Battleground Schools: Reflection

 This piece was interesting because I had never thought of Math as a politically charged subject. We hear in the news about english and history being re-written due to political influence, but math seemed to be one of the untouchable subjects. I see now that this is untrue.

I saw many similarities to the concepts of instrumental vs relational understanding. As has been mentioned in class, instrumental understanding is more quickly grasped, while relational understanding leads to better applications of one's knowledge. It is apparent that society chooses which type of understand to prioritize depending on the society's needs. 

The most difficult part of this reading is that it is apparent that there is no right answer, and there is no pleasing everyone. It is humorous that parents who wish to protect their child's future will argue about which path forward is best. Parents will argue both for and against changes to the curriculum, both with the exact same end goals.

My big takeaway is that becoming a math teacher does not make you immune from politics. It will be my duty to fight for what I see to be in my student's best interests. 

TPI Reflection

 I am very happy that I was categorized as Developing and Nurturing. I related very strongly to these categories after reading their summaries.

Developing means that I meet students where they are and work with them towards their goals. My experience focuses on those who need concepts broken down into simple terms before being able to approach the complicated ones.

Nurturing means that I recognize the emotional aspect of teaching, keeping students engaged by supporting their abilities and congratulating their efforts.

I think this both of these qualities stems from my origin as a tutor for failing students. They required me to teach with a developmental mindset, for the were only in the emerging category. They also required me to teach with a nurturing mindset, because they thought they were terrible at math and therefore had lower self confidence. 

I am unsure how this profile will change after I teach in a classroom. I think that as my class size increases, and as the skill level of my students increases, my profile will change.

My Mini-Lesson Theme

 I'm going to teach the class the fundamentals about installing swing stage scaffolding. I used to do this work.

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.