Math/Art Assignment Group Reflection

Group Members: Sarah Dicastri, Tiffany Gong, Kia Prezeau 

Original Artwork: Sudoku Without Numbers by Dru Horne and Shannon McKillip 

We decided to remake and extend the original piece using cardstock and markers instead of fabric, as none of us had experience with quilting. This proved challenging in that it was quite time-consuming to cut out every individual piece and glue them all together. Additionally, we needed to glue all background squares together in a way that was structurally sound, taking us 6 hours to make just the extended art piece. Coming up with a unique concept for each layer was also difficult to decide on, as we wanted to select distinct features that layer well on one another and still show the other elements beneath. We ultimately settled on adding features like coloured borders, coloured squares, and hand-drawn icons that could fit around our larger icons without blocking the background. 

We first focused on extending the piece mathematically by scaling the canvas to a 7x7 matrix and layering 6 different mutually orthogonal latin squares. We arrived at this number of layers by using theorem 7 as described by Ballif (2008), which produces 6 as the maximal number of mutually orthogonal latin squares that can be determined from this matrix, given that 7 is a power of a prime. We also used this theorem to construct a full matrix of six layers in a numbered sequence, by assigning a number of 0 through 6 to each element in each latin square to construct the full picture that ensured all entries were distinct. This part of the process required some care and could be a point of challenge for others interested in recreating, as any mistakes could result in duplicate entries. 

We then extended the art by focusing on how we could tie it into learning from place, specifically by having one of the latin squares be icons from Coast Salish symbols for different local animals (sources were cited in our presentation). This was beneficial both for ourselves to explore more local Indigenous art and artists, as well as to tie in the BC curriculum’s goal of incorporating Indigenous ways of knowing. 

We designed our interactive activity to be a smaller/simpler version of the mutually orthogonal latin squares that we constructed. Specifically, we created a 4x4 matrix with two fixed elements and a movable third that students can use to layer their own latin square. We hope this can cement an understanding of orthogonality between latin squares, as well as spark discussion on how many possible arrangements exist when they compare with other students. We decided on a simpler matrix to have confidence it can be completed during the short allotted time during lecture, while still giving everyone a chance to have hands-on experience with these concepts. It also brought the content difficulty closer to the high school level – as combinatorics is only introduced in Foundations of Math 12 – instead of the university-level math the original piece employs. 

Original digital mockup

Progress Pictures

Final Product 

Bibliography:

Ballif, S. (2008). Mutually orthogonal Latin squares. [Lecture notes]. Department of Mathematics and Statistics, Dalhousie University. https://www.mscs.dal.ca/~janssen/4370/Orthogonal_Latin_Squares_text.pdf

Battleground Schools

The first idea that made me stop was realizing how polarized the views on mathematics education have been between progressivists and conservatives. I had assumed that, although people might disagree about how math should be taught, there would be more generalized public view about balancing understanding with procedural fluency, and the difficulty with doing so. Instead, the reading showed how strongly these ideas could be separated, with progressivists emphasizing understanding and inquiry while conservatives placed more emphasis on fluency and procedures.

I also didn’t realize how early the progressive movement in mathematics education began. I had associated inquiry and student-centred learning with more recent years, so learning about John Dewey’s influence during the 1910-1940 period made me rethink how long these ideas have been around. I also hadn’t really considered how much large political events could influenced what students were taught in mathematics. Reading Susan’s part, I realized how much the article was framing math schooling as a key to global competitiveness; the most important part about math in school was to create more scientists to win the global race.

Finally, I found the discussion of American students’ international mathematics rankings interesting. Seeing that U.S. eighth-grade students ranked 28th in the world made me wonder how Canada compared at the time. Overall, I realized that debates about how mathematics should be taught are much older and more complicated than I had thought.

What is meant by 'curriculum'?

When reading, one idea that made me stop was the section on rewards. I’ve always liked the idea of giving students small rewards because I assumed it would encourage them to enjoy learning even more. I hadn’t considered that relying on external rewards could actually reduce a student’s intrinsic motivation over time. It made me rethink how I want to motivate students because I’m aware that many students dislike math classes. 

I was also surprised by Eisner’s discussion of how schools create competition through honours programs and differentiated classes. It reminded me of my own experience being in Mini School in high school and how I would always tell people it wasn’t a class of smarter or more capable people; we just so happened to have some classes together as a cohort. But it did always feel like there was some segregation. I hadn't really thought about those structures as part of the curriculum before, but they definitely teach students beyond the content they learn in lessons. 

When I first thought about curriculum, especially in math, I pictured a list of topics that students are meant to learn. I think the BC Math Curriculum connects with Eisner's ideas through the core competencies, since students are expected to develop communication, thinking, and personal and social skills alongside the curricular competency. This reading expanded my idea of ‘curriculum’ because it made me realize that I'm not only teaching academic content, but also shaping the students. That feels like such a big responsibility and a bit scary to think that I can really affect students. 

The locker problem

To approach problems that don’t exactly have a clear start, I always like to do a smaller example base case, which I decided on a sample of 6. 

I noticed that only 1 and 4 were closed, while the other lockers were open. Intuitively, I could tell the factors of each number were determining the end result of each locker; thus, I wrote out the factors of each one and noticed the numbers with an odd number of factors were the ones locked. I added a few more examples (7,8,9) to make sure. 

Plus, when I was thinking of what bigger numbers could have an odd number of factors, I could only think of numbers where a factor is repeated, like 49, 7x7. So another interesting aspect is that all the locked lockers are likely square numbers, as they would need a factor pair of n x n. 

Overall, I concluded that the open lockers are lockers with an even number of factors, and the locked lockers are numbers with an odd number of factors.

Favourite and least favourite math teachers

 My favourite math teacher was my Grade 9 or 10 math teacher. The first thing I remember is that she taught us to take notes by folding our paper and using one side for basic notes and the other side for example questions. Although I didn’t continue using this method after her class, I thought it was a unique way of taking notes and learning. What made her most memorable to me, though, was that she had a blog. She only updated it during the year that she taught me, but at the time, especially before COVID and all the online teaching, I felt like it was pretty rare for a teacher to have a blog. I liked that it gave us a way to get to know more about her life outside of the classroom. I also remember having a conversation with her about how handing in homework was optional in her class. I never did it because math came pretty naturally to me at the time, but she explained that this probably would not work in more advanced math because the concepts would not always be as intuitive. She was right, and I ended up taking her advice and doing my homework more consistently in later math courses. I think what made her such a memorable teacher for me was the connection she had with her students and the effort she made to get to know us.

I also remember my mom teaching me math and making me memorize my multiplication tables when I was young, which I was really appreciative of, especially in high school because I felt like I had a strong foundation in math and was a little ahead of my peers in elementary school.

I can’t really think of a math teacher who taught me particularly badly, because I tend to learn better independently and through practice problems. My least favourite math teacher was a high school teacher who I had a negative experience with outside of the actual lessons, so my opinion of them was more influenced by my personal experience than their teaching. Overall, this made me realize that when I think about my favourite and least favourite teachers, I remember my personal connections with them much more than how they actually taught math. As I take on the role of a teacher, I think building connections with students will be really important. The small things, such as having a blog or taking the time to have conversations with students, can make a teacher much more memorable and have an impact beyond just the math they teach.

Skemp on two approaches to teaching and learning mathematics

 I definitely agree with what Skemp is arguing about the importance of relational understanding. I think having a relational understanding is especially important when learning more complex topics because it allows students to understand how ideas connect and apply their knowledge to further topics. This also made me think about my experience learning in IB, where making connections between concepts and applying knowledge in different contexts is emphasized, so I think relational understanding is especially important in an IB classroom. However, I also think instrumental understanding can be a useful starting point and has its own advantages, as Skemp discusses. 

When I was tutoring math, I would often adjust how I explained a topic depending on the student and how much time they had. For example, if a student came in the day before a test on a topic they had not covered much, I might explain it more instrumentally by making sure they knew the formula and could apply it correctly, rather than spending time explaining the reasoning and context behind it. If a student was being introduced to a new topic and brought in their schoolwork, I would usually take more time to explain the reasoning behind the concepts and things they might not have learned in class. Both approaches definitely have a place depending on the situation. 

Hello World!

This is my friends cat that I drew :)