Course Reflection

The course started with a discussion on “why do we teach mathematics history?” and we are ending with a group art project presentation on a historical mathematics topic not covered in the course. This course is an interesting guided journey that gradually answers the question of why we should teach mathematics history.

In terms of developments, we come to understand the hard works and smart ideas of the mathematicians of the long past. We come to feel the details of how ideas were developed, and the difficulties faced by ancient peoples. Our story of mathematical development started around Mesopotamia/Babylon (today Syria, Iraq, Iran, Eastern Turkey), Egypt, and then Greek, Roman, and Islamic empires, Asia, Europe, and other places. In terms of the time period, we have attempted to cover perhaps four thousand years. If the set of all known and unknown mathematicians who have contributed to our body of knowledge is a bird, the journey of this bird is thousands of years in time and tens of thousands of kilometers in width. The journey is filled with progress, stagnation, difficulties, improvements, and multitudes of applications both happy and sad. Yet, the journey is probably still at the beginning. The bird has to continue flying into the long future.

In class, we wrote blogs out of readings, imitated ancient methods, developed ideas based on old mathematics. Understanding some details of the development certainly will help us gain more maturity in basic mathematics. This in turn, together with our continued study, would assist us in assisting our students to become responsible citizens equipped with necessary skills. Perhaps, few of our students would become contributing mathematicians. 

The course has a lot of room for flexibility. In addition to numbers, geometry, and algebra, ancient people also developed logic and logical reasoning which are inseparable from math and everyday matters. Logic is an unbiased method of reasoning toward a conclusion that can be tested. We can also learn how mathematics has been applied in various industries. In fact, mathematics was responsible for not only the building of civilization but also for warfighting, injustice, and environmental destruction. Educators’ goals in teaching math should include “peace”, “justice”, “diversity” and “environment”. In democratic education, we should enjoy the freedom to discuss what happened (both good and bad) in the past with respect to the use of mathematics (technology). There are plenty of lessons to learn from. By understanding and knowing all forms of the application, we would be more concerned about our future, our students’ future, and the future of human beings on this planet Earth. We can pass on this concern and need for development toward a peaceful society to our students. Overall, as a class, we overcame the obstacles given by the current situations. The flow of the course is great and I have witnessed many incredible projects/research from my peers. I would like to say thanks to Susan who tried her best to ensure the quality of this course (offered online) as well as her support and valuable feedback. I wish everyone a safe Christmas and a happy new year.

Assignment 3 Reflection

 

It was a pleasure doing this research. Time is one of the most important variables in this universe. Keeping track of time accurately has been therefore crucial to people's day to day lives. The sun is the source that provides life on this planet. Due to its property of being steady and consistent (relatively speaking), the sun is what we can count on to understand time. Hence, we looked into the history of time telling among several geographical locations. The sundial from different places may look different and function differently. That is why it was fascinating to see different perspectives. We are absolutely impressed by the genius of our ancestors. And making a sundial can be used as an activity in a high school classroom! We did run into some technical issues when putting together the art piece. Since we are doing this virtually, it is really hard to exchange and edit drawings.  It would be much easier if we can draw on actual paper or a poster. However, this is a good learning experience for all of us because we will likely get benefit from this in the future.     

Assignment 3: History of Sundials

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

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

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

 

Group Project: History of Sundial (Draft)

 Topic: The history of sundial

Art format: One painting and one hand-made sundial 

Reference list:

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

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

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

[4]: Ancient Chinese Sundials Kehui Deng, 2015

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

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

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

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

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

3 things from "An Introduction to The Mathematics of The Golden Age of Medieval Islam"

 1.  “In the tenth century the scholar al-Sijzī, writing from an unnamed locality, complained that where he lived people considered it lawful to kill mathematicians. (Perhaps this was because most mathematicians were also astronomers, and hence astrologers.)… … mathematicians and astronomers in Islam could expect both honor and support… … ’’.

This is interesting to know. I think the development of any subject would require support from various channels. I cannot believe that during the tenth century, it is considered lawful to kill mathematicians. People at that time relate mathematicians to astronomers to astrologers to fortune tellers and then perhaps to witchcraft. This indicates that the general public at that time was scarcely exposed to any type of science, even the policymakers (who made it lawful to kill mathematicians). On the other hand, I do see that many examples of great mathematicians died either prematurely or from unnatural causes, namely Fourier, Archimedes, Galois, Gödel, Cardano, Abel, Ramanujan, Riemann, Ramsey, and more. This makes me wonder about the reasons for such a visible trend because mathematics is considered the backbone of all science subjects.   

2. “Al-Khwārizmī’s achievements in geography earn him a place among the ancient masters of that discipline… … Al-Khwārizmī’s contribution went beyond this to assist in the construction of a map of the known world … … Among al-Khwārizmī’s achievements in his geographical work The Image of the Earth were his correction of Ptolemy’s exaggerated length of the Mediterranean Sea and his much better description of the geography of Asia and Africa’’.

 I always know that Al-Khwārizmī contributed hugely to mathematics and astronomy, however, I never knew that Al-Khwārizmī also made a huge contribution to the subject of geography. It makes total sense since math can be essential to geography because map-making involves relating spherical geometry and plane geometry. Even small tasks such as mapping a portion of the surface of a sphere onto a plane need theories to support and needs time to verify (for example by travelers and sailors) the precision and accuracy. Having an accurate map at that time would strategically put the ruler/caliph at an advantage when expanding/controlling his/her territories.   

3. “‛Umar is admired more as a poet than as a mathematician, and yet his contributions to the sciences of mathematics and astronomy were of the first order… … he was able, in 1079, to present a plan to reform the calendar then in use… … and produced a length for the year closer to the true value than does the present-day Gregorian calendar”.

‛Umar al-Khayyāmī is the only poet–mathematician I have ever encountered in the literature. I think normally people don’t associate something so sentimental (such as poetry) with something so rational (such as mathematics). I guess what poetry and math have in common is the requirement of being imaginative and creative. Mathematics is not just crunching numbers and doing computations, it is the way for which “logic” expresses itself. Similarly, poetry is how a poet expresses him/herself.

Another thing I was surprised about is that ‛Umar al-Khayyāmī produced the length for the year closer than the present-day calendar. This was quite impressive since it was almost 500 years earlier than the Gregorian calendar!!   

Trivium & Quadrivium

 Three quotes from the article that made me stop and that surprised me in some way.

(1) “Logistic was practical and utilitarian, a study for children and slaves; logic was a liberal art, a study for free men” (Dorothy V. Schrader , 2018, Page 266).

 This tells us something about the social classes of medieval Europe. The slaves (I guess peasants and other workers) worked to produce food, did the construction and maintenance of buildings, etc. These workers studied logistics. The free men studied logic and arithmetic not for the purpose of practical works but philosophical.

Those who prayed (clergy), those who fought (knights), those who worked (peasants)

According to Wikipedia, the period between 1000AD and 1347 AD was an expansion of the population. By estimate, the population grew from 35 million to 80 million. Around 90% of the population was rural peasants, many of them settled into villages called manors. Peasants paid noble overlords rent for places and for services. So one wonders who went to the schools or universities to study logic and arithmetic to know some properties of numbers, proofs and some formal demonstrations, etc (Dorothy V. Schrader , 2018, Page 266)? Meanwhile, good nutritious food production by peasants might be a contributing factor in population expansion.

 (2) In the first arithmetical period (5^th to 10^th century approximately) , the emphasis was on the art of computation especially on the method of establishing the date of Easter (Dorothy V. Schrader , 2018, Page 267).

 Free men went to study logic in the lower level, and then arithmetic in the upper level. Almost all they did, in terms of applications, was to learn how to correctly calculate the date of Easter.  The date of Easter was one very important matter during that period. Many important days of the Christian Church were dependent on the date of Easter. We can imagine the difficulties these free men faced partly due to the Roman numeral system they were using. 

 Only in the second arithmetical period (end of 10^th to end of 12^th century) all four arithmetic operations on abacus were possible due to improvement made by Gerbert (Dorothy V. Schrader , 2018, Page 269). Compared to the scribes doing math on base 60 positional system in Babylonian period more than 2500 years earlier (~ Hammurabi in old Babylon, 1700 BCE), it seems that development of mathematics went through several darknesses.

 (3) The third arithmetic period (end of 12^th century to the end of middle ages ~ late 15^th century) was one of great activity and great change, an almost intellectual revolution in Europe. The Hindu Arabic number (positional decimal number system used today) was introduced and zero was added to the previously zero-less world. It was possible by translation into Latin from Arabic and Syriac (old Syrian language) (Dorothy V. Schrader , 2018, page 267).

We have learned that, by evidence, positional system with base 60 was used in Babylon of Mesopotamia (current Iraq) in 1700-300 BCE (Victor J. Katz, 2009, page 10-12).  Greeks used this base 60 positional system for astronomy but it is not clear if they used this system in other situations. Even though earlier ancient Chinese used base 10 system, the clear evidence of positional system including zero in China came from 12^th century (Victor J. Katz, 2009, page 198). In India, Syria and Cambodia earliest evidence of the use of decimal place value system came from 7^th century.  Meanwhile, until the third arithmetic period, European were using the cumbersome Roman numeral which requires a great number of symbols to write a large number. Calculations with Roman numerals are difficult.

 It took more than 2500 years for human being to finally adopt the positional numeral system after first  discovering it in Babylon. If the positional numeral system was a bird species, it was an amazingly difficult journey dotted with deaths and rebirths for that species to spread its seeds around the world for humans to make good use of.

Reflection on Assignment 1

 In our first assignment for this course, we talked about Babylonian arithmetic using base 60 sexagesimal system. We learned about the history and Babylonian way to multiply and divide in base 60 system as well as the modern way interpretations of multiply and divide in the sexagesimal system. Overall, I think the organization of our presentation was reasonable. We covered some basics at the beginning so that audience have good background for the later content. One thing we would consider to improve in the future is to include a slide for summary and wrapping up. We shouldn't have ended our presentation abruptly. We had rehearsed several times before the presentation to make sure we had good pacing. We didn't use much of materials from the media but we did have activities to engage the class with some interactions. I would say that we did a satisfactory job and we will take what we have learned for the betterment of our future studies.