Assignment 3 Topic, Draft References, Artistic Format Declaration

I will be partnered with JJ, and we will be working with the history of cellular automata. The format of our artistic piece will be electronic/digital art through the use of code (most likely Python).

Draft reference list:

Berto, F., & Tagliabue, J. (2017). Cellular Automata (E. N. Zalta, Ed.). Stanford Encyclopedia of Philosophy; Metaphysics Research Lab, Stanford University. https://plato.stanford.edu/entries/cellular-automata/ 

Poundstone, W. (2024, October 24). John von Neumann. Encyclopedia Britannica. https://www.britannica.com/biography/John-von-Neumann

Sarkar, P. (2000, March 1). A brief history of cellular automata. ACM Computing Surveys, 32(1), 80–107. https://doi.org/10.1145/349194.349202

Schiff, J. L. (2008). Cellular Automata: A discrete view of the world. Wiley-Interscience. https://doi.org/10.1002/9781118032381 

Wolfram, S. (2002). A new kind of science (pp.876-878). Wolfram Media.

Assignment 2 Reflection

Link to the slides: https://docs.google.com/presentation/d/1R6IOM_hyLE2_RbFfOASSaiuSfAKEa13GPtx9JwKw0Tk/edit?usp=sharing

My slides are 25 - 35, unless anyone decides to change the slide count before my slides.

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First off, I wanted to say that I did not expect to go as hard as I did for this assignment. The assignment started with me flipping through the physical Math Through the Ages book that Susan put on a table and seeing the word "pi". "This should be relatively simple to do, let's go with this topic," I said to myself. Of course, I forgot that I have trouble keeping things short, so that would prove to be a challenge... which strangely enough, ended up working out during the presentation! More on that later though. I also ruminated on the idea of coordinate planes and grids because of its own rich history and also how surprisingly recent it was, but ultimately I decided to go back to pi since I felt that I could do it more justice.

In terms of what I took away from the project: I gained a new lesson segment to use, which is great especially since I will be assigned a Math 8 class for my long practicum! Since pi will be brought up in that class, I will most certainly incorporate the presentation that I did in my lesson. I wanted to spend more time letting my classmates watch the pi Plays Pokémon livestream a bit longer because I really found it interesting and that it should've been more than a 15 second afterthought. After all, isn't it cool that we can make a statement about the true infiniteness and inclusiveness of pi by seeing how long it takes to beat a video game? The statement that "pi contains every finite sequence of numbers" is honestly fascinating, because there are real implications with it. I think that this project really put into perspective just how wild pi can get, and I would love to use that sense of unsolvable mystery to really strike a chord within my students. One statement that I wish I could've articulated better during the presentation was how over the times, people tried to capture infinity by finding pi, even though it was unnecessary. It's amazing how a simple ratio between two parts of a circle cannot be captured by anything except for a Greek letter and a mathematical definition or three, even though it's conceptually so simple. I also suspect that this unfortunate truth shattered some ancient mathematicians and their dreams harder than sqrt(2) did for the Pythagorean era. What do you mean the ratio is irrational? But I digress. I also would've liked to spend more time to just watch the Numberphile video where they used pies to calculate pi. It's also a good demonstration that students can use for the activity that I had the class do where they had to find pi.

On the note of the activity, I found that it is very adaptable if you change a few parameters. One of the parameters is the amount of time you give the students to complete the task you give. Is it going to be a 3 minutes and 14 seconds quick pi-finding contest? Is it going to be a longer pi-finding contest with 314 seconds? Perhaps you want to give them an inquiry-style assignment where they have to come up with a unique way to find pi within the span of 3 days and 14 hours/minutes/seconds? With the creative ways one can arrange the numbers of pi, the time parameter can influence the scope of the rest of the activity, assignment, or project. There would also be a scaffolding parameter; how much inspiration will you be giving them? I've considered the idea of having students do a take-home project on how they discovered their value of pi, and having some kind of fun reward component (like Leon's Pokémon pins) for whoever can experimentally find their value of pi the most precisely. The amount of materials available to the students is another thing: do we want to give them striped/line paper and tape and bendy rulers and all these fancy things? Or do we want to force the students to get a bit creative like I did in my presentation? I saw the creativity of my peers as they used paper plates, wrapped paper, and were only provided basic 30cm rulers to make measurements. Could students be as creative within the span of 3 minutes and 14 seconds? I think for something like that, I would need at least 314 seconds. Finally, another parameter is the group size, but honestly -- 3 is a good group size. I realized as I watched the class doing the activity that the Thinking Classroom encourages groups of 3 as well, so this activity lines up perfectly. Wow! Even more beauty has been found in pi. Amazing.

There are a few notes I have regarding my slides. I remember locking in pretty hard, so there is a bit of pride in them.

• The background was a random idea that I had. I actually did a Discord server poll for pi Day once, and these were the options:
  
  
  
  Remembering this, I decided to add the verbose attempt to capture pi in its entirety to the slides by putting it into the background. It was a bit difficult to work with Google Slides text boxes auto-adjusting, but I think it was worth it.
• It would have been nice to elaborate a little further on the method of exhaustion and Liu Hui's algorithms. I remember being stuck on figuring out how exactly the algorithms worked for a while, but eventually I realized that there's some geometry going on in both that is accessible to high schoolers (though they'd have to be at an extending spot in order to fully grasp it; the Liu Hui algorithm is particularly abstract).
• I intentionally chose to make the last historical slide shorter, just because many of the processes behind the methods (infinite series, literal supercomputers) are far beyond the scope of Math 8 or even high school math. That being said, those parts could've used a bit more elaboration on my end, just a bit of entertaining the ideas behind them just so the audience could get a full appreciation and some basic understanding of what's going on. Possible leads for future iterations of the presentation, I suppose.

All in all, this was another pat-on-the-back worthy presentation for me. I will see you all on the final assignment.

EDIT: I should probably take the time to ***briefly*** talk about my thoughts on the other presentations. In general, although I'm stating the obvious a little here, every topic had so much history behind it, whether it was a silly ratio like pi or the golden ratio, limits (it was particularly cool how history and curricula go in opposite progressions from integration to limits -- thanks JJ!), trigonometry, quadratics, coordinate planes, or something else that was presented. There were also topics that weren't explicitly tied to any part of the BC curriculum but could still fit in, like shadow reckoning and sacred geometry, which I thought fleshed out the overall composition of the resources created by our class.

As someone who sometimes feels a sense of self-doubt in terms of ability to think of activities, I feel like there were a good handful of activities I could take for various topics. Since I'm most likely teaching Math 8, 9, and Pre-Calculus 11, I think the most beneficial ones would be the golden ratio for the 8s (Leon's activity is hilarious, and I may need to prepare by sharpening my jawline to ensure the authenticity of my iteration of his "yay or nay" activity), quadratics and square completion with algebra tiles and diagrams, and even the idea of doubling a radical (though that might be more for FoMPC10 rather than PC11; it would still be a neat recap activity, but I'd need to be considerate about it). Generally though, I am very appreciative of the resources that everyone put together, and I imagine they would serve well as quick yet sentimental references if I need historical fun facts for my students.

Also, long live Desmos golf.

Response to "Dancing Euclidean Proofs"

The first of these stop-and-thinks came relatively early on, as they usually do: I did not know what Euclid's Elements were already being used in dance routines well before the Euclidean proofs dances that we are covering now. Even though it was not done for pedagogical reasons, the fact remains that the minimalistic beauty of Euclid's proofs were beautiful enough to capture the artistic vision of dance companies around the world. Regardless, it is interesting that this whole time the dances were not done pedagogically, so there is certainly a creativity being described here. I did wonder though: how did these dance companies appear when they did these dances? Were their dances of a similar nature, using the arms as circle radii and having solid objects on the ground to represent givens? Were there multiple people on the floor, or just one person (both unlike the dances exhibited by this paper)? Regardless of those two possible body counts, it does make me wonder about how others did their dances. Was there any educational value that could be garnered from these other dances? Did they fully incorporate The Elements into entire dance routines, or were bits and pieces simply taken to construct a larger dance? So many questions, but also props to the two education students preceding us for still making something original by having pedagogy as the intent. When attempting something similar in class, my group of 3 with Leon and Zain chose to choreograph the same equilateral triangle proposition as the first dance in a way that took inspiration from very pose-heavy media. (The inspirations were the fusion dance from Dragon Ball, where two fusing characters would arch their bodies and connect their arms, and the general exaggerative posing that is common in children's cartoons. The difference is that instead of connecting fingers like in the image, we overlapped arm circles) The cool extension here is that at least for myself, I took inspiration from something completely unrelated and incorporated it into Euclid's mathematics here.

(Screenshot taken from Dragon Ball Z anime.)

The second of the pauses was when the article described the first of the three dances. In the article, they describe an intuition-based, embodied way of illustrating an equilateral triangle through dance. The quote "Since all arms are equal lengths, we may trust that this triangle is indeed equilateral", however, made me do a double take. What? But arms aren't equally long. While it worked out when Leon and I, who are basically the same height, joined our arm circles together to set up a central triangle, we witnessed as Jacob and Jasmine had to stretch their dimensions and bend their wrists to make it work for them. Because of this unreliability, I would argue that such an embodied way of proofs is not particularly sound as a proof method. That being said, there is still value in using it to provide holistic understanding; there is nothing invalidating about doing the embodied dances to consolidate the ideas behind the proof. Thus I would conclude that the order of ideas would still have to be: 1) Elegant, beautiful Euclidean abstract proof, and then 2) Get a holistic grasp of the truth by doing cool bodily dances to drive home the point.

Regarding the question on how this type of embodied activity could be helpful for math learning and understanding math history? For the latter part, I will have to admit that the dancing portion feels a bit arbitrary (not so much in the sense of Hewitt's usage of the word), and doesn't provide too much historical value aside from simply bringing Euclid's Elements to the attention of students. Additionally, Euclid's Elements is but a single part of mathematics history, which I don't think the embodied activity fully does justice to the whole mathematical history of the world. Are there fragments of dances from different cultures that we can incorporate? Sure, but it certainly won't appear as obvious to the average student. A lot of pointing out the intent would need to be done for students to be made aware of what is actually being attempted. Regarding math learning in general though, I could see merit in embodied activities as an additional access point for students who may not be fully convinced or understanding of concepts or proofs in class. Embodied activities have been shown to be applicable to Euclid's where the dancing could be fairly one-to-one with the structure of a proof, based on what I've seen here. I've used my body before to illustrate coordinate grids by making a + symbol with my arms as well, which is a lot faster than pulling out a writing tool and depicting something. I imagine that students would take inspiration from my physical modelling and have a silent conceptual tool they can take with them into a test environment, for example.

In terms of constraints though, there are a couple that I can think of. The most important one is that if we compare possible activities in a mathematics classroom to types of dishes based on how often you would eat them, something like embodied proofs would be more along the lines of a fancier dinner that you'd eat once in a while. I feel that after short practicum, many students actually prefer a more simple way of learning mathematics akin to the rice and vegetables that you can eat every day, and movement can feel weird to them especially if you don't fully buy in to the idea of embodied learning. For me, I don't think I am exactly there yet, and still see it as just an additional scaffolding or explanation tool that I can use if existing strategies are not effective to communicate ideas (as opposed to getting the whole class moving and experiencing it). I think that depending on the class, dedicating precious minutes to do embodied learning can throw off the schedule. I've experienced first-hand just how little time there can actually be for teachers to get a point across, and I feel that there are other things that I value more that compete for the precious time, such as group work activities and even lecture/prompt style teaching. To say I'm dismissing the idea of embodied learning is not true though, but the way I see its usage is for the most part relegated to being an additional back-up explanation and consolidation tool if students really need it. (From my own experiences in class with dancing Euclidean proofs, I think I would've understood the proof even without the embodied version.)

Response to "Was Pythagoras Chinese?"

"If we consider a bird and a bat, and realize that they both discovered the way of flight separately, who can we really say discovered flight?" -- Me (2024)

This quote was drawn up based on my own thoughts regarding two relatively independent entities coming to the same discoveries and applications, despite having no influence on each others' progress. I was going to use a bird and a fly instead, but I realized that there was likely an evolutionary arms race happening here where either the bird influenced the fly to learn flight or vice versa. Once again, I digress. The idea is that in history, with all the isolated peoples of the planet and their own form of mathematics and numerical/spatial/et cetera reasoning, there are likely many things that could've ended up being discovered independently, with the "Pythagorean theorem" being the star player of this blog post. Thinking about it, I find it somewhat silly to try and attribute the credit to any singular person, and suggest that a good compromise is to allow the theorem to exist under multiple names. HOWEVER, there is a practical downside to this in that it would be very confusing. So the question goes: "If we don't want to have this confusion, who should we go with? The one who said it first? The one who said it louder?" If I were being honest, I would prefer to keep calling it the Pythagorean theorem by virtue of it simply being so ingrained, and I say this as a person of Chinese background. (Also because "Gougu theorem" doesn't roll off the tongue as well.)

Does it really make much of a difference to our students' learning if we acknowledge/don't the non-European sources of mathematics? My initial thoughts before reading the article would have me telling the reader that it probably doesn't make much of a difference. The students can still achieve the same "necessary" mathematics regardless of the acknowledgement (if we take a page from the recent reading on Arbitrary vs. Necessary from EDCP 342). For example, if we never put a name to this magic a^2 + b^2 = c^2 theorem, it doesn't mean that people can't reason with the ideas. It doesn't mean that we can't show the right triangle to a bunch of students, have them discover the relationships of a right triangle on their own and make an absolutely groundbreaking discovery in geometry that would have the ancient Greeks and Chinese flabbergasted by the fact that an 8th grader could discover and prove something so integral, and have the classroom collectively dub it the "Hoang theorem" after their loving, caring, and absolutely fantastic mathematics mentor. (I am absolutely yanking at chains here.) That being said, it is a lot more standardized to just call it the "Pythagorean theorem" instead. As nice as it would be to have my students using my name for such an important theorem, it would likely be very awkward as they try to explain to their contemporaries the name that stuck with them. So to that, I would say there isn't really a big difference in the necessary process. However, by acknowledging all the different sources of discovery, it does allow for a more diversity-conscious classroom and learning process. This could be a motivating factor for students who may be from a part of the world where these sources are; I did find it cool that the Chinese (along with many other civilizations) were able to come to the same conclusions as Pythagoras.

In terms of my thoughts on the naming of the Pythagorean theorem, I'm just happy that there is a standardization that everyone (for the most part) was able to agree on. Standardization has its clear benefits, and I think that despite the lack of crediting, we still receive the dividends of a universal term (unless I'm missing something here and the modern Chinese call it the Gougu theorem). I still think that despite my whole spiel earlier, it's okay to allow a necessary concept to go by multiple names as a way to credit the many minds behind them (although we should still have a principal name, just so people don't get confused). It did raise questions on the sequence of events for me: how good were the Chinese at crediting themselves anyways? Sure, we knew some of the names behind the ideas such as Liu Hui and Chongzi, whereas Euclid is a completely mysterious entity who allegedly lived in Greek-controlled Egypt. However, I feel that because of the long-standing processes of Eurocentrism persisting the ideas of the Greeks -- The Elements -- throughout the world, it really is just as if they "said the joke louder" and got the last laughs. There are many such cases of the Eurocentric side saying the figurative joke louder which ends up obfuscating the global history of mathematics, but that just means that the process of historical discovery is paradoxically easier for us as educators who simply need to search for a few minutes to find fun trivial facts about who else discovered/proved/reasoned about a certain theorem or mathematical idea.