Final Reflection Blog Post

Slides: https://drive.google.com/file/d/1w-pcnnZEBbRNehiXtjZtgs8PBVHCP445/view?usp=sharing

I thought that although I wouldn't particularly teach a lesson on the guitar fretboard in my math class, learning about the mathematics behind it was pretty cool and fascinating. The lesson required a lot of background knowledge in music theory and had a lot of content. I think I should have cut down on content with the alotted amount of time. 

Regarding the history of the guitar, its fretboard, and the music theory behind it, I thought it was extremely fascinating. As a guitar player, I've never studied the mathematics history behind it and was surprised to see how complex the math can actually be if I decide to go further in depth to the topic. It was also fun to expore the various patterns on the guitar fretboard and their relation to the circle of 5ths (vertical columns of the art above) and the chromatic scale (horizontal rows of the art above). In addition, I used recursion to space out the fretboard relative to the size of the piece of cardboard in the art above. It was pretty cool to learn that the guitar fretboard spacing can be calculated using recursion. 

I thought that assignment 1 was a well thought out project. Evan and I were able to teach chords from two different perspectives and were able to make it interactive, involving the whole class. I thought that teaching chords would be great in a trigonometry unit for math 10, to show that using both the pythagorean theorem and SOH CAH TOA can be used to find lengths of chords. On the other hand, I personally disliked the dancing Euclidean Proofs blog post. I thought that the dancing aspect felt forced and was not exactly sure what to take away from it.

Overall, I thought that the class was a little bit too lectury. I felt like I was sitting for extended amounts of time and it felt boring and unengaging. In addition, althought this is a history in math course, I felt like the course tried too hard to force history of mathematics into the curriculum. I feel that history in mathematics should not be forced, but only incorperated when deemed necessary and relevant.

Assignment 3 topic: Mathematics behind the guitar fretboard

Reference List:

Fretboard.com - The History Of The Guitar. (n.d.). Www.fretboard.com. Retrieved December 4, 2023, 

            from https://www.fretboard.com/guitarhistory.html

Garofalo, J., Corum, K., & Rutter, J. (2022). engineering - a context for learning mathematics: the case of 

            guitar fret spacing. Technology and Engineering Teacher, 82(2), 14.

Language Log» The Musical Origin of the Seven-Day Week. (n.d.). Retrieved December 28, 2022, from 

            https://languagelog.ldc.upenn.edu/nll/?p=54528

Passy. (2012, September 20). Guitar Mathematics. Passy’s World of Mathematics.       

            https://passyworldofmathematics.com/guitar-mathematics/

Schulter, M. (1998). Pythagorean tuning and medieval polyphony. Web source: www. medieval. 

            org/emfaq/harmony/pyth. html.

The Guitar Fretboard’s Mind-Blowing Mathematics. (n.d.). Www.youtube.com. Retrieved December 4, 

            2023, from https://www.youtube.com/watch?v=WxAf3NgF4jk

Lugg, O. (2021). How Pythagoras Broke Music. YouTube. Retrieved December 10, 2023, from https://www.youtube.com/watch?v=EdYzqLgMmgk.

Artistic Format: Undecided

Medieval Islam Mathematics

 1) One stop for me was the mention of the introuction of the concept of 0 in the Islamic world. I was surprised because I thought it was India who introduced the number zero and brought it to the rest of the world. I did not know that the Islamic world also came up with the concept of the nunmber zero independent of India or other nations.

2) Another stop for me was the construction of a geometric representation of complete the square (first one). I found it so interesting that Al-Khwārizmī was able to come up with an alternate method for complete the square than we are used to. This shows that mathematics has room for creativity and imagination, and that there are multiple ways to get to the same solution. This concept that there are multiple ways to get to the same answer in mathematics is something that is not taught in schools nowadays. 

3) I thought that the construction of a parabola using circles on a grid was very interesting and a concept that I have not seen before. This method offers a unique perspective on how geometric shapes and lines intersect and converge to form the distinct curve of a parabola. It blends geometry rules and creativity, showing how math and visuals connect. The step-by-step process involving circles, perpendicular lines, and intersections illustrates a precise yet visually engaging method to depict the elegant structure of a parabola on a grid.

Maya numerals

1. Consider the so-called Hardy-Ramanujan number 1729, and the story of Taxicab Numbers, retold on Wolfram Mathworld at the link above. Hardy is quoted as saying of Ramanujan that "each of the positive integers was one of his personal friends". What do you make of this in terms of Major's paper?

Ramanujan's story tied to the Hardy-Ramanujan number 1729 aligns well with Major's exploration of the personalities linked to numbers. His attachment to numbers, seeing each as a personal friend, resonates with Major's insights into Ordinal Linguistic Personification (OLP).

Major delves into how some people attribute human-like traits or personalities to numbers through OLP. Ramanujan's statement about integers as personal friends showcases a deep and intimate relationship with these numerical entities.

2. Is this something that you might want to introduce to your secondary math students? Why or why not? If you would use these ideas in your math class, how might you do so?

I think that personifying numbers is a pretty cool way to incorperate the First People's Principle of storytelling. Teaching exponents, for example, through personifying numbers can help students remember the rules and methods better. For example, I can tell my students that the "negative" 4 in the exponent is being very negative, grumpy, and sad, and ask them how we can make it positive and happy. They will answer my question by instructing me to bring it to the denominator.

3. Do numbers have particular personalities for you? Why, how, or why not? What about letters of the alphabet, days of the week, months of the year, etc.?

I think as we learn mathematics, we innately label numbers with distinct personalities to help us remember content. The author calls this phenomenon OLP. For example, certain numbers are "nice" numbers and thus "happy" numbers as well. For example, any number that ends with a zero is a nice and happy number to me, most likely because they are nice numbers in the base 10 system. For letters, I guess common letters like vowels, s, t, etc. are nice and happy letters? Regarding other things like days of the week and months of the year, I often associate them with my mood. Obviously, Mondays are sad and depressing because it is often the first day back to school/work, and Fridays are all happy. For years, I love the summer so June and July are like happy months while January and Febuary are sad months.

trivium & quadrivium

 I found it extremely interesting that back in the day, priests could not be ordained if unable to use arithmetic to compute the day of Easter and teach it to others. I have always wondered that without technology how history and dates were able to be retained, and it nbever occurred to me that mathematics was involved.

In class, we learned that multiplication and multiplication tables date back all the way to the Babylonians. However, it was interesting to read that methods of division was only developed during the medieval times. This begs the question of what took so long for scholars to find the counterpart of multiplication. 

Recorde's method of multiplication really intrigued me. At first glance, I was confused and did not know what he was doing. It is really interesting as to why and how he has came up with such a method, that at first glance, looks like he is going in circles. It was also interesting how the text mentioned that the cross he does when multiplying could be the first glance of the multiplication symbol.

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. 

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.