Monday, October 23, 2023

Dancing Euclidean Proofs

Having watched the video and read the article, I do not think I would incooperate dancing in particular to my future lessons. However, I did think that I was cool how in the video, she talked about using combinatorics to count the number of different types of dance moves. Overall. I do think that it is always good for students to take abstract math and do something that allows them to move around and explore the outdoors. It allows them to see math in a different light. Rather than doing dancing in particular, I think I would incoorperate projects that allow students to explore an area of their choice in a mathematical context. I remember taking a geometry class in grade 9 where I was able to do it completely on minecraft, using minecraft blocks as the unit of measurement. I thought it was pretty cool.

The article mentions that "dancing spreads Euclidean constructions across space and time, and as a written or drawn proof may symbolize the mathematicians’ work, so dancing may draw attention to the math learners’ work." When thinking about designing projects for students in class, it is important that the topic of choice (dance, architecture, sports, etc.) allows students to critically think through the mathematical process. For dancing in particular, it takes the whole "proof" and puts it into slow motion. That allows students to think critically about the process and helps explain why certain things are done the way they are. It also helps the concepts resonate more with them, making it more memorable / easy to remember.

I also liked the way the author concluded the article. He concludes by saying "It [engaging knowledge differently] involves welcoming a wider conversation between the land, the dancing body and the mathematical mind; to us, that is where art truly takes place." This reminds me of Indigenous integration and the accomodation of various learning styles in the curriculum. This quote encompasses the first two Indigenous First Principles and implments them implicitly. It underscores the significance of acknowledging the land we occupy and honoring the ancestors who once inhabited it. 

Tuesday, October 17, 2023

Euclid and Beauty

 The fact that Euclid and Euclidean geometry have been studied for over two millennia shows how much societies over decades value his work. I did some digging on the content of his book "The Elements" and some of the theorems in it include the pythagorean theorem, the law of sines and cosines, the area of a triangle, and more. In number theory, I also covered more of his work like the Euclidean algorithm, Euclid's lemma, Euclid's theorem, and more. His proofs were very rigorous and thorough. So why is Euclid and his book "The Elements" still celebrated and studied today? Simply because his theorems and work, especially those in geometry, encompasses so many real world applications and is the basis for so many mathematicians after him.

In my opinion, beauty is subjective. I might think an outfit looks beautiful on a girl but another person may not. That being said, there are different types of beauty. Appearance is a type of beauty, but there is also beauty in elegance, depth, simplicity, and harmony. In math, I see a beauty in a concise and clean proof. That being said, yes I do beauty in Euclid's work. Logical clarity plays a role in elegance in Euclid's work. I just find it fascinating and inspirational that Euclid through his timeless work "The Elements," laid a foundation for the rigorous area of geometry, leaving a lasting legacy that continues to shape our understanding of mathematics today.

Wednesday, October 11, 2023

Was Pythagoras Chinese

Does it make a difference to our students' learning if we acknowledge (or don't acknowledge) non-European sources of mathematics? Why, or how?

Acknowledging non-European sources of mathematics broadens the perspectives of students, showing them that mathematics is not exclusive to a single culture. By allowing for cultural diversity and inclusivity, students see that mathematics is an option they can pursue and see that it is not restricted to race. This breaks the wall of stereotypes.

Going back to our class discussion about gender roles in society, we mentioned that it is important that students feel represented in the classroom. It is important that females, for instance, see that they are representated among scientists, doctors, and mathematicians. This allows them to see a viable career path for themselves and encourages those who are gifted in the subject area to pursue that. Representation of race, gender, etc. is important in the classroom because it allows students to see themselves, their experiences, and their cultures reflected in the learning environment. It promotes confidence, engagement, and gives students a sense of belonging in the classroom.


What are your thoughts about the naming of the Pythagorean Theorem, and other named mathematical theorems and concepts (for example, Pascal's Triangle...check out its history.)

I think that naming is merely a matter of tradition and self-recognition. While Pythagoras is credited for his theorem, it is also important to teach students the broader context and origins of mathematical concepts. Students have the right to know that the origins of the Pythagorean theorem and its ideas traces back to ancient Babylon and China. 

Overall, it is important for students to see that the history of mathematics and the development of theorems is often collaborative. It is also important for students to know that different societies were capable of coming up with their own variations of theorems (just like how the Chinese had their own variation of the pythaogrean theorem) despite societies separated from one another and growing at their own rates. This speaks to the overall intelligence of human beings and tells students that intelligence is not a matter of race or gender, but that anything is possible if you pursue it and put your heart and effort towards it.

Tuesday, October 3, 2023

Assignment 1 Reflection

This project has shed some insight towards the need to understand why certain math concepts were looked at in the first place. My presentation topic was the Egyptian method of chords. In all honesty, I was always taught the math behind finding the length of a chord but was never taught how and why it is used in the real world. I always saw chords as irrelevant because most of the time, we have the radius, which has been generally taught as more versatile mathematically. Hence, it was interesting to learnthe importance of chords both in the context of ancient Egyptian civilization and modern day times. I also thought the the activity we used to illustrate why chords are sometimes used over radii would be a good tool to use in the classroom, as it allows the students to move around, giving them a stretch break and a good laugh in the classroom as well. Oftentimes, the most memorable moments in school are the ones that were the most entertaining. Overall, I felt that using the method we used today could be a good way of teaching high school students real life applications of circular geometry.

For the extension, I struggled with it for weeks. I tried out different ideas and proofs for various theorems and found them irrelevant or out of the scope for the project. I then looked at the whole idea chords being used for road making, and then it dawned on me. I was thinking of how city planners estimated how much material is needed to make a road turn and the whole idea of arclength came up. Thus, I looked into the relationship between arclength and chord length. Initially, I was stuck because nowadays, people use trigonometry and angles to find the arclength. However, I thought to myself, "are angles always accessible? How did the ancient Egyptians provide estimates for arclength in the past?" Looking into mathematics I've learned in my undergrad, we learned about many estimates and approximation techniques without learning about why they are important, especially since now we can obtain exact answers for most things. But can we obtain exact answers for all things? Not quite. With that thought, I looked into how the pythagorean theorem can be used to create a lower and upper bound for arclength and how doing so provides a good estimate of how much material to gather for a project. I was pretty happy with my results, because such an extension is applicable and uses simple concepts that grade 8s and 9s can understand. Hence, it is practical and can be easily used in the classroom setting to teach 8th and 9th graders how to critically think as well. It also ties in to the pythagorean theorem unit and teaches its applications. Overall, this project allowed me as a prospective teacher to stretch my creativity as well as shed insight on a topic that most high school teachers do not go deeply into.

Final Reflection Blog Post

Slides: https://drive.google.com/file/d/1w-pcnnZEBbRNehiXtjZtgs8PBVHCP445/view?usp=sharing I thought that although I wouldn't particula...