Math/Art Assignment Reflection

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I believe this art project truly changed my view on integrating art and mathematics. Before completing the project, I was hesitant because I considered myself someone who was not very good at art. However, having a supportive group and maintaining a positive attitude toward the assignment changed my perspective. I felt comfortable with the mathematical component of the project and enjoyed looking more deeply into the mathematics and creating an extension to the original work that challenged us to think in new ways.

The artistic component initially made me nervous because I am a perfectionist and was worried about making mistakes. However, I ended up thoroughly enjoying the painting process. My group and I were able to use the experience as an enjoyable opportunity to get to know one another while stepping away from our usual school mindset. It was especially satisfying to see the artwork gradually come together as we continued adding to it. This experience showed me that I can enjoy and be successful in something that initially felt outside of my comfort zone.

As a teacher, I think this project taught me the importance of encouraging students to step outside of their comfort zones. Just as art initially felt unfamiliar to me, mathematics may feel outside of some of my students’ comfort zones. Integrating different subjects can provide students with new ways to engage with mathematical concepts and demonstrate their understanding. It also showed me that there are many connections between mathematics and other disciplines that students may not typically recognize.

As I begin working with my own classes, I can see how projects like this could help students experience mathematics in a more creative and engaging way. The affordance of this type of work is that it allows students to collaborate, think creatively, apply mathematical concepts, and develop a more positive relationship with mathematics. At the same time, there are limitations. Art projects can be time consuming and may be difficult to fit within the regular classroom schedule. They can also require materials that may create financial barriers for some students. As a teacher, I would need to consider these factors and find ways to make projects accessible to everyone.

Overall, this project was a meaningful experience for me as a student and as a future teacher. It encouraged me to see mathematics beyond traditional problem-solving and recognize the value of creativity, collaboration, and exploration. I will definitely consider incorporating the beauty of art into my future mathematics classroom.

Math/Art Assignment Group Post

Group members: Alina, Emily K, Keith, Tara

Regina Bittencourt: One by one (The original version)

Our remake of the original

In remaking the artwork, we chose to use acrylic paints on paper. During this process, we wondered about why the artist stopped at a maximum of 9 rows (before descending). Exploring this informed our group activity of attempting to make a 10th row, observing that we no longer have the same palindromic pattern, and discussing why that happens. We tried adding a row ourselves and soon found that the “place holders” get in the way and the pattern breaks. Inspired by the lesson on Babylonian place holders, this prompted us to wonder if there was a maximum number of rows for other bases as well. After trying out a few it became clear that this maximum is one minus the base you’re working in. For example, base 10 has a maximum 9 rows while base 2 has only 1 row before the pattern breaks!

Our extension

The extension that multiplies 9x9, 99x99, 999x999, … does not follow quite the same pattern as with the 1s, and so we worked to find out why this one has a pattern of its own. Of course the painting does not have 90-degree symmetry like the original, so we decided to adapt it slightly, “flipping” the bottom half to produce an image with 180-degree symmetry to give it more visual appeal. In making the extension, we had to make decisions regarding whether the color representative of each number should be the same as the original. We ultimately decided to keep the colors consistent between both works, in the hopes that it would make it easier for viewers to easily switch back and forth between viewing. Since 0 does not appear in the original, we chose the color to represent 0 (burgundy) based on what we thought would look best with the other colors in the piece.

In designing a class activity, we wanted to offer a way to further explore the idea of the One by one piece. At first we considered using sticky-notes or tiles, but decided this would be difficult to manage in the short time for presentation. We decided to create a visual handout with the images of our artworks on one side and an activity on the reverse. Providing this visual handout means the class can see the detail of the painting (the paintings are on 12” x 12” paper, so it would be difficult to see the colours from the back of the room).

For the activity, we ask the class to consider what comes next. That is, what would happen if we attempt to add a tenth row to the painting? We’ll work this out together on the paper and on the whiteboard. Then, we ask the class to consider what would happen if the painting were done in a number system other than base 10.

No action shots available of our painting because we were too in the moment. 🙂 

To set the vibe we played the September 17th NTS Breakfast Show w/ Flo while painting and learned a lot more about each other's interests and hobbies.

 

Battleground Schools

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Before reading, I had little knowledge of the complex and often contentious history of mathematics education and curriculum in North America. I was shocked by the negative public perceptions of mathematics, particularly the suggestion that those who enjoy math are typically male, struggle with human interactions, and are not fully mentally competent. This surprised me because today I associate an interest in and understanding of mathematics with people of all genders. I was also surprised by the suggestion that talented mathematicians struggle with human interactions, as mathematical ability does not necessarily determine someone's social abilities or relationships.

Another point that made me pause was the discussion of how, during the post-war period, mathematics became a focus of educational anxiety in the United States. This is interesting because it highlights how mathematics education can become connected to broader political and national concerns, rather than being viewed solely as a subject taught in schools. 

I was also surprised by the extent to which the media influences public perceptions of education. I believe this remains highly relevant today, as the way information is presented can shape people's opinions, even when that information is inaccurate or misleading. The reading raised an important point about how governments may use the media to present a positive image of the education system while drawing attention away from its challenges. Today, with social media playing such a significant role in our lives, I think it is especially important to question the information we encounter. Even from my own experience, I recognize how easy it can be to believe something immediately after seeing or hearing it in the news.

What is meant by 'curriculum'?

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While reading, one thing that made me stop and connect was Eisner’s discussion of how visiting different university campuses allows us to recognize both similarities and differences. This reminded me of touring universities before committing to my undergraduate degree, and I was comparing their academic programs, rankings, and environments. Some campuses were located in busy cities, while others were more secluded, and every university had its pros and cons.

Another idea that made me pause was the concept of the null curriculum. I found it interesting that schools have consequences not only through what they teach, but also through what they neglect to teach. This made me reflect on how teachers’ decisions about what to include or leave out can shape students’ learning experiences and opportunities.

This reading expands my understanding of what is meant by the word “curriculum,” as it makes it clear that "curriculum" is more than a set of learning outcomes. Curriculum is also shaped by teaching methods, learning experiences, and the decisions made by teachers. Every classroom is unique, and students may experience the same curriculum very differently depending on the circumstances including the teacher, classroom, etc. 

The mandated BC Provincial Curriculum connects with Eisner’s ideas because it outlines the learning standards, however, teachers have flexibility in how they choose to design and deliver the class content. Their instructional choices, assessments, and decisions about what to focus on, influence what the students truly learn. This highlights the importance of the decisions teachers make when implementing the curriculum, as they are trying their best to ensure that the students have a variety of opportunities to learn.

Introspective Writing

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I have had many math teachers throughout my life and have experienced a wide variety of teaching styles. My least favourite math teacher was a professor at the university where I completed my undergraduate degree. He was extremely monotone, did not use slides or a PowerPoint, and often did not explain why we were doing the mathematics we were doing. It was a very difficult fourth-year course, so I understood that the material was supposed to be complicated. However, there was very little explanation of the reasoning behind the mathematical processes. As a result, the math often felt random, and I found that I did not know how to approach a problem if the scenario was changed. Additionally, there were no PowerPoint slides or lecture packages to follow along with. Everything was written on the board as he taught and then erased when he needed more space. When I asked questions, his explanations were often unclear and sometimes left me feeling “dumb” for not understanding. This experience taught me how important it is for students to understand not only how to solve a problem, but also why they are solving it and how the concepts connect.

My favourite math teacher was a different university professor who taught my Calculus course. He was enthusiastic, welcoming, organized, and made the class enjoyable. He would sometimes make math jokes in the middle of a lesson that made everyone laugh, which helped create a positive classroom environment. He also provided a printed booklet containing the course content and practice problems, which made it easy to follow along and stay organized. Most importantly, he was extremely approachable and genuinely believed that there was no such thing as a “dumb” question. He was very good at explaining the mathematics and the reasoning behind each step, as well as showing us how different scenarios could change the way a problem was solved. He still challenged us, but he created an environment where being challenged felt motivating rather than discouraging. Reflecting on both of these experiences has shown me that as I take on the role of a teacher, I want to create a classroom where students feel comfortable asking questions, understand the reasoning behind the mathematics, and feel challenged while also feeling supported.

The Locker Problem

 

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To approach this problem, I started writing out the end result (whether the locker remained open or closed) starting at locker #1. As I continued the process, in my head I would have to recall what had happened to the locker previously to determine the end result. Recalling what had happened previously to the locker, and what number of student did it, made me realize that changes to the locker happened at numbers that that locker # is divisible by. If the locker # has an even number of factors, then the end result of the locker is open. If the locker # has an odd number of factors, then the end result of the locker is closed. Thus it is possible to determine the end result of the locker if you know how many factors that locker number has.

Relational Understanding and Instrumental Understanding

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To begin, while reading, I found Skemp’s music analogy interesting. It mentions that some students simply memorize the notes EGBDF, while others associate the written notes with the actual sounds and relationships between them. This connected with me because when I took piano growing up, I knew where the notes were on the piano, but reading music was the difficult part. Once I knew the notes, I would often memorize the finger movements needed to play the song seamlessly rather than fully understanding the music I was reading. This reminded me of the difference between memorizing a mathematical procedure and actually understanding why it works. I also found it interesting how Skemp acknowledges that even people who understand mathematics relationally will sometimes use instrumental thinking. This made me realize that relational and instrumental approaches are not necessarily completely separate. Instrumental thinking can be useful within relational understanding, however, instrumental understanding alone can be limiting. I also connected with Skemp’s discussion of over-burdened syllabi and how quickly mathematics can be taught. I experienced this in my own undergraduate mathematics courses, when professors sometimes moved through complex concepts very quickly or assumed certain ideas were common knowledge. As someone who was less experienced with the material than the professor, this sometimes made it difficult to build the deeper connections necessary for relational understanding.

Overall, I agree with Skemp’s argument that relational understanding should be emphasized in mathematics education. I believe there are some situations where instrumental understanding is useful, especially when students need to quickly apply a familiar procedure. However, students should not be limited to memorizing rules without understanding the reasoning behind them. As a future mathematics teacher, I hope to prioritize helping students understand both what they are doing and why they are doing it, while still recognizing that instrumental thinking has a place in mathematical learning.

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