Course Reflection

This is my first time learning the history of mathematics.  This course provides such an in-depth and enriched learning experience in discussing how the notions and methods of mathematics evolved throughout history and across continents.   From our class readings, group projects, and presentations, I was able to discover many fabulous mathematicians’ great works and their interesting life experiences.  The assignment which focused on connecting curriculum math concepts with mathematical history provides an amazing and enriched resources for me as a future math teacher.  Our classmates have put a collective effort in gathering the mathematics history through this assignment gives us the best resource which we can always use as reference in our future teaching.  In addition, many ancient methods of doing mathematics are really fascinating.  It will be a good way to show appreciation to ancient mathematicians by showing the ancient methodologies in solving modern math problems.   

            One suggestion that I have is that we may start a timeline for the history of mathematics at the beginning of the term.  As someone who didn’t know the background of the math history, it was difficult to grasp the whole picture as what had happened in the history.  I think it could be helpful if we can talk about the whole history with a detailed timeline and in a geographic sense before the reading, so that it is easier to understanding the articles.   However, as complicated as the history of mathematics, it might be difficult to really provide that whole picture.

 

 

 Reflection on Assignment 3 

                                    The Beauty of Telling Stories through Tangrams      

When we (May and I) started this project we didn’t really have any idea except we knew that the tangram is an excellent puzzle for kids to play.  As we delved into the topic more, we found that the tangram actually has a very interesting history which can be traced back to the ancient China.  In addition, the tangram is also full of mathematical ideas which can be used in math classroom.  Due to its hands-on nature, tangram can be served as a perfect tool in teaching geometry in a classroom. 

The tangram was said to have roots in a set of furniture of the Song Dynasty, however, the earliest known printed reference to tangrams appears in a Chinese book dated 1813, which was probably during the reign of Jiaqing Emperor, according to Tian.  During the 19^th century, tangram was introduced to Europe and the North America through trading ships. Interestingly, another source said that the rearrangement puzzle roots can be traced back to the 3^rd century BC when Archimedes, the Greek mathematician designed a tangram like puzzle called Loculus Archimedis or Ostomachion.  The puzzle consists of 14 flat pieces of various shapes and the area of each piece is commensurate with the area of the square in the ratio 1:48. Isn’t that so fascinating to know that the history of mathematics from different continents have so many similarities.

            Tangram can be a really fun activity for students.  I can use tangram to teach ratios, fractions, areas, perimeters, and different geometric shapes.  The lovely part of tangram is that students can make animals, birds, houses, boats, mankind, and other geometric shapes. Like Tian said in her article, “give me a set of tangrams, I will reveal the image of the world with geometry” (Tian, 2012).  Since students can visualize all the artistic work they made, it provides a feeling of success and the hands-on experience will make the class time less boring and more engaging.  Another valuable part is that we can tell stories through tangrams as our presentation showed.  Not only we can integrate mathematical history into tangram story telling, we can also integrate Indigenous culture into it.  One thing I found really difficult was how to teach math but also teach Indigenous culture using numbers.  Although we can create a word problem with some Indigenous concepts, students hardly know the meaning and the history behind these concepts.  It seems to me that this type of integration of Indigenous teaching stays at a superficial level and there is no in-depth investigation for our students.  However, if we could use tangrams to tell the story of Indigenous people, not only we will deliver the mathematical concepts, but we are able to let students really learn the Indigenous culture and its history.  Story telling through tangrams is one of the most fascinating way to use in our classroom.

 

 

Tian, X.X. (2012). The art and mathematics of tangrams. Bridges, 553-556.   

 Final Art Presentation Google Drive Link

https://docs.google.com/presentation/d/1_dVG3csojlbZqqCm7hGdqxALpLiw4IEm29FPVQ9jCRw/edit?usp=sharing

Blog Post   Due 09^th

It is a very interesting to know where the custom of naming children come from in the Islamic world. The reading suggests that a Muslim family will name their children use some common names such as Muhammad, Husain, Thabit, etc.  Then what follows after the name is the “son-of so and so”, written as “ibn”.  It really helps people to identify the child’s family background. If we were to follow the tradition, we can name the child by using “the son of so and so” continuously and the child’s name can be really long.  

I think the work that al-Khwarizmi had done really made a significant contribution in the field of mathematics.  However, I didn’t know aside of being the father of algebra, he also had worked in the areas of geography and astronomy.  It was really important that he introduced the Hindu methods to the Islamic world.  Especially in his arithmetic work The Book of Addition and Subtraction according to the Hindu Calculation, he provided the very useful algorithm from the decimal positional system.  Another thing was his contribution on the science of cartography.  I had very little knowledge about al-Khwarizmi until I did my research on division and polynomials.  The reading suggested that he was also an astronomer.  By that time, ancient mathematicians could estimate the size of the earth by multiplying the length of one degree by 360 because they had known that the earth is spherical.  In addition, al-Khwarizmi’s contribution in assisting the construction of the world map by then.  He was able to use astronomical observations and computations to find the latitude and longitude of the earth.  No doubt he was such a wonderful mathematician and had left such a legacy for Islamic world. 

Although I am not familiar with the history of mathematics about the Islamic world, it is vey useful to learn this part of the history and be able to tell stories to my students.  The several great mathematicians and their great works have mentioned in this book are worth remembering.  However, it is challenging to pronounce their names correctly.  In addition, the names of the places are difficult to make connections for me as the cities were not in my knowledge of geography.  Telling the students about this part of the math history is actually where I can draw students’ attention after a lecture or maybe when they feel bored or tired.  However, I need to really work on it if I intend to do it in the front of the students.

  

 

                                    The Art of Tangrams and its Mathematical Implication

1. Our final project will be working the art of tangrams and trace back its history to ancient China

2. References                                             

Read, R. C. (1965). Tangrams: 330 puzzles. Courier Corporation. URL:

https://books.google.ca/books?id=80yRBQAAQBAJ&lpg=PP1&ots=EzOW0rhuWt&dq=history%20of%20tangrams&lr&pg=PP1#v=onepage&q=history%20of%20tangrams&f=false

 

Russell, D., & Bologna, E. (1982). Teaching Geometry with Tangrams. The Arithmetic Teacher, 30(2), 34-38. Retrieved December 6, 2020, from http://www.jstor.org/stable/41192134

 

Siew, N. M., Chong, C. L., & Abdullah, M. R. (2013). FACILITATING STUDENTS'GEOMETRIC THINKING THROUGH VAN HIELE'S PHASE-BASED LEARNING USING TANGRAM. Journal of Social Sciences, 9(3), 101.

 

Tian, X. X. (2012). The art and mathematics of tangrams. Bridges, 553-556.

 

Weng, C., Otanga, S., Weng, A., & Tran, K. N. P. (2020). Effects of tangrams on learning engagement and achievement: Case of preschool learners. Journal of Computer Assisted Learning, 36(4), 458-467

 

Grandfather Tang’s story:

Mr. Boyd reads: "Grandfather Tang's Story" - YouTube

Fractions with Tangrams

GOHMATH ~ Fractions & Tangrams 3 ~ MTEL, PRAXIS, NYSTCE, CBEST MATH ~ GOHACADEMY.COM - YouTube

 

Blog Post   Due Dec.01

Trivium & Quadrivium- Liberal Arts, the Study of “Free Men”

“[Plato] would have the first twenty years spent on gymnastics, music, and grammar, and next ten on arithmetic, geometry, astronomy, and harmony, and the next five on philosophy” (Schrader, 1967, p.264).

I have studied a philosophy course at SFU during my undergraduate study and the course fouced on continental philosophers and their related study.  It was a course I considered more difficult to understand than mathematics.  The terms and their behind meaning are complex and requires abstract thinking. When I read the name Plato and Aristotle, it reminds me of this course. Although their life stories were interesting, it was difficult to really understand their philosophy.  Long time ago when I studied in China, I learned that Plato’s philosophy was extremely abstract and he was the primary Greek philosopher.  However, his works was very popular and used as required reading for many centuries.  Aristotle was influenced by Plato and his works were the basis for both religion and science through the middle ages. One thing that surprised me about Plato was that he also studied gymnastics and music.

“Later, when Christianity gained the ascendency over paganism and the pagan schools no longer a danger, the pagan educational methods were re-examined and were eventually adopted by the Christians” (Schrader, 1967, p.265).

My second surprise was when the article talks about the influence of pagan educational system to Christianity.  I have bible studies for several months with two Christian friends. My understanding was that they were really against pagan religion (that’s the word they used to teach me) and they told me paganism was evil and originated from devil.  It was a shock to me that the close relationship between Christianity and pagan world. From the article, it seems like the study of the seven liberal arts was the prerequisite to the study of theology and pagan educational methods were adopted by Christianity.

“Throughout the Middle Ages, university instruction was based on a lecture-disputation method. …..[t]here 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 lecture” (Schrader, 1967, p.272).

In my opinion, this lecture-disputation is very advanced. It sounds more like our modern day “inquiry” learning style. It was really nice that they didn’t use exam to evaluate students’ learning outcomes. That really surprised me!  In the Middle Ages, students also had “after-class discussions, reviews and recapitulations of the lecture by the young bachelors” (Schrader, 1967, p.272).  Does that sound similar to our tutorial sessions run after the lecture in universities?

 

Reference

Schrader, D. V. (1967). The arithmetic of the medieval universities. The Mathematics Teacher, 60(3), 264-278.

 

Head variant” glyphs for the numbers 1-13 are seen in the inner circle of this plate, created by a local artist at Lake Atitlan.

Blog Post    Due 24^th

The Personified Maya Civilization

First of all, I really enjoyed the presentation given by Myron Medina.  It was fascinating to see Maya civilization from mathematics perspective.  While I was young I heard stories about Maya calendar and Maya prophets. It was full of myth and as a young person, it was a really interesting story to hear.

The way the Mayan record their first 20 digits number using head variants personifies and gives life to each number.  Although Mayan’s bar-and-dot system seems more efficient and practical in recording numbers, it is not as interesting as the head variants system.  One article I have read claimed that the Mayan forgot the source of numb13 and 20 because the Mayan didn’t have a written language but relying on oral legacy for recording their history.  It was said that the glyph for 13 was based on the number of major joints in the human body.  Other sources claimed there were 13 levels of Heaven in Mayan cosmovision, and that the 13-day sub-cycle within the lunar cycle might be the source.

Major explains that “[c]reativity is the ability to make remote connections in the brain… and the ability to make cross-model connections that resonate with other people” (Major, 2017). Relates Major’s point to Hardy-Ramanujan number 1729 and Taxicab number, the seemingly “a rather dull number” turns out to be a very interesting number - “the smallest number expressible as the sum of two positive cubes in two different ways” (Hofstadter 1989; Kanigel 1991; Snow 1993; Hardy 1999, pp. 13 and 68).  Ramanujan, with a curious and creative mind and adventurous attitude, has given life to number 1729.  In his eyes, number 1729 is something with live. 

Ramanujan was a poor math whiz with no formal education and lived in Indian.  Hardy was a prestigious, static math professor who taught in Cambridge university.  The two had nothing in common expect their love for mathematics.  If it were not for the letter from Ramanujan, Hardy would pursue his steady and repetitive academic professional career for the rest of his life.  However, Ramanujan and his mathematical whiz have completed changed Hardy’s life.  I admire the facts that their mutual interesting in mathematics has brought these two people from distinctive social and cultural backgrounds together.  As a teacher, one inspiration from their story is that the importance of being supportive to my students.  In addition, it is crucial to provide opportunities to allow my students to explore the theories and the logics behind each mathematical concept.  It will be beneficial to let them make their conclusions rather than simply give conclusions beforehand.

The most important numbers to me are my daughter’s birthdate. She was born on the last day of summer, which was June 21.  It turns out to be a significant number in Maya’s calendar because it is the longest day in a year.

 

References

Major, A. (2017). Numbers with Personality. In Bridges 2017 Conference Proceedings (pp. 1-8). Tessellations Publishing.

Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999.

Hofstadter, D. R. Gödel, Escher, Bach: An Eternal Golden Braid. New York: Vintage Books, p. 564, 1989.

Kanigel, R. The Man Who Knew Infinity: A Life of the Genius Ramanujan. New York: Washington Square Press, p. 312, 1991.

Snow, C. P. Foreword to Hardy, G. H. A Mathematician's Apology, reprinted with a foreword by C. P. Snow. New York: Cambridge University Press, p. 37, 1993.