Assignment 2: Math History Reflection - Blog Response 15

My presentation was focused on the history of limits to infinity. It was really interesting to research on how it all started, especially before a time of mathematics where Arabic numerals and algebraic notation were absent. The majority of my presentation was focused on the findings of Archimedes and the area of a circle via infinitesimals. It incorporated a clever use of a proof by contradiction method, which somewhat surprised me as I thought that would have been more of a modern notion. Due to time constraints, it was impossible to cover the entire history of limits with every detail. It stretches over a millennia and the most interesting part is that the order of development started with integrals, then derivative, and finally limits. This is the opposite order of what is being taught in schools! 

One of the main things I want to take away from this and show students is that infinity is not mere value that we plug in the limits notation. It seems like due to the extensive amount of analytic work that students have to in a calculus class, they get desensitized to the concept of the idea and ignore the beauty and complications that come with it. Infinity was deemed as beast by mathematicians, and even the very best failed to wrestle with such concepts. Perhaps by presenting these historical moments, students will be able to relate and realize why it's okay to find calculus confusing. We have prime examples of moments of epic collaboration and battles between pairs of mathematicians like Kepler and Galileo, and Leibniz and Newton. And we can't forget that none of their work wouldn't occur without the work of Ancient Babylonians, Greek, Egyptian, and Indian mathematicians that pioneered the early foundations. 

Dancing Euclidean Proofs - Blog Response #14

 Although I consider math as art, I never considered the medium of conveying it through dance and human body movements. I was surprised and enlightened to see that there is a cultural pursuit in mathematics that utilizes both body and mind. One of the statements posed in the article is that the body's natural symmetries can be used to make simple geometric shapes like circles and triangles. I find this interesting because its such an obvious and unobvious fact at the same time. Perhaps the beauty we were looking for in mathematics was always right in front of our very eyes, or body I should say. 

I appreciate the amount of visible effort that was put into the choreographs of the dances. When watching the video, the performances were very fluid and smooth in transitions, which makes excellent clarity of the proofs that they were depicting. As the article describes, the process of going through various iterations of dance moves, concern in maintaining mathematical integrity, and the deliberate choice of the environmental location and sounds. All these considerations brought harmony in the arts and math, making the final product visually and auditorily pleasing, without reducing the technical contents of Euclid's proofs. 

Incorporating such activities in a high school classroom will definitely spark interest in students. Physical activity is almost never seen in a math classroom, so this could be an excellent alternative for students. Availability of classroom or outdoor space may pose as a constraint, as well as funding for specialized camera equipment like drones if top view shots are desired. 

Was Pythagoras Chinese? - Blog Response #13

 I definitely do think that acknowledging non-European sources of mathematics makes an impact to students' learning. Often times, students and even myself take for granted on the origins of mathematics that we get taught in the classroom. There is so much opportunity to create discourse in a classroom and promote diversity amongst students, especially in a math classroom where it's rare to see such actions take place. Leaving out important details of history and only focusing on European contributions can distort a student's view of mathematics. When we talk about the origins of how math came to be, and not just the techniques on solving problems, it provides a more immersing learning experience and gives students with diverse backgrounds a chance to have moments of realizations and possibility relate their own cultures. 

When I see naming's of mathematical concepts, the immediate question that I always have is why are some theorems named after mathematicians, while others are named by functionality, like the Fundamental Theorem of Calculus. While I do understand that recognizing the individual who discovered the theorem is done out of respect, we often have inconsistencies like we have seen with Pascal's Triangle and Pythagoras Theorem. Sure, we can take it with a grain of salt and accept it as it is, but it still feels wrong to eliminate a story that represents decades if not centuries worth of collaborative work across the globe. Perhaps we should try to aim theorems to be named by descriptors or we as teachers should clarify names of theorems whenever necessary. 

Euclid and Beauty - Blog Response #12

Euclid's Elements has proven itself to be one of the greatest mathematical textbooks to have ever exist. I believe that one of the  the reasons why Euclid and his book have been so important and popular for centuries is that it is able to uncover ground without circular reasoning. I speculate that there must have been frustration amongst mathematicians before Euclid when it came to logically proving ideas, despite how blatantly and intuitive it seems. For example, if one were to look at a diagram of a square, how do you know that it is indeed a square? I think that it's the general concept of not taking things for granted, and making sure we always pay attention to our surroundings is what we have valued throughout these years. Also, if I remember correctly, Euclid's fifth postulate raised many eyebrows until hyperbolic geometry was studied. This is a great example to always question the obvious!

There is elegance in the way we prove that things that occur in the world.  Is there beauty in defining the given truth? I believe so. It's quite fascinating how we can fix a couple axioms, and then construct complex theorems out of them. Euclid's work was mainly based on the 2-D plane yet it served as a basis for relatively and related ideas like determining the shape and dimension of the universe. The beauty comes from how we can set some simple, intuitive facts (axioms) and develop them into sophisticated and crazy ideas through logical reasoning and proofs. 

The Dishes puzzle - Blog Response #11

 Here is my approach in solving the dishes puzzle without the use of algebra. 

First I will interpret the information and relationships that are given to us.

• Every 2 guests shared 1 dish of rice
• Every 3 guests shared 1 dish of broth
• Every 4 guests shared 1 dish of meat
• There are 65 dishes used in total

Okay, then to make my calculations easier, I see that 12 is the common multiple of 2, 3, and 4. Then I can say that for every 12 guests there are:

• 6 dishes of rice
• 4 dishes of broth
• 3 dishes of meat

Then for 12 guests, there are 6 + 4 + 3 = 13 dishes. 

Then I interpreted the question like this: If a group of 12 guests enters the restaurant and requests 13 dishes, how many groups are required to reach a request of 65 dishes?

After some quick guess and checking, I see that 13 $\times$ 5 = 65, so 5 groups of 12 is 60 guests in total! 

I think that it makes a difference to our students to offer problems that come from diverse cultures. Especially when students see examples that relate to their own backgrounds, they may feel a personal connection to the material. Maybe it makes them more interested in the problem, or it maybe it's just the simple thought of seeing their culture being recognized in the classroom that makes them happy! To me, it's always the little things that matter and if incorporating storytelling in the classroom brings the students joy, I see no reason why we shouldn't do it! 

For similar reasons as above, including fun and informational images can contribute to a lot. I can imagine a student looking at that image of the banquet and going "Mmm..., that makes me hungry." Even if its a small difference, it still has a lot of power to bring colour in a classroom. 

Assignment #1 Reflection - Blog response #10

My group and I tackled problem 1.2.4 from 5000 Years of Geometry. 

Here is the link to our presentation: 

https://docs.google.com/presentation/d/1FsASoA7QqRUaOuKEkFRpz4ao2k-pwjKoL-1nAPdiKfA/edit?usp=sharing

When I first looked at the problem, it seemed rather simple. When observing the provided formulas of calculating the diagonals of 3 by 4 rectangle, it seemed to just work magically. That's when the deceptiveness of the simplicity shocked me. How did the Babylonians manage to calculate the diagonal of a rectangle from its sides without resorting to the Pythagorean rule? When we showcased the modern solution, its quite obvious that we use our handy $a^2 + b^2 = c^2$. The challenge came when we hand to decipher how the Babylonians approached this problem. 

After researching the clay tablet in question, I realized that the Babylonians, given their limitations of modern math tools, figured out ingenious methods of calculating diagonals and had a wide list of rectangle sides written in terms of linear combinations to produce the correct diagonal length. I still have some unanswered questions of regarding their discovery of their methodologies, but it astonishes me nonetheless. 

When we were doing our computations to show our work, we used the calculation of expanding binomials was prevalent. As an extension, I wanted to showcase how we can intuitively teach students on how we expand binomials through pattern recognition and imagery. Although we as teacher candidates have explored deep into math concepts and are aware of how handy Pascal's Triangle is when it comes to expanding binomials, students in high school often times do not get to learn these concepts. If you observe the BC curriculum and the list of topics that are taught, combinatorics and probability no longer exist. I noticed that senior students in math classes only know how to expand binomials through FOIL. 

I believe that combinatorics and probability have excellent content when it comes to developing thinking strategies to solve puzzles and play games, which is listed as a BC curricular competency. My aim with this extension is to remind us that we can teach students solve seemingly painful computational problems through creativity and restore teachings of certain content that has been lost in the past few years. 

The market scale puzzle - Blog Response #9

Since we are given a two-pan balance, given some weights, we have 3 actions to utilize:

1. Place a weight on one side
2. Place a weight on the opposite side with the herbs
3. Not use a weight at all

This gives us a systematical way to give value to our weight such that:

1. We can multiply by 1 to the weight's value to represent placing a weight on one side
2. We can multiply by -1 to the weight's value to represent placing a weight on the opposite side
3. We can multiply by 0 to the weight's value to represent not using a weight at all

This implies we use a base-3 system, given our three choices of 1, 0, and -1. 

Let $W$ be the weight of the herbs. We also choose the four weights to be $1g$, $3g$, $9g$, and $27g$. Note that these are the powers of 3. 

Then we can express $W$ as a combination of these weights such that,

$$ W = (a_1 \times 1) + (a_2 \times 3) + (a_3 \times 9) + (a_4 \times 27) $$ where each $a_i \in \{-1,0,1\}$.

and indeed the sum of all the weights $ 1 + 3 + 9 + 27 = 40$ as required. 

For example, if we want to measure out $W = 25g$ of herbs, 

$$ 25 = (1 \times 1) + (-1 \times 3) + (0 \times 9) + (1 \times 27) $$

Since we place the $1g$ and $27g$ weights on one side, and then place the $3g$ weight on the opposite side with the $25g$ of herbs. to balance the two scales. We do not require the $9g$ weight. 

Another example, if we want to measure out $W = 15g$ of herbs,

$$ 15 = (0 \times 1) + (-1 \times 3) + (-1 \times 9) + (1 \times 27) $$

Since we place the $27g$ weight on one side, and then place the $3g$ and $9g$ weight on the opposite side with the $15g$ of herbs to balance the two scales. We do not require the $1g$ weight.

Now if we are given a one-pan scale, we can apply a similar approach. This time we only have 2 actions:

1. Place a weight on the scale
2. Not use a weight at all

Again, we will represent each action by either multiplying the weight's value by 1 or 0. We choose the five weights to be $1g$, $2g$, $4g$, $8g$, and $16g$, which are the powers of 2. 

Similarly, we represent $W$ this time as, 

$$ W = (a_1 \times 1) + (a_2 \times 2) + (a_3 \times 4) + (a_4 \times 8) + (a_4 \times 16) $$ where each $a_i \in \{0,1\}$.

and the sum of the weights is $1 + 2 + 4 + 8 + 16 = 31$. 

For example, if we want to measure out $W = 11g$,

$$ 11 = (1 \times 1) + (1 \times 2) + (0 \times 4) + (1 \times 8) + (0 \times 16) $$

and we can follow a similar procedure for any $W$ from 1 to 31. 

When it comes to topics related to number theory, we notice that the concepts of representing numbers in different bases and balancing weights can be thought of in modular terms.

We can ask students to determine which weights are needed to weigh items and have a remainder of $1\mod 3$. Then we can have students group weights into congruence classes to understand how different combinations yield the same result in modular arithmetic. We can also change the maximum weight and number of pans. For the stories involved in these problems, we could have students research how different civilizations historically measured things and explore the development of measurement.