Personal reflection for EDCP 442 and the future

EDCP 442 was one of the most interesting courses I have taken at UBC. I think one of the most overlooked aspects in learning mathematics within secondary and post-secondary education is learning about the history of mathematics and origins for different ideas. Knowledge about early civilizations and contributors to the field of mathematics is very beneficial to developing a stronger sense of understanding of how these ideas became theorems and rise of conjectures and proofs.

Moving beyond well-known mathematicians such as Euler, Euclid, Archimedes, Gauss, Fermat, Fourier, (to name a few), and learning about ancient civilizations such as the Babylonians and non-European roots of mathematics was refreshing. It was very thought provoking to learn about how “different” solving mathematical problems in the past were as there were no variables or algebra used how we use them today. Babylonian multiplication, classical Mayan mathematics, Alcuin’s recreational word problems, and problems from medieval Islam have changed the way I see how math problems could be presented. For example, using head/mask variants or different strokes to represent different counts of numbers, or using base 60 to count differently have changed the way I see different representations for numbers. This course has helped me consider different avenues to explore and incorporate in my teaching so that students can get a “bigger picture” of mathematical history beyond well-known mathematicians.  

A suggestion for the course in the future, could be to include more mathematics from an Indigenous perspective. I would be interested in learning how Indigenous cultures use numbers to record different aspects of their lives and how mathematics is rooted in their artwork and designs.

I'd like to thank Amanda and Susan again for all their hard work, feedback, and instruction during this course. I learned a lot and had so much fun attending class! 

Ada Lovelace Project Reflection and Take-aways

This was a really fun project and I enjoyed presenting the information researched in the format of a video. It seems almost “full-circle” in the sense that we researched about a person who contributed to the creation of the first “programmable computer” and created a video on the history using a computer. Due to time constraints and other factors, we were unable to record voice-overs for the video. Nonetheless, we as a group were satisfied with the outcome of our project.

Interestingly, this project has overlaps with other groups such as the short story on Alan Turing and history of coding and computer algorithms. The thought process in deciding on the topic surrounding Ada Lovelace for this group was because we wanted to uncover histories of a mathematician that was not very well known for their contributions to mathematics. Ada Lovelace was a mathematician I had not heard about in any years of my learning, and yet computers and technologies alike were taken completely for granted during my studies and entertainment. I hope that as a teacher, I can give voice to many more not-so-well-known mathematicians, especially female mathematicians, to help inspire more women to enter the fields of STEM and STEAM.

Ada Lovelace “draw-my-life” video

Ada Lovelace “draw-my-life” video: https://drive.google.com/file/d/11r9fq8ZYN3fp1VBWtcU23RsmgoG1JArr/view

Assignment 3 Topic

Zoe, Ivan and I will be researching the history and life-story of a not-so-well-known mathematician: Ada Lovelace and present the topic in the form of a “draw-my-life” video.

Bibliography:

Famous Scientists The Art of Genius, n.d. Ada Lovelace. Website. Retrieved from https://www.famousscientists.org/ada-lovelace/#:~:text=Lived%201815%20%E2%80%93%201852.,much%20more%20than%20just%20calculations.

Wolfram, S. 2015. Untangling the Tale of Ada Lovelace. Wired. Retrieved https://www.wired.com/2015/12/untangling-the-tale-of-ada-lovelace/

San Diego Supercomputer Center, n.d. Ada Byron, Countess of Lovelace. Retrieved https://www.sdsc.edu/ScienceWomen/lovelace.html

Fuegi, J., Francis, J. 2003. Lovelace & Babbage and the Creation of the 1843 ‘Notes’. IEEE Annals of the History of Computing. Pp. 16-26. Retrieved from https://pdfs.semanticscholar.org/81bb/f32d2642a7a8c6b0a867379a4e9e99d872bc.pdf

Hollings, C., Martin, U., Rice, A. 2017. The early mathematical education of Ada Lovelace, BSHM Bulletin: Journal of the British Society for the History of Mathematics, 32:3, 221-234. Retrieved from https://www.tandfonline.com/doi/pdf/10.1080/17498430.2017.1325297

An introduction to the mathematics of the Golden Age of medieval Islam

This reading introduced several Islamic scientists and mathematicians including: al-Khwarizmi, al-Biruni, Umar al-Khayyami, and al-Kashi. Content from this week’s reading would be useful to use as a background to provide students when introducing topics on algebra or other related math topics.

It was interesting to learn more in-depth about al-Khwarizmi, as his name appeared in my individual math history research on exponents and powers. Al-Khwarizmi contributed to four areas of science: arithmetic, algebra, geography, and astronomy. From this reading, I noticed that many early mathematicians were astronomers. I also learned that algebra comes from the Arabic word al-jabr. As a math teacher, I think it would be interesting to teach a math lesson bridging with astronomy like how early mathematicians studied the stars. One idea I have could be for students to learn and trace out planetary orbit movements; we can tie an apple to a string and tape a marker to the apple and swing it over a piece of paper to map out it’s path of movement. From this activity, we can see interesting patterns form on the paper.

I learned that Umar al-Khayyami found that the ratio of the diagonal of a square to side (sqrt 2) and ratio of circumference of a circle to its diameter (π) and wanted to consider these as new kinds of numbers. These became known as irrational numbers. This history could be interesting to mention to students when teaching the number system. An idea I had was to design a “just for fun” assignment that get students to create their own way to classify and group numbers into a system like the number system. For example, students could great a classification for just prime numbers and another classification for numbers that at first glance look like prime numbers, when are in fact, composite numbers (eg. 2431).

Al-Khwarizmi and al-Kashi both correctly estimated the value of 2π to the first 16 decimal places. Al-Kashi solved a cubic equation to correctly obtain the value of sin(1^o) = 0.017452406… Al-Kashi developed a calculator method that can repeat a procedure each time using the previous result to obtain numbers near the true value of the root of an equation. This reminds me of Euler’s method, which a technique used to analyze a differential equation using the idea of linear approximation by calculating the result using the previous result.

 

Reading:

Berggren, J.L. (2016). Episodes in the Mathematics of Medieval Islam. Springer. 2nd Edition. 1-23.

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.