar solid….]

When I was creating the artwork, the most significant part was thinking about how to construct each stroke. If I want to draw a triangle, I have to think about which method I should use, and how to make it not arbitrary. I think this activity could help to develop high school students' creative skills. They can get experience reasoning about axiomatic system. Teaching student compass and straightedge construction can also help them develop logical and geometric reasoning. 

I think this project is very meaningful to me. I will definitely use it in my future teaching practice.  

Assignment 3: Explication of the Artwork

 Slides: 

https://docs.google.com/presentation/d/1M9hQBBKYPPtYrK5JE_7VumyGOTVbhnkPVEgzco1Y4Xk/edit#slide=id.gaffa94aba8_0_309

The artwork is called Silent Night. This is a reasoned and reliable picture since every stroke was drawn by using a compass and straightedge and based on the lines or circles that have already been constructed. All straightedge and compass constructions consist of repeated application of five basic constructions (constructing the line segment, the circle, the intersection of two lines, the intersection of a line and a circle, the intersection of two circles) using the points, lines, and circles that have already been constructed. 

The first problem is how to draw the first horizontal line on the paper. This line has to be constructed based on a horizontal line that has already been constructed. A-ha, the edge of the paper is a good choice. The interesting part of creating this picture is that nothing is arbitrary during the process of creation. 

Based on my research, I learned its restrictions and rules, its history and relevant stories, learned how to draw five basic constructions and how to draw parallel lines and perpendicular lines by using different methods. I also learned how to draw a pentagram in three different ways. It was used by the ancient Greeks as a symbol of faith. Some books show us how to prove the method. They show the way to find the value of cos72(degree) in a unit circle by using the idea of the geometry of complex numbers. Although it is not easy to understand the mothed of constructing a regular polygon with 5 sides, the ancient Greek mathematicians already knew how to do it 2000 years ago. 

In this drawing, the methods I used include: constructing an equilateral triangle, constructing a parallel line through a point, constructing a perpendicular bisector of a line segment, constructing a 45-degree angle, constructing a 90-degree angle, bisecting an angle, and construction a pentagram. When constructing the parallel line and the perpendicular line, I tried different ways to do it according to different theorems. 

For example, to construct a parallel line we could use the method of copying an angle according to the following theorem in reverse. 

There is a theorem: two lines are parallel if they are cut by a transversal such that two corresponding angles are congruent.    

We also could use the rhombus method. I prefer this mothed. It is simpler. 

It is meaningful to know compass and straightedge construction. We can get experience reasoning about axiomatic system. We could teach students logical and geometric reasoning by teaching them Compass and straightedge construction. A person stated on-line that as he had not learnt to use the compass straightedge construction, he didn't know compass can be used to measure distances so that he always thought that a circle is something round. He didn't know the most important property of a circle: a circle is a list of points at an equal distance from a central point. 

Article response: Episodes in the Mathematics of Medieval Islam

 "His (Al-Khwārizmī ) other famous work, written before his Arithmetic, is his Kitāb al-jabr wa l-muqābala (The Book of Restoring and Balancing), which is dedicated to al-Ma’mūn. This book became the starting point for the subject of algebra for Islamic mathematicians, and it also gave its title to serve as the Western name for the subject, for algebra comes from the Arabic al-jabr. " (p.9)

Now we know why "algebra" is called algebra. I think it is great to tell my students how we name this subject as "algebra". By telling this story, the name of this subject "algebra" is no longer arbitrary. All children like to hear stories, at the same time, they could easily remember the word "algebra" as well as "al-jabr". "al-jabr" has the meaning of balancing. This makes sense. We are trying to keep the equation in balanced when finding the unknown numbers.

"The other is his (‛Umar al-Khayyāmī) suggestion that the idea of the number needed to be enlarged to include a new kind of number, namely ratios of magnitudes. For example, in ‛Umar’s view, the ratio of the diagonal of a square to the side (square root of 2) or the ratio of the circumference of a circle to its diameter (π), should be considered as new kinds of numbers. This important idea in mathematics amounted to the introduction of positive real numbers and, as was the case with the parallel postulate, this was communicated to European mathematicians through the writings of the pseudo-Naṣīr al-Dīn al-Ṭūsī." (p.16)

This is another thing that I did not know before. It was Umar who suggested to include more types of numbers, such as some irrational numbers. I could also introduce this historical story to my students when I draw the Venn diagram as below:

"The observatory, as the scientific institution we know today, was born and developed in the Islamic world. Here is a part of the sextant (or perhaps quadrant) at the observatory in Samarqand where al-Kāshī worked. It was aligned in the north–south direction and was 11 meters deep at the south end. Thus an astronomer sitting between the guide rails could have seen the stars crossing the meridian even in the daytime while assistants sitting on either side held a sighting plate through which he could observe the transits of heavenly bodies. It was at this observatory that the greatest star catalog since the time of Ptolemy was compiled" (p.21)

Amazing! I didn't know that the observatory was born and developed in the Islamic world. I believe that children all like to visit the observatory. I will show them photos or let them watch videos about this observatory. I am always thinking of how I can integrate some astronomical knowledge while teaching mathematics. For example, space systems can be taught when teaching patterns and cycles. Many interesting Math problems are related to astronomy. I think I will design some interesting practice questions which combine mathematics with astronomy for my students.

Assignment 3---Draft 1:

Topic:  Geometric Structures Using Compass and Straightedge

The artistic format: Drawing（Only use compass and straightedge)

Draft reference list: 

1. Moti, B., 2019, Surprising Constructions with Straightedge and Compass, Creative Commons Attribution-ShareAlike 3.0

https://www.weizmann.ac.il/sci-tea/benari/sites/sci-tea.benari/files/uploads

/rwcourse/construct-en-v1-0-0.pdf

2. Richeson, S. D., 2019, Tales of Impossibility, Princeton University Press

3. Lee, G. T., 2018, Abstract Algebra An Introductory Course, Chapter 14 Straightedge and Compass Constructions, Springer International Publishing

4. Kendrick, D. (2015). The Basics: Geometric Structure.Sarhangi, R. (2007).

5. Geometric constructions and their arts in historical perspective. In Bridges Donostia, Conference Proceedings, The University of the Basque County, San s  Sebastian, Spain, Reza Sarhangi and Javier Barrallo, eds. London: Tarquin Publications (pp. 233-240).

6. Lim-Teo, S. K. (1997). Compass constructions: a vehicle for promoting relational understanding and higher-order thinking skills. The Mathematics Educator, 2(2), 138-147.

http://math.nie.edu.sg/ame/matheduc/journal/v2_2/v22_138.aspx

It has a lot of fun!

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?

Eye of Horus and unit fractions in ancient Egypt

 

The most interesting thing about Eye of Horus to me is its missing fragment. The legend says Horus' eye was broken into 6 pieces. Ancient Egyptians gave each of them a fraction as a unit of measurement. They are:  1/2,  1/4,  1/8,  1/16,  1/32,  1/64. In the legend, the eye of Horus was restored and made whole. So, the sum of these 6 unit friction should be equal to 1. However, we know the sum was short of 1 by 1/64. There was a fragment missing. 

There are some assumptions on this point. 

Some people believe that the missing part was withheld by Thoth's magic. 

Some people said it could mean that nothing is perfect. 

Some people explained that if we see 1/2,  1/4,  1/8,  1/16,  1/32,  1/64 as a well-known Calculus 'infinite sequence', then this infinite series can be added up to exactly 1. The missing 1/64 was supplied by Calculus. 

He Tu 河图 ( River Chart )--Chinese Cosmology

He Tu is also called Yellow River Map. "He" 河 means River, river of stars or galaxy. It is cosmological diagrams used in ancient China. It was used for the movement of the stars through the Nine Palaces.

The Nine Palaces were the groupings of stars that were identified

that traversed the heavens.  

It also used in geomancy. The table shows the meaning of numbers when it serves to explain the correlation of universe and human life. 

Reference:

https://thekongdanfoundation.com/lao-tzu/the-heru-luoshu-and-nine-palaces/

Constructing a Magic Square

please work on figuring out this 3X3 magic square, where each number from 1 to 9 is used once, and where all the rows, columns and diagonals add to 15!

 My solution:

1. The first thing I thought was to put 5 in the center because I found these pairs (1, 9), (2, 8), (3, 7), (4, 6) plus 5 is equal to 15.  

 2. 9 is the biggest number, start looking at this one. I can only find two combinations with 9 that sum to 15 : 1+5+9=15, 2+4+9=15. So 9 cannot be placed at corner. Then 1, 2 , and 4 can be placed in the square.

  3.  Then the rest of the numbers can be placed in the square by simply using addition.

Reading response----Was Pythagoras Chinese- Revisiting an Old Debate

Pythagorean theory is one of the most important theories in Euclidean geometry. When the Chinese high school Math teachers introduce the gou-gu theorem (another name of Pythagorean theory in China), they always emphasis that, "No, the gou-gu theorem is not invented by Gou Gu, no such a person called Gou Gu. Gou represents the long leg, and Gu represents the short leg, Xian represents the hypotenuse. Gou three, Gu four, and Xian five." 

Students always try to get some information more or less from the name of the theories or laws. For example, Newton's laws of universal gravitation, and Newton's laws of motion. Students could know that these laws was firstly formulated and published by Newton. The theories, such as Euclid-Euler theorem and Galileo's Principle, get named after their originators probably because people could easily reference them by  their names. For example, Galileo Principle was given by Galileo in his second book " Dialogues concerning Two New Sciences" published in 1638.

For Math teachers, I think it is unnecessary to intend to avoid acknowledging non-European sources of mathematics. If the source of mathematics is appropriate to support their teaching practice, they could use it to enrich students' knowledge base.  There are hundreds of proofs of the Pythagorean theorem. Different proofs use different mathematical theories. For example, Garfield's Proof used the knowledge of area of trapezoid.  Chinese Zhaoshuang's proof used the knowledge of squares of binomials. Teachers could choose those proofs to support their teaching if the proofs are relevant and appropriate.  

The method of 'false position' (estimate, check, adjust)

Problem:  Tom has a glass of water. He pours out three eighth of the water, then he has 20 gallons of water. How many gallons of water does he have at first?

The modern solution:

   x  -  3/8 x = 20        5/8 x = 20       x = 32

Egyptian " False Position" solution:

Try x = 8,    if originally Tom has 8 gallons of water

3/8 x = 3,   then he pours out 3 gallons of water

 8 - 3 = 5,     then he has 5 gallons of water. 

However, in the problem, Tom has 20 gallons of water left, which is 4 times as much  as 5 gallons of water. So this time, we could try 32 ( 4 x 8 = 32 ).

If x = 32,  3/8 x 32 = 12 , 32 - 12 = 20 

Assignment 1-- Extension about Egyptian Fraction

Reference:

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fractions/egyptian.html

(I put the reference at the beginning is because this is a great webpage about Egyptian fraction. I highly recommend you to visit it if you are interested in the Egyptian fraction. Using copy and paste if the hyperlink doesn't work. )

From Egyptian Mathematical Papyrus, we found that ancient Egyptians could handle of fractions 4000 years ago. They only have notations for Unit Fractions, but they can represent more general rational numbers as a sum of distinct unit fraction.   

                                

Here are two algorithms for finding Egyptian Fraction (during my presentation, I only introduced one):

Method 1: Using Splitting Equation

Example:     2/6 

Decompose a fraction into the sum of unit fractions

               2/6 =  1/6  +  1/6  

(All the unit fractions should be different. That's because when ancient Egyptian repeated the process of dividing, the reminder gets smaller and smaller.)

Convert one of the repeated unit fractions into the sum of distinct unit fraction by using the splitting equation:

               2/6 = 1/6  + 1/7  + 1/42   

                         

Method 2: Fibonacci's Greedy Algorithm

Fibonacci proved and gave this method in his book Liber Abaci, the same book for Fibonacci Number.

Reflection (written on Nov 17th):

Due to some technical issue with my computer, my part of the presentation went not well on that day. After I shared my screen, the audience looked at a screen that was different from what I looked at. We realized this problem when I almost reached the last piece of slides. I think if I were not that nervous and talking slowly, we could notice that problem earlier. 

Fortunately, the two most important and interesting slides, "Egyptian solution" and "Fibonacci’s Greedy Algorithm" were presented without a problem. There are so many interesting topics under Egyptian Fractions, and I just presented the tip of the iceberg of it. There is so much fun to learn about the history of mathematic, so I think I will introduce it to my students in the future. 

Response on The History of the Word Problem Genre

Teaching and Learning word problems could help students better understand Mathematics concepts and real-life situations. However, not all word problems reflect reality, or not all of them have the practicality. Some word problems might be designed for further exploration, or just for practicing mathematical strategies, or just for fun. From this point of view, word problems have generality and abstraction. Not only symbolic manipulation could be abstract, word problems could also be abstract, especially for those artificial word problem. " Pure " mathematics and "applied" mathematics cannot be completely separated, since even the most abstract mathematics could have applications.

Above statements are true for word problems either of Babylonians or of modern world. Although ancient Babylonians lived in more than 4000 years ago and their representations are limited by language and notations, they have the same wisdom as the people living in modern world. Our familiarity with contemporary algebra could not change the nature of the mathematics.

Entry slip (Sep 28)----Babylonian "algebra"

 

I could not imagine how Babylonian solved those puzzles before the development of algebra. I wonder how did they figure out these procedures? 

For example, in example 4.6 in the article, the third step of Babylonians' solution is " Take the reciprocal of 0; 33,20". I do not understand how they knew to do so. What is their logic to take the reciprocal in this step? Without the unknown variable x, what do these independent numbers 0; 33, 20 and 1; 48 mean to them? 

Another example is about solving quadratics. I am surprised at Babylonian's solution of Example 4.7 shown in the article. Their approach was equivalent to our quadratic formula. I cannot imagine how they figured it out. Their first step is " Halve 7",  but it does not make sense to me. Why halve 7? Even though they got the right answer, how could they explain logically to others? 

I guess they made many tables to summarize the rules and laws about   algorithms. They might not have a general mathematical principle, so they need to study and record many cases as the samples. 

I think mathematics is mostly about abstraction and I hope it could be more about generalization. Once we deal with numbers and do calculations, we are entering the Math world, and it is abstract. For example, you have 5 apples, and I give you another 5 apples, if I ask you how many apples do you have, you will think about 5 plus 5 equals 10. When you doing this calculation, there is no apple in your mind. You are using some symbols and rules, and there is nothing to do with apples. When you get 10, you jump back from abstract math world to the real world, and tell me you have 10 apples. So if mathematics could be more about generalization, we could use and apply it more easily. 

Without algebra, I think sometimes it is not easy to state general or abstract relationships in any areas of mathematics. Since if it is an general or abstract relationship, we need to use some symbols to represent every thing in such category. Algebraic expressions is used to represent mathematical relationships. So I think using algebra is a good way to state relationships. 

Class notes:

Here is a good piece elaborating on the origins and uses of base 60 by Mesopotamian/Babylonian mathematicians, from the excellent University of St. Andrews math history database. 

https://mathshistory.st-andrews.ac.uk/HistTopics/Babylonian_numerals/

A Babylonian word problem:

Entry slip (Sep 25.)---Babylonian Table of Multiplying to 45

 

Class notes:

What might these tables mean?
The first column multiplied by the second column equals 60. Or, 1/2 = 30/60, 1/3 = 20/60, ...1/10=6/60, this is a table of unit fraction. 
Note that, in this notation, commas separate place values (for both whole numbers and fractions).

Can you figure out the common theme here?

Why are certain numbers missing from the left hand column? For example, there is no 7, 11, 13, etc.
Because 60 divided by these numbers will get recurring decimal numbers. 
How do fractions in the Babylonian style connect with our fractions? (Keep this in mind as we learn about ancient Egyptian fractions later on...)

Entry (Sep23)---- The Crest of the Peacock

 

" The Crest of the Peacock", what a beautiful and miracle book. When I read it, I feel like entering another world. From Egypt and Mesopotamia to Europe, from the Dark Ages to Renaissance, plenty of pictures appear in my mind. Mathematics as the bridges connect these cultural areas and eras, and also connect me and you.  

The four-thousand-year-old Babylonian clay tablet with the value of n^3+n^2 struck me. This clay tablet implies that "Babylonians may have used these values in solving cubic equations ". I wonder how and why they invented it. Is it used for calculating the orbit of stars or trying to figure out the relationship between tracks of sun and moon?

When I read about the cross-cultural contact between India and China, I recalled a famous classical historic Chinese novel named "Journey to the West". This story is about a Tang dynasty Buddhist monk traveled to India to find Buddhist scriptures. This novel is so popular that most of Chinese have read it. It is also an evidence of a cross-cultural contact between India and China. In Yuan dynasty, a Chinese mathematician Zhu Shi Jie wrote a Math Book. Some every large numbers used in this book were from India's Buddhist scriptures. 

One of the features of mathematical activity through the ages makes me feel admiration for ancient mathematicians' spirit of pursuing knowledge. This feature is stated as "the relative ineffectiveness of cultural barriers(or 'filters') in inhibiting the transmission of mathematical knowledge." Around 2500 years ago, Greek Mathematicians such as Pythagoras and Eudoxus need to travel on foot to another country to learn knowledge. One can imagine their journey must be full of difficulties. Nowadays, people could work and learn online at home, and obtain many resources remotely. We should cherish what we have today, and remember what they have done for us yesterday.  

Link of the book:

http://www.ms.uky.edu/~sohum/ma330/files/Crest_of_the_peacock.pdf

Entry ( Sep 21) ----- Base 60

 

Thousands of years ago, the Babylonians used a number system based on 60. One of the reasons to use 60-based number system is because of its convenience. From 1 to 100, 60 is the only number has factors including 1, 2, 3, 4 and 5. That means if an ancient people has 60 fish, he could equally distribute these fish to 2, 3, 4, or 5 person, and if he has 2 times 60 fish, he still could evenly divide fish into 2, 3, 4, or 5 parts. With this property of 60, they divided many things including the time, the year, and the circle. 

Especially in the astronomy measurement, (yes, ancient people like to watch the sky, me too), people often need to divide angles. If they use Base-10, it would be difficult to divide an angle into 3 parts equally.  

In human history, there existed the base-8, base-12, base-16 and base-20 number systems. In China, people are still using the idiom "Half catty eight taels" to describe two things which have no difference to each other. It proves that Chinese people ever used base-16 number systems in the past. 

Nowadays, the base-10 number system is mostly used  in our daily life. Comparing with the base-60, the base-10 needs less symbols to represent digits, and can be easily learned. However, when we tell time and measure angles, the sexagesimal plays an irreplaceable role. Even today, in Chinese traditional calendar, they still use traditional way to  count year based on the sexagesimal. 

The base-60 number system embodies the ancient people's wisdom. Through writing this blog, I have an opportunity to learn and show my appreciation to those great endeavor.   

Below is the link of the reference article:" Why is a minute divided into 60 seconds, and hour into 60 minutes, yet there are only 24 hours in a day?"

https://www.scientificamerican.com/article/experts-time-division-days-hours-minutes/#:~:text=The%20Babylonians%20made%20astronomical%20calculations,the%20first%20six%20counting%20numbers

Course Note:

Our detailed look at the history of mathematics will start in an area known as Mesopotamia (meso: 'between', potamia: 'rivers' -- between the Tigris and Euphrates Rivers), in what are now the countries of Iraq, Syria, parts of Turkey and Kuwait. The area is sometimes called the 'fertile crescent' because the river systems flowing down to the Persian Gulf made agriculture and cities viable.

In accounts of mathematics history, the mathematics of Mesopotamia circa 3100 BCE - 300 BCE is usually called Babylonian mathematics. But if you look more carefully at the history of Mesopotamia in this period, there were several different peoples and nations that governed this region, including the Sumerians, Akkadians, Assyrians and Babylonians. 

For our purposes, we will use the term 'Babylonian mathematics' to refer to the fairly unified mathematics traditions over this 3,000 year period (starting approximately 5,000 years ago). 

Babylonian writing was done with a wedge-shaped reed stylus in wet clay tablets the size of a person's palm, and then dried in the sun or in a kiln. Their ingenious writing system was known as cuneiform: 'wedge-shaped', and it was possible to write words, numbers and other symbols with just these wedge-shaped forms. Because these baked clay tablets are very durable, we have many of them in museum collections to this day -- and quite a few of these seem to be teaching tablets for scribes learning mathematics to take government jobs in the Mesopotamian cities!

Here is an example of one of the existing mathematical clay tablets from ancient Mesopotamia. Your job is to figure out what is written here, and how the writing system for this mathematical tablet works!

First reading (Sep 9)----Why teach Math history?

 

I firmly believe that Math teachers should mention some interesting historical Math stories when teaching a certain theory or concept. For example, when teaching the Gauss formula many Math teachers would tell a story of Carl Gauss about how he amazed his teacher with finding the sum of the integers from 1 to 100 when he was a young boy. Through telling stories, students will know the names of mathematicians, uncover how and why these concepts have been invented, and most importantly feel happy and relax. Students love stories. Teachers could tell the story of zero when explaining why zero cannot be the denominator, the story of the Pythagorean Theorem, the story of numbers when teaching place value, and so forth. Telling Historical Math stories is the only approach I knew about integrating history of mathematics in the classroom. Since the history is normally imparted chronologically, how to connect these pieces together to show students a whole picture of the history of Mathematics makes me feel I am not teaching history.

After reading the article of "Integrating history of Mathematics in the classroom: an analytic survey", I realized that "integrating the history of mathematics into the educational process" is so significant. The authors of the book elaborated Why from five aspects. Among them, "The appreciation of mathematics as a cultural endeavor" is the one that I strongly agree. Without Math,  technology, medical treatment, transportation and many other things wouldn't be possible. To show our sincere appreciation to Math, we need to know its history more or less.

Another part of the article inspired me is the Historical Problems. Problems with no solution or problems unsolved could show students what is the tenacious and perseverance. These are the spirits of Mathematics we want to pass on from generation to generation. Be courageous, do not fear the problem that looks hard. It is hard, and the only thing we need is taking a little bit more time on it.