Binary Tree Art Project

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.

Here is the drawing of the original version of the binary tree with 8 iterations, ratio of 0.63 and 60 degree angle.

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).

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 🌳

Battlegrounds

When I was reading the article, the first thing that stopped me was when the article mentioned how people from various backgrounds have the assumption that there aren't any other ways of "rigorous mathematical training" for new generations other than the fixated, stereotypical assumptions people have about math. I find this very intriguing and relatable because growing up, people around me always thought of math as boring and dull. Many view the subject as purely memorizing and applying equations to problems given on assignments and exams. To add on, I attended classes where I received conservative learning similar to what the article talked about. Teachers would often give us homework and exams without putting much effort into explaining the concepts behind them. A lot of the time, I was memorizing and writing down answers without understanding where they came from. A teacher who lacks sufficient subject knowledge can create difficulties for students. This can cause lot of people to become math-phobic, and I want to make sure that doesn't happen in my own classroom.

The other thing that caught my eye was Dewey implementing the inquiry-based approach of teaching into the system. This struck me because in the past when I was learning math, it was always the conservative way of learning and it seemed as though no other approaches could work. It makes me wonder what non-conservative approaches I could implement when I start teaching. The progressive way of teaching also involves students moving around and talking with peers for social interaction and reflection, which help students improve in different aspects, not just their math knowledge. I was interested in how Dewey proposed that students engage in doing math as part of reflective inquiry, perhaps allowing students to reflect and test hypotheses on questions would be more beneficial to students in the long run.

The last thing that stopped me was the focus on the political aspects of math education that sparked debates between different groups on how math should be implemented in school systems. It was very interesting to see how the clash between multiple parties contributed to the development of the education system we know of today, from "New Math", which attempted to reshape math curriculum in order to prepare students in competition with the USSR, to the backlash against standards-based curriculum in the 1990s. With so many influences on math throughout our history, the back and forth arguments in the article between different groups make me wonder what are some of the possible ways we can incorporate everyone's perspective on learning into one to develop a better curriculum for the future.

What is meant by curriculum?

When I was reading the article, the first thing that made me stop was when Eisner talked about how schools analyze the implicit curriculum and use it to teach students. We often hear how people mention schools prepare students for their future careers, but outside of knowledge and skills, what else do they offer? The article mentioned examples including how most jobs depend on routine that students learn while they're in school by following their class schedules, as well as how most people depend on extrinsic motivation, such as money, to sustain enough interest to work while they receive rewards and good grades for performing well. While I was aware of how school functions, the unwritten part of how it prepares students for their future through these small details was something that has never crossed my mind before.


The other thing that stopped me was when the author talked about null curriculum and how it is as important as what schools teach. Eisner mentioned how there are subjects and areas that schools emphasize such as Math, English and Science, but others such as Economics and Law are neglected. This connected me because when I was growing up, I was taught the way that Eisner mentioned. Now looking back after growing up and entering the workforce, other subjects that were neglected by schools such as economics are as important because they teach us how our social systems work. This made me believe that recognizing null curriculum is important when deciding what should be included in our school system.


After reading the article, I now believe that 'curriculum' shouldn't just be a set of expectations, learning outcomes and knowledge of subjects that we give to students. Other things, such as the importance of life skills and social norms of society that students learn from curriculum, are equally as important. The BC Provincial Curriculum emphasizes the "understand", "know" and "do" part which aligns with Eisner's ideas when he talked about meeting the learning standards. However, the author also encouraged us to look beyond the explicit curriculum offered to students. I believe that considering Eisner's idea of implicit curriculum and null curriculum within the BC Provincial Curriculum would be a step forward for the education system. While we focus on how to prepare students for the future, we also have to think about the importance of renewing curriculum when necessary.

Introspective Writing

Throughout my years as a student, there was one math teacher who was my least favourite, and that was my Grade 9 math teacher. To begin with, he showed up to the first day of the class seemingly unprepared for the coming semester. The teacher handed out the course syllabus and briefly went over it without explaining or going into detail about some of the expectations he wanted in our class. To make matters worse, my Grade 9 math teacher did not provide any meaningful lessons throughout the semester, as he would give us worksheets or assign our class textbook questions and expect us to do them without explaining the concepts. Often, the teacher expected classmates to sit together and form small groups to help each other while he did not do the teaching. A lot of my classmates ended up getting a grade they weren't satisfied with because the expectations and material weren't delivered properly. 

On the other hand, my favorite teacher was my Grade 11 math teacher. The first thing I remembered was that he gave us each a syllabus with clear explanations of the due dates, absences, assignments and exams. Me and my classmates ended up enjoying his class because he was clear with his expectations. In addition, he would give us fill in the blank worksheets that corresponded to each unit he was teaching. The teacher would go over each point on the worksheet and demonstrate the work for each question, explaining each concept. On top of that, he would also wait and ensure all students finished copying down what he wrote on the board before moving on. His patience and attention to detail were other reasons why I enjoyed his math class.

Some things I can potentially learn from both my Grade 9 and Grade 11 math teachers are the importance of coming prepared and having a lesson plan for the class so students know what they are learning on the day of. On top of that, I need to ensure that I make expectations clear for the class so students know what to expect for the rest of the school years. Last, but not least, I would deliver lessons before forming students into small groups, that way the students already have a basic understanding of the material and can understand the specific areas that they struggle on better. These experiences with both of my math teachers in high school have taught me that being a good math teacher isn't just about understanding the material, but about the way it is delivered, whether students can understand it as well as being patient with students who need extra support.

Math Puzzle

When looking at the problem, I first looked at the pattern by writing out whether each locker is closed or open.

I wrote down a diagram of ten lockers and based on the result, I can see that the first locker, fourth locker and the ninth locker are closed. 

I then dug deeper by writing out numbers up to twenty and I found that all the lockers whose numbers are perfect square are closed while the non perfect square number lockers are open. Not only that, I also found that the perfect squares have an odd number of factors. For example, 4 has three factors:1, 2 and 4. The next perfect square 9 also has three factors: 1, 3 and 9. On the other hand, non-perfect squares have an even number of factors, such as 6 has four factors: 1, 2, 3 and 6 while 12 has six factors: 1, 2, 3, 4, 6 and 12. 

Therefore, we can conclude that numbers with an odd number of factors are closed, which are the perfect squares in this case, while non-perfect squares remain open.

Set of lockers:

1st closed (1 factor)

2nd open (2 factors)

3rd open (2 factors)

4th closed (3 factors)

5th open (2 factors)

6th open (4 factors)

7th open (2 factors)

8th open (4 factors)

9th closed (3 factors)

10th open (4 factors)

11th open (2 factors)

12th open (6 factors)

13th open (2 factors)

14th open (4 factors)

15th open (4 factors)

16th closed (5 factors)

17th open (2 factors)

18th open (6 factors) 

19th open (2 factors)

20th open (6 factors)

Not only that, I also noticed that the number of open lockers is increasing by two between each set of closed lockers.

To prove this, I tried writing out an equation for this result. Let's say 2 x n, where n represents the position of the set of open lockers, the first set of open lockers is two because we know 2 x 1 is qual to 2, and we can attest that by seeing there are two open lockers between the first two locked numbers. The next number n = 2, which represents the second set of open lockers, we plug it into our equation 2 x n and we find that it is equal to four, so we know that there are four open lockers between the next two locked numbers and so on. By doing this, we can also see that each locked numbers we end up with are perfect squares.

I can safely assume that for lockers up to 1000, every perfect square lockers are closed while every other lock numbers are open.

Response to Skemp's approaches to learning and teaching

While I was reading Skemp's article, the one thing that stood out to me the most was when he mentioned how teachers want their students to learn relationally but instead, students are learning instrumentally. It struck me because often times students were not taught relationally at a young age which resulted in them memorizing equations and answers as they grew older, which raises another challenge because when the questions are worded differently, students may not know what to do. Another thing that caught my eye was when Skemp was talking about the "Devil's Advocate". Skemp mentioned the three advantages of instrumental advantages that are so beneficial that a great number of teachers are using them. While reading the "Devil's Advocate", I related to this because this was what I did with my students when I was tutoring math: giving them equations and telling them the steps to do. The students were satisfied that they received quick answers felt confident about themselves. However, since a large number of educators are using instrumental teaching, it makes me wonder if most educators simply want the easier option and whether it is ideal for students in the long run? The last part that stood out to me was when Skemp talked about why many children are only taught instrumental mathematics in school. It amazes me how even though relational teaching is a better method in the long run, most still choose instrumental teaching and it makes me wonder if there is a way where we can convince educators to use relational teaching on a more consistent basis.

After reading Skemp's article, I agree with his standpoint about the issue where relational teaching is not used as often as instrumental teaching, and at the same time, relational teaching is superior to the latter, such that the comprehension of concepts developed from the basics is able to contribute as students grow and learn newer materials. To expand on what Skemp mentioned near the end of his article, he also talked about how relational teaching takes too much effort and is too difficult to achieve. However, I believe that there are circumstances where relational teaching can prove to be useful, such as implementing it at an early stage when students are still young and the learning materials are not as complicated. By doing this, when students grow up they would be able to build on what they've learned previously and apply the knowledge as they learn new concepts. So, I do think both instrumental and relational teaching are beneficial, depending on the stage of when each is used.

Hello World

Hi everyone, I'm Brandon

Excited to work with everyone for the coming term!