Trivium and quadrivium

The first stop I had when reading this article was from the discussion from Plato in which he believed that education should be the sole occupation of the first 35 years of a man’s life, with the first 20 years spent on gymnastics, music and grammar; next 10 years on arithmetic, geometry, astronomy, and harmony; and the last 5 years on philosophy (Schrader, 1967, p.264). Essentially, that could be equivalent to today’s time of a person going to school from Pre-K to grade 12, then immediately going into a 5-year post-secondary program, 2-year master’s program, and 3/4-year doctorate program with still 11 years to go. If a person today spent the first 35 years of their life devoted to education, they would have so much time for mastery learning of subjects before moving onto new topics. There were various views by Greeks and Romans; Greeks who were concerned with the education of free men (non-slaves) as future citizens and Romans who expected a boy to be ready for advanced work before he turned 16.

I did not know that arithmetic was considered an essential part of the curriculum in cathedral schools and was taught in all monastery schools (Schrader, 1967, p. 267). They emphasized computations as they were seeking to determine the date of Easter. It was also mentioned that numbers were identified with various gods and that odd numbers were considered male and even numbers female (Schrader, 1967, p.267). This reminds me of Alice Major’s (2017) paper from last week that discussed Ordinal Linguistic Personification (OLP), which was the automatic, involuntary tendency for individuals to attribute personal characteristics to sequences like numbers.

Finally, the comparison between university during the Middle Ages and present-day was interesting. Schrader (1967) mentions that university instruction was based on lectures and there were no exams; to qualify for a degree simply required a student to defend or oppose a proposition by another student (p. 272). Schrader concludes that most of material for medieval universities is equivalent to common third grade knowledge today and much of arithmetic taught on ratio and proportions are taught in modern eighth grade mathematics. Though this should not discount the fact that medieval arithmetic concepts are just as challenging today as they were back then.

 

 

Reading:

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

Alice Major on Mayan and other Numbers

A taxicab number, Ta(n), is the smallest number representable in n ways as a sum of positive cubes. They derive from the Hardy-Ramanujan number Ta(2)=1729 = 1³ + 12³ = 9³ + 10³.

The Hardy-Ramanujan number 1729 appearing in Futurama. 

The Hardy-Ramanujan number story comes from G.H. Hardy telling Ramanujan that he noticed a taxicab with number 1729 and noted it being the smallest positive number expressed as a sum of two positive cubes in two different ways (Weisstein, n.d.).  Hardy was quoted to say, “each of the positive integers was one of his personal friends.” In terms of Major’s (2017) paper, this is the idea of Ordinal Linguistic Personification (OLP), which is the automatic, involuntary tendency for individuals to attribute personal characteristics to units in ordinal sequences like numbers. Hardy personifies the taxicab number of 1729 as being his friends.

The ideas of OLP and Synesthesia; individuals making deeply automatic, stable connections between different kinds of mental experiences, proposed by Major would be interesting to introduce in a math class to act as inspiration or imagination for students. The first time I heard about synesthesia was from a Youtube video called, “Math Genius Computes in the Blink of an Eye” (ABC News, 2010). The video tells a story of a man named Daniel Tammet who memorized the first 22,000 digits of pi and recited it in front of an audience lasting over 5 hours. Tammet saw numbers as a three-dimensional shape and creates pictures in his mind with these numerical shapes. Another video by Numberphile describes synesthesia.

Numbers do not come to me as shapes or colours. When working on math problems, I see numbers as simply numbers and manipulate them mathematically enough to solve the problem tasked with. However, I do think of odd numbers as “odd” and unsatisfying. As a child, I felt that I had to do things an even number of times. For example, when watching TV and adjusting the volume, I felt that I had to always leave the volume at an even number or when playing hockey, I had to stick handle the puck an even number of times before shooting. Over time this became less of a worry for me, though I still find myself unintentionally leaving the volume at an even number whenever I watch tv or adjust the volume on my laptop.

 

Readings and Supplementary Material:

Major, A. 2017. Numbers with Personality, Bridges 2017 Conference Proceedings, 1-8.

Weisstein, Eric W. n.d. "Hardy-Ramanujan Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Hardy-RamanujanNumber.html

ABC News, 2010. Math Genius Computes in the Blink of an Eye. [Youtube video]. Accessed November 18, 2020 from https://www.youtube.com/watch?v=Xd1gywPOibg.

Numberphile, 2013. Synesthesia – Numberphile. [Youtube video]. Accessed November 18, 2020 from https://www.youtube.com/watch?v=dNy23tJMTzQ.

Assignment 1 Reflection

I really enjoyed this project as it was an interesting look at how ancient Egyptians likely solved mathematical problems. It demonstrates that there are different methods to solve a single problem, and limited by techniques developed at the time, we can see that it is still possible to solve this problem using the same method they solved in the past. I can see this activity be incorporated in a classroom learning about surface area and volume, which can teach students to think creatively “outside the box” when determining frustum volumes by separating the frustum into different three-dimensional shapes.

I am a very visual learner. I like to visualize how we can “cut” shapes into other familiar shapes. Visualizing the frustum as a pyramid with a “smaller top pyramid” cut off allows me to easily understand the shape of a frustum. Alternatively, I learned another way to construct a frustrum through adding a cube, four smaller corner pyramids, and four triangular prisms. This example alone demonstrates that there are multiple ways to visualize the same shape. For the non-visual and pure math learners, this exercise of calculating the volume of a frustum through similar triangles was another approach that demonstrates different methods arriving at the same solution. Ultimately, it demonstrates how there is no definitive method to solving mathematical problems. As an educator, it is my job to present several possible ways to solve problems to students, and allow them freedom and flexibility to choose which method works for them!

Dancing Euclidean proofs

One moment that stopped me during the reading was through the ‘ah-ha’ moment realized by Milner, Duque & Gerofsky. When dancing the proof of Euclid’s Proposition 2, Milner and Duque included the environment around them such as drawing in the sand and using points marked by rocks and shells. They referred to this as the mathematical necessity and constraints of the bodies, which opened a new dimension for the project (2019). The dance was symbolic and connected mathematics to nature and human. I also liked how Milner et. al describe including natural elements in Dance 2 stemming from the idea that mathematics is found as much in the body as in the natural world. I feel like the connection between math and the body is easily more forgotten than the connection between math and nature. When I think of the connection of the golden ratio to the real world for example, I immediately think of snail shells or fractals in tree branches instead of the golden-ratio proportions found in our bodies. I also appreciated how Milner et. al said that a vital aspect of embodied knowledge is to acknowledge the space that shapes their movements. When dancing outside, they realized how little in control they were of their surroundings; weather and other users of the space were things they could not control.

Another moment that I enjoyed in the paper was the perspectives and dimensions of ‘being in the proof.’ Deciding to film the dance in a top-down perspective allowed the viewer to witness the embodied movements in a two-dimensional plane rather than 3D perspective. This replicated the point of view of a geometric diagram. Milner et. al (2019) note that loss of perspective is not always negative and uses the example of musicians sometimes closing their eyes while playing to focus on the sound. Coming from a music background myself, this thought resonates with me. It is the performer listening intently to the music and allowing the rhythm to take control. It is like Csikszentmihalyi’s idea of flow, where the musician enters an effortless and spontaneous state and is completely immersed in the activity and has a sense of timelessness and inner clarity.

I was completely mesmerized by Miler and Duque’s bodily movements in the video produced by Gerofsky (2019). Watching that dancing video after the reading ties everything all together and gives the audience a stronger visualization and understanding of Euclid’s first three proposition in Elements.

 

References:

Milner, S., Duque, C.A., Gerofsky, S. 2019. Dancing Euclidean Proofs: Experiments and Observations in Embodied Mathematics Learning and Choreography. Bridges 2019 Conference Proceedings.

Gerofsky, S. 2019. Dancing Euclidean Proofs. Vimeo. Accessed November 9, 2020 from https://vimeo.com/330107264