Reflection on Assignment 1

 Our group presented on Serpinski Triangles. We focused on two aspects their design, Cellular Automoton and the use of different cells. Cellular Automoton was taught by showing the class the different rules that generate different triangles, while the concept of creating fractal shapes one level up was taught by having the class play around with physical cells that we cut out.

This project was a great experience for me. Every time that we hit a road block, it felt more like a puzzle to solve than a challenge to overcome. I had genuine fun with this assignment. I understand how much more enjoyable math is when it is in the form of visual art, contrasted with its traditional form. The best part of this project was that I did not have to practice at all. Typically when presenting on a topic, I have to rehearse. For this project I was so engrossed in the subject that I learned everything on my own time. Once we began presenting, the ideas just flowed out of my brain to the class. This happened because I chose to learn the topic in side and out, as opposed to memorizing a script. I could improvise everything because I had relational understanding, not instrumental.

Teachers cannot prepare for every possible outcome. The tiling activity did not at all go according to plan, but went better than I could have dreamed. While they were not fractals, the patterns which the class came up with had an incredible mathematic beauty. When I am a teacher, I will be very open to ideas that students propose, and will improvise around them. I want to promote inquiry in my classroom, and that is best done when students are allowed to take the reigns.

I have no current ideas for projects at this time, but because of this project I will have the eye to see them in my daily life. Both math and art exist around us all the time, and I now that I can recognize them everywhere I will bring them into my classroom.

Letters from Future Students

 

Hi Mr. Braun,

 

I just wanted to send you a quick shoutout and say thanks. While math wasn’t always easy for me, you were able to help me get through. My mom was so happy that I was able to make it through high school, even though I never continued with math in academia.

The reason I’m reaching out is because I just finished an exam for my plumbing apprenticeship. It wasn’t the math that you are probably used to, but studying for it ended up being really easy. I remember what you said about pattern recognition and was able to build upon that to ace the exam. None of my friends in the program know anything about math, but I was able to help them out.

Thanks for never giving up on me!

 

Hi Mr. Braun,

I just wanted you to know the repercussions of your words. I remember how much you I spired me to follow skills and further my math knowledge. Well guess what? I graduated and got a job in a highly math related field and it is terrible. My life is terrible. You never told me math used in the real world is so soul sucking. Sure it’s fun in the abstract ways of beauty, but what good is that when I have to pay the bills?! And guess what? My job is being automated next week because it’s all algorithm based anyways. Going into a field which emphasizes social skills would have been a better investment, but because you showed me the “beauty of math” I got bewitched and am now here.

Thanks for ruining my life.

**UPDATE**

I worry that my excitement about math may inadvertently deify the subject to some students. My enthusiasm for math is one of my strongest assets, and I feel that I will make math more attainable to many students who would have otherwise been scared of the subject. However, I worry that my excitement could bring a level of fanaticism that is not compatible with our current society. In my experience in the corporate field, I found that math used in the real world really was soul sucking. I obviously don't want to tell this to students because I want them to be engaged, but I don't want to lie to them.

Math is an art, but when art and a capitalistic society merge the art can become tainted. My experience with engineering was exactly this. I will try to guide students down proper paths that allow them to play with the art of math more earnestly. 

The dishes Problem

 My first step was to reword the problem and understand what was happening. When the chef says “every 2 used a dish of rice” it means that each guest had half a dish of rice. Another way of looking at this is that, of the total number of guests, half of them had a dish of rice, one third had a dish of broth, and one quarter had a dish of meat.

Let’s represent the total number of guests by twelve groups of one twelfth. This is because we can easily figure out halves, thirds, and quarters of twelve. Half the guests had a rice dish, that is 6 twelfths. A third had a broth dish, that is four twelfths. A quarter had a meat dish, that is three twelfths.

Rice

Rice

Rice

Rice

Rice

Rice

Broth

Broth

Broth

Broth

Meat

Meat

Meat

 

You can see that we have an extra twelfth of dishes. Because there are 65 dishes served, there is an extra twelfth. The problem becomes “What number, when increasing that number by one twelfth of itself, becomes 65.” Traditionally, algebra would be used at this point. However, a culture could guess and check until they arrived at the correct number of 60 guests.


While presenting itself as a problem of Chinese origin, the Chinese culture does not exist in the problem outside of the choices of food. However, the existence of the problem as a piece of history, going so far as to provide the original historical text, is very important. It shows that the traditional western view of mathematics is incomplete by giving a concrete example of maths existing outside that realm.

The fact that this is presented as a story does make the problem more enjoyable to solve, almost as if we are being transported back to an ancient Chinese restaurant. It is still nonsensical, one would simply count the number of tables and average table size before counting dishes. But if one can understand the problem as a fun test of our math abilities with a flair of the dramatic, it becomes much more enjoyable. 

Evan's Reflection on Lockhart

The points Lockhart makes about math and culture in this article were amazing. The analogies to the musician and artist’s dreams were beautiful. I agree that the art of math has been lost due to a societal fascination with utility. This obsession can also be seen in the lack of funding awarded to arts programs. Mathematics is an art who functionality is more widespread, but that does not mean that math and its function are inseparable. I loved Lockhart’s point that marching bands may make armies more efficient, but music exists outside this concept. I also appreciated Lockhart’s elegant definition of math as the art of manipulating simple, imaginary things and seeing how they react.



I was enamoured by Lockhart until he began his critique of “the curriculum,” after which I began to disagree with him more and more. The idea that math does not build upon itself is non-sensical. Even if one were to buy into Lockheart’s idea that students should learn their own math, some concepts will still require students to know the solution to other simpler problems. One cannot imagine the abstract nature of multiplication as groups before understanding the abstract nature of incremental increases. There is no correct way to structure the math curriculum, but I would argue that some structure is necessary. I would use a metaphor of driving from BC to Halifax; sure there are many different roads and paths that you can take, but to contrive of a path that jumps from BC to Ontario to Alberta to NWT to Nova Scotia is nonsensical.



This article relates heavily to Skemp’s idea of instrumental vs relational mathematics. However, Lockhart takes a very radical approach that I do not necessarily agree with. Lockheart seems to argue that not only is instrumental mathematics bad, but teaching it is a moral failing that would make Euclid and Pythagoras roll in their graves. I prefer Skemp’s position of neutrality on this issue, recognizing that there are benefits to both types of instruction.



Moving forward in my teaching career, I will strive to coax the beauty of math out of the hollow shell Lockhart describes as "The Curriculum." I understand that I will not be able to change the curriculum myself, but I believe that by knowing the beauty is there, my excitement and appreciation will be contagious enough to invigorate my students. Hopefully they will learn that math is not the scary beast of boredom, but the beauty of our universe.

My Best and Worst Math Teachers

My best math teacher comes easily to mind. He was my math teacher for grade 11 and grade 12 honors math. In grade 12, I originally had a different teacher. However, I was so unhappy with this new teacher that I begged the administration to switch me into my old teacher’s class. He was one of the most formative male role models in my life. This role modeling existed outside of the classroom. I not only admired him for his intellect in and passion for mathematics, but also for his attitude towards life. I will admit that I was not a morally good person in high school. However, I went through some personal trauma in grade 11 and this force me down a path of self-reevaluation. He recognized this in me and took active measures to guide me down a better path. I would not be as caring or empathetic today if not for him. He educated me in ways well outside the math curriculum.

My worst math teacher was an unorganized mess. Not only were his courses unorganized, but his expectations of us were also unorganized. I felt that I was constantly struggling just to understand where I was, so to understand where I was supposed to go was an impossible task. I missed assignments due to his lack of organization, and even when I handed them in, I missed marks because of his unclear rubrics. It frustrated me and I began doubting my self-efficacy. Like a wet blanket on a fire, he almost extinguished my passion for math.

Response to the locker problem

Instead of looking at all the lockers and using brute force to see a pattern, I looked at a specific locker. I asked “What would cause this locker to be open/closed.” I realized that a locker would be closed if an odd number of people interacted with it, and open if an even amount of people interacted with it.

If each locker was numbered from 1-1000, the number of people interacting with the locker would be equal to the number of factors the locker number has!

If the number of factors is even, the locker is open. If the number of factors is odd, the locker is closed.

This begs the question, which numbers have an odd number of factors? My first thought was prime numbers! Because they would only be visited by their own number. I recognized my mistake quickly; primes are divisible by ONE AND themselves.

I recognized that squares have an odd number of factors, or at least an odd number of distinct factors. For example, 9= 3x3, but 3 is distinct! In more depth, the factors of 9 are 1, 3, 9; an odd number!

I did a quick check by drawing out a list of lockers, and my prediction was confirmed. The square numbers were left closed.

Evan's Reflection on Skemp: Relational vs Instrumental Understanding

 This article has summarized in words ideas that I have had for years. Through my time tutoring students who were failing math, I experienced the two types of understanding. These families wanted a passing grade, and the students wanted instrumental understanding. I would often teach two lessons in parallel, teaching instrumentally first, relationally second. I prefaced the in-depth (what I now know as relational) lesson with “I have taught you the raw ideas of how to do this, now this is how/why the math works.” I provided relational knowledge as a bonus, for them to learn from or tune out if they choose.

 

My favourite part of the article was the analogy to the music lessons. This encapsulated what I have been trying to tell my non-math friends for years. It was such an elegant analogy that my jaw dropped. I cannot wait to share these ideas with my adult friends. It will be the key to make them finally understand why I love a subject that they learned to hate.