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.