Art Math Project Summary

 Brandon, Henderson, Jacklyn: Georgina Ryan’s Binary Tree

Georgina Ryan used cotton fabric and cotton embroidery thread on a 21.5 x 20.0 x 1.0cm plastic embroidery loop

Georgina Ryan: Binary Tree (Original Version)

When remaking this project, we first decided to approach the binary tree through the lens of fractals instead of combinatorics. This meant we had to simulate the appearance of many more iterations than the 8 that Georgina Ryan did.

The first step in making a binary tree is deciding on an angle between branches and a ratio with which the branches shorten at each iteration. In an attempt to mimic the proportions of a “classic” tree, we chose an angle of 30° and a ratio of 1/2. We found the branches got too short too fast, making it very difficult to do many iterations and resulting in no interesting fractal geometry. We then decided to increase the ratio to 2/3. With this change, we found we could create more iterations and got some overlap in our later iterations because the branches got smaller slower. Choosing the length of the initial iteration was also a problem we faced. We tried sketching out a couple versions with different lengths, but struggled with finding a satisfactory length. We ended up committing to a length that resulted in the ends of the tree being a little off the canvas, but it gave it an interesting aesthetic so we are not upset. All this exploration was done through a combination of drawing fractal trees on paper, sketching out our tree on the canvas, and using an online binary tree generator (https://homo-deus.com/lab/mathematics/fractal-tree).

Computer generated tree with our parameters

The original art piece was embroidered using thread. In order to make this project our own, we decided to make a binary tree out of trees themselves! We looked around the garden and found sticks of varying thickness. We then cut them to the proper proportions, attempting to pick thinner sticks for each iteration of the binary tree. We then hot glued the sticks onto a canvas (30.05 x 40.05 cm)  in order to recreate an actual binary tree! In order to simulate the appearance of the chaotic ends of a binary tree with many iterations, we used moss. Adding the moss also had the effect of making the art piece look even more like a tree found in the wild 🌳

Exploring binary trees in the sun ☀️

For our interactive activity, we decided we will split the class into groups (preferably those they are sitting with), and assign them each different angles to explore with their own binary trees. Within each group, each person can choose a different ratio and everyone can draw a binary tree. Afterwards each group can show their drawing to the rest of the class. The goal is to get a visual representation of how binary trees change depending on different ratios and angles, along with instilling a general confidence in how binary trees work. 

The final product 🌳

Battle Ground Schools

 The first spot I really paused is when this text compared the approach we take while teaching mathematics to the blind obedience to authority and lack of independence of mind that run rampant in the face of fascism. This really made me think of how the way we teach math can impact students and their broader social values. We want everyone in society to be able to think for themselves, to form their own thoughts, create their own art, and come to their own conclusions. If we center math around simply memorizing and regurgitating an algorithm, we are reinforcing the digestion of information without questioning it. Students and individuals in general should be able to look at concepts, dissect flaws, and learn to apply them to different situations. These are skills we want people to have so it would make sense to ensure students develop them across all educational disciplines for a more comprehensive education. 

The “New Math” section as a whole made me question its validity as an approach in the first place. I feel as if a curriculum designed out of a desire to outcompete another country cannot truly be focused on optimizing student learning. I feel as if pumping K-12 full of abstract material and difficult concepts university students tend to tackle is the equivalent of getting students to just regurgitate information without a solid understanding. If anything, I feel like it’s more important to tackle basic concepts and ensure students can apply them to numerous situations and contexts. Although this method did value understanding over literacy, its main purpose was to pump out a few top-notch students from high schools while leaving everyone else behind. A teaching philosophy based on competition focuses on the wrong topics. K-12 should be more about building problem-solving skills and critical thinking rather than introducing students to every topic they may encounter further on in their education. 

Finally, I don’t think I’ve considered math in relation to politics, but it makes a lot of sense when pointed out. Individuals often get stuck on “math is math”; it can’t change, so why are we changing the curriculum or how we teach it? Once again, I believe this stems from a prevailing belief that math is all about algorithms and memorizing how to get to an answer. When you take a second and ask why, it opens up a lot more discussion and possibilities. However, the general public may see this as an “attack” on science because it's not the way they learnt things. I feel like, in this day and age, we need to understand that just because that is how something has always been done, it doesn’t mean it's right. There is always opposition to change, even in math, and I find it quite interesting that the way in which we have taught math throughout the years coincides with the current political state of the world.

What is meant by "curriculum"?

 I found this reading really interesting, as it expanded on ideas I feel like I’ve seen before. I’ve heard of the hidden curriculum before from sociology classes I’ve taken, and it made sense that schools also instilled general societal values indirectly in their students. However, I’d never heard of the null curriculum before- the curriculum or rather content, schools don’t teach. When Eisner mentioned that schools typically do not teach artistic reasoning and imagination exploration, it made me pause. It seems weird that we don’t teach something that has been such an integral part of human interaction and culture throughout history. The fact that music, art, and other artistic expression is so integrated in our daily lives, yet it is barely touched upon in school, makes sense. I understand an emphasis on the sciences, as they are very important subjects, but it does make you think about how much potential is wasted not exploring other areas of knowledge. There are so many students who would benefit from exploring other modes of learning, especially in relation to the arts and artistic reasoning, and I think Eisner makes a good point to acknowledge the absence of such a curriculum in schools.

When Eisner mentioned that kids are conditioned to follow directions, schedules, and orders, i.e. the hidden curriculum, it made me think of what else teachers unintentionally teach kids about how they should think of themselves in comparison to others. Simply approaching this from a feminist perspective, I thought of how whenever something needed to be moved or carried in school, the teacher always said: “Are there any strong boys that can help me move this?” This directly insinuates from a young age that girls are as “capable” as boys, and that there are different tasks for the different genders. Or another example is boys teasing girls and the teacher's response being “They have a crush on you.” I feel like this teaches young girls that they are just going to have to put up with inappropriate behaviour from boys simply because they “can’t” express themselves differently. Even though these are never direct lessons taught by a teacher, they do heavily influence how students view gender roles from a very young age and can impact their self-image. 

Eisner implied throughout this reading that there should be many different ways of teaching and learning, which I feel the BC curriculum has made an effort to incorporate over time. Simply from my experience during my practicums, even math class has been utilizing art and other modes to teach concepts. We gave our one grade eight class a Pythagorean proof assignment, where each student picked a different proof and had to explain how it worked. You could do so visually, or using math and words. I found that all the students did well on this assignment, as they could approach it in whichever way made more sense for them. There were a few beautiful projects where students ended up doing grade twelve math but presented everything artistically. In fact, these projects typically came from students who showed no interest in math class, so it was cool to see them get so invested. Although there’s definitely still work to be done on expanding modes of learning within school systems, I think the BC curriculum does a good job of providing freedom in how teachers choose to approach different concepts. This ultimately allows for diverse learning styles to take place, and for kids to lean into their strong suits.

Locker Problem

  I needed to draw out the problem in order to visualize the pattern. Once I had a set order, it made sense to look at perfect squares and what properties they have in particular that make the doors closed. 

 

Favourite and least Favourite Math Teacher

 My favourite math teacher I’ve had was probably my pre-calc and calculus teacher. She had very structured classes, where we would start almost every class with a lecture. She explained the concepts of the day, and then proceeded to work through a few examples with us, increasing in difficulty. Then we would get a work sheet of questions to practice with which always had solutions posted already. I really enjoyed this style of teaching as I feel like I learn by observing. I almost always need someone to work through a problem before it makes sense to me, or simply explain the thought process they use to approach a question. I find I learn a lot better when I can just attempt all the problems and then identify the areas I’m lacking in and need to work on. This teachers structure allowed me to do that perfectly! During our work time, we were always allowed to ask questions and were encouraged to come in before and after school to solidify concepts. To be honest, although I really enjoyed this teaching style, it was probably more of the one ion one help I received when I came into school early that helped me the most. Being able to ask specifically about small parts of problems that caused confusion helped me to really grasp concepts as a whole instead of just “tricks” and “rules”. 

My least favourite math teacher was probably a university math prof. He would only really go over definitions and proofs in class. He would never link the concepts to problems we were working on, so when it came to homework questions I was always so confused. I understand it’s important to understand why you solve a problem the way you do but it felt like I was expected to understand and be able to apply all these new and difficult concepts without support. There were rarely examples done in class, and if there were, he wouldn’t actually solve the problem but just asked students who did understand what the answers were. He would rush over the nitty gritty of each question and skip straight to the solution since a small portion of the class understood. By the end of the semester I didn’t even feel like it was worth going to class because it felt no more helpful than reading definitions online. Even when asking for help in office hours, he would just ask me another question and tell me to think about things. I know it’s necessary for students to figure out questions on their own, but when you’re completely lost, sometimes you just need a step by step walk through of a problem which he refused to do. I felt like I was always behind and that I had to pre-learn concepts before class to get any benefits from them. In this class, I always felt stupid or incapable in this class because I couldn’t learn the way he taught, and I feel like I ended up internalizing a lot of those feelings. 

Richard Skemp Reflection

 As I was reading this article, I found the football analogy very thought provoking. I feel as if it explains a huge problem in mathematical education. When I was in junior high and high school, I never understood why my friends were struggling with math, it just made sense to me. However, thinking back on things, I think my friends just possessed mostly instrumental learning. They could do a problem that was shown to them with the exact steps the teacher did, but the moment you changed one aspect of the problem they threw up their hands. It was “impossible” because they hadn’t learned it before. I think this way of learning is more about regurgitation than actual problem solving and applying concepts. I also thought that the map analogy was very insightful. I think it describes perfectly why a relational mathematical understanding is typically better suited for solving a wide range of problems. As I mentioned previously, students often get lost when a part of the question is unfamiliar. But if students are given the tools and techniques to attack all different kinds of problems this probably wouldn’t happen as often. I think it’s important for students to understand this as well. I feel like a lot of kids just want to get the right answer with minimal thought but don’t consider how this method of attacking problems fails when difficulty increases. It is essential students understand that although there are “shortcuts” or “tricks” to get the right answer faster, it is still important to understand the underlying concepts to genuinely understand the material. Having a mental map of ways to approach and simply questions is in my opinion a much stronger method of solving problems. 

I do generally agree with Skemp. I think relation mathematics should be the goal when teaching students, it gives them the most comprehensive understanding of material and helps with problem solving skills instead of just testing memorization. However, there is the concern of time and resources. As Skemp mentioned, sometimes explaining the concepts of a topic is far beyond the level you are teaching, and students need an easy way to approach problems for tests. I think this issue stems from those who decide the curriculum, and what their goals are for students. There is also the concern of reaching every student. For some students, things will click immediately, but others may need longer to wrap their heads around things, but how do you balance this? You still need to get through an entire curriculum so sometimes simplifying problems is the only thing you can do. I think it’s a tricky issue, in theory it would be ideal if all math was taught as relation math, but in practice this isn’t always feasible.  

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