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.

 

 

 

Blog Post Due 17^th

Dancing Euclidean Proofs!

If I were not taking this course, I would never have known that mathematics can go hand in hand with dance.  It is even more striking to me that we can dance Euclidean proof!  The experience I have with Euclidean was from Number Theory.  To me, that course was dry and full of abstract concepts.  I often had hard time to understand the mathematical logics behind each of the theory or theorem.  When I saw Oliver Byrne’s edition of Euclid, I knew that if I were taught with a pictorial way, I would have understood Number Theory much better that I have now.  Taking one step further, Dr. Gerofsky and her students were able to incorporate dance with these proofs!

If the audience were to read the pictures of Olive Byrne’s edition of Euclid, they would be amazed to see that the actual dance of Carolina and Sam were beautifully and accurately representing the propositions.  The dancers’ goal was to present the most beautiful body movements and to present the proofs most accurately with their bodies.  As Dr. Gerofsky describes “[s]ometimes our bodies felt at odds with the geometric abstractions…and felt like trying to connect two south poles of magnets” (Gerofsky et al., 2019).  The process in designing the dance often times requires “rethinking and repositioning” where they “found the math and the dance actually fell together quite naturally…” (Gerofsky et al., 2019).  And as the dancers used their body parts to represent circles, lines, and points, they finally integrated land into their dance as well.  This decision has freed their body constraints and allowed them to rethink how the proofs could work by adding new elements such as sand, rock, or shells.  This creative and constructing process enables a beautiful combination of mathematics, dance and the nature!

The “dancing proof” has shown a new methodology in studying mathematics - a dynamic, cooperative and visual way.  To study while dancing, we become “the active agents responsible for the making and understanding the representation” (Gerofsky et al., 2019).  I think this innovated way of learning mathematics is applicable in secondary math teaching.  We can design such a project and teach our students in the same way.  I believe the math concepts learned from this “dancing proof” experience will eventually internalize and become part of our students’ permeant knowledge.   

 

Reference

Milner, S. J., Duque, C. A., & Gerofsky, S. (2019). Dancing Euclidean Proofs: Experiments and Observations in Embodied Mathematics Learning and Choreography. In Bridges 2019 Conference Proceedings (pp. 239-246). Tessellations Publishing.

 Reflections on Presentation of Ancient Puzzles in Ancient and Modern Way 

It is fascinating to know that ancient Egyptians had so many mathematical word problems recorded in Rhind Mathematical Papyrus. It was a good experience to have the opportunity to do some research on this problem topic. By looking into both the ancient and modern solution of the same problem, I really appreciate the mathematicians who has contributed their efforts and intelligences in advancing mathematics. 

The Ahmes' Loaf Sharing problem is easy to be solved if we use the formula of arithmetic sequences and series.  However, using ancient method to solve the same problem will require more time and the trial and error method sometimes can be trivial. Nonetheless, the ancient method allows us think more about how to teach students to solve one problem with different approaches. 

 

Blog Post Nov 08^th

 

Euclid Alone Has Looked on Beauty Bare

Euclid of Alexandria was the most famous Greek mathematician and was referred to the “father of geometry”.  His most announced work was The Elements which recorded a system of rigorous mathematical proofs.  The Elements covers his work on geometry, algebra and number theory which is seen as one of the most influential mathematical work in history.

In Edna St. Vincent Millay’s poem, she refers Euclid as “the second coming of Christ” (McGee, n.d.).  Millay’s admiration for Euclid has made her set his’s intelligent insight above the rest of the mankind.  She claims that Euclid is “alone” the one who can see beauty of mathematics with the rest as Praters.  This conclusion comes from Euclid’ genius contribution on the beautiful mathematical proofs in pure mathematics. Using biblical reference, the poet alludes that Euclid has opened the door of a new way of thinking for mathematics, which put him in a position of “mathematical savior” and using The Element to bring a new light to people (McGee, n.d.).

In response to Millay’s work, David Kramer has dismissed the idea of “Euclid as the Jesus of Mathematics”.  He questions the ignorance of Millay for not seeing great contributions from other intelligent mathematicians in history, but rather, putting one man in a holy position.

I admire the intelligence of Euclid but believe that the advancement of mathematics is more of a collective effort made by many mathematicians throughout history.  The mathematics we know today has taken hundreds of years of evolution to formulate the ground-breaking theorems and theories.  We can easily name some distinguished mathematicians and their works in history: Issac Newton (Newton’s Law of Motion &Calculus), Leonhard Euler (Euler’s Equation), Carl Friedrich Guass (Guass’ Law), Fibonacci (Fibonacci Sequence), and Bernhard Riemann (Riemann Sum) etc.  It is hard to imagine that one genius’ work would form the whole system of mathematics, although Euclid does change the thinking and approach to mathematics to his successors.   

 

Reference

McGee, Ryan. (n.d.). A Mathematical Jesus: An Explication of Edna St. Vincent Millay’s “Euclid Alone Has Looked on Beauty Bare”. Life Orange.  Retrieved from

https://lifeorange.com/writing/writing2.htm