The locker problem

First, I tried smaller numbers (like 10 or 20 lockers) and looked for patterns in which lockers stayed open. I noticed that only lockers with numbers 1,4,9,16..... were open. Then I thought about why, I found that each locker is switched once for every student whose number divides that locker. That means the total number of switches equals the number of divisors of that locker. Normally divisors come in pairs, so lockers get switched an even number of times and end up closed. But perfect squares have one unpaired divisor (like 4 with itself in 16), so they are switched an odd number of times and end up open.

From this reasoning, the general conclusion is: after 1,000 students finish, the open lockers are the perfect squares up to 1000, what means there are 31 of 1000 lockers are still open.

                                          

Math Art Project - Individual Response

In this group project, I really learned a lot. It showed me how many math concepts can actually be found inside these artworks. Even though our group only picked one project that we were most interested in, and we didn't get the chance to try others, I already saw how much potential there is in bringing these kinds of hands-on activities into a math class. This feels very different from just teaching math concepts directly or giving students word problems that don't connect to their lives. When students actually try to make something, the clear steps help them see those math concepts in a more direct way. They also get space to be creative and discover math through their own ideas.

But this requires teachers to try these activities first and think from the student's point of view. The projects should keep students curious but not be so hard that they lose confidence or spend too much time stuck on it. Just like in our group project, it's only after we try different art projects, understand the ideas behind them, and connect them to the BC curriculum that we can design activities that work for students and get them involved in learning.

I really enjoyed this activity. My favourite part was when everyone finished their own small piece and signed their names on it. At the end, when all of our pieces and names were connected together to form a larger artwork, I really felt a strong sense of connection with the whole class.

Math Art Project - Group Write-up

 Group members: Damanjit, Elvie, Helin, Yuki

Original Artwork and Artist: Flowering Grid by Eric Gjerde by Tejom Patel

After our group collectively chose this artwork as our project topic, we faced a big issue of not having resources that lets us perfectly remake the original artwork. The artist, Tejom Patel, had creatively expanded their flower tessellation (based on Eric Gjerde’s work), by adding different folds to produce a new, unique piece. We started off attempting to perfectly mimic Patel’s art. However, due to our lack of knowledge on how the folds and designs work, we had to start off by choosing a design that had guides and tutorials. This was the Spread Hex Tessellation.

Similar to the original artwork, we kept the idea of hexagons and reflectional symmetry, but folded a tessellation where the hexagons overlap and pile up.

 

Here is the link to the video tutorial of the spread hex tessellation: 

https://youtu.be/3BTu2Hih39A?si=jRJpq3Fw6cG6oauz 

Through our research phase, we came across Eric Gjerde’s book Origami Tessellations: Awe-Inspiring Geometric Designs. This resource provided detailed folding tutorials for many origami tessellations built from triangles, squares, and hexagons, and also explained key techniques such as Pleat Intersections, Triangle Twist, Square Twist, and Hexagon Twist. With this reference, we gained a deeper understanding when looking back at our own work, and it also gave us the idea to design an activity more suitable for a short classroom session.

Our interactive activity with the class was a hands-on origami activity where each student folds their own piece of flower that will then combine to create a big multi-piece flower tessellation. Since abstract origami tessellations take a long time to fold, we designed it such that all prep is done (fold lines created beforehand) and students are to follow instructions while helping each other to collectively create one piece of art with the class. 

Here is the link to the origami flower we made in class:

https://youtube.com/shorts/tQMteMhp1Dk?si=Hxtkz7Yhg_-RoIt7 

In addition to experimenting with hexagon-based tessellations, our group created a variation called the Layered Compass, which is folded from square paper rather than a hexagonal grid. This shift gave us a chance to explore the mathematical flexibility of tessellation design. Whereas hexagons naturally lend themselves to 120° rotational symmetries and interlocking flower-like patterns, the square base highlights 90° rotations, reflections, and layered symmetry. By adapting the same folding principles—pleats, twists, and repeating units—to a different polygonal foundation, we were able to compare how tiling properties change with shape and how symmetry groups are expressed through origami art. Using square paper also made the process more accessible, since it is a common format and easier for classroom folding activities. Through this variation, we not only made the project our own but also deepened our appreciation of tessellations as a versatile mathematical art form that can be reinvented through creative folding choices.

 

Favourite and least favourite math teachers

My favourite math teacher was my high school calculus teacher. He also graduated from the education program at UBC, and he was one of the reasons I decided to come here. 

I liked his class simply because I always wanted to challenge him. Even when my ideas were not full developed, I would raise my hand, walk up to the board, and think through the next step while writing out the solution.  He often gave us very difficult calculus problems with multiple layers of integrals that included functions like sine cosine, exponentials, and inverse trigonometric functions. These problems made me want to try and he made me feel like I could beat him, like I could solve these problems faster than he could.  At the time I really believed this, until graduation day when he showed us his UBC transcript filled with A's, can't find a single B on that paper. Now, as I study Education myself, I realize how hard and special it is when a teacher can make students feel smarter than him.

In class, he often shared stories from his university days. One line that stayed with me was: "When you are in university, don't always eat lunch with the same group of people." He explained that the campus was full of people from different backgrounds, and by hearing their stories, you could open up your perspective.

To be honest, I never really had a math teacher I disliked, because I truly love math. But some teachers were more inflexible. They didn't care if you had a clever idea or a different way of solving a problem. They just gave you one set method, the quickest way to solve the answer. Day after day we repeated the same kinds of problems, and it felt like there was no interest or motivation.

Responses to Second Reading: What is meant by 'curriculum'?

As I was reading Eisner's article, I was struck by his discussion of the explicit, implicit, and null curriculum. The first thing that caught my attention was how described the explicit curriculum as an "educational menu." On the surface, schools seem to let students choose the subjects they like. But when he raised the question of the null curriculum, I started to think more carefully. I recognized that I had never asked myself during my own schooling why my school didn't offer courses like law or economics. Which now seem to me some of the most practical skills an adult need to survive in society.

The second point that stood out to me was his explanation of the implicit curriculum. He gave the example of students reading books to reach a smiling face and then getting a goldenrod ticket. Three tickets allowed them to leave for lunch five minutes early. This example really struck me because I also had similar experiences in school, and I can still see these practices in BC schools today. I used to think this was a good way to make students interested. But now I realize there is a big difference between being interested in learning itself and being interested only in the reward. This is a reminder for me as a future teacher. I want to help students build a true interest in math, not just work for small rewards or grades.

Finally, this article gave me new perspectives on things I used to see as normal and ordinary. Eisner also pointed out that schools often miss not only subject content like law or economics but also ways of thinking. This makes me ask myself how I can plan classroom activities, especially tests, to guide students to build creative thinking and to truly enjoy math.

Responses to First Reading: Relational Understanding and Instrumental Understanding

As I was reading Skemp's article, I felt genuinely inspired. When he contrasted instrumental and relational understanding, it was like he was describing my own experience as a student. In class, professors often just introduced a formula with a typical example, which made it seem perfectly fitted. But when I later worked on assignments and quizzes, I often ran into "surprises" where the formula did not quite apply. At that point I had to practice many similar but slightly different problems, and by comparing them I would slowly discover new patterns. That is why Skemp's " 20 cms by 15 yards" example struck me. He mentioned about instrumental understanding involving "a multiplicity of rules rather than fewer principles of more general application" also helped me realize why math education often relies so much on endless practice, because no single example or formula can really cover it all.

What also stood out to me was Skemp's analogies. The football story is interesting, but the music one made the most sense to me. Thinking about students just memorizing notes without ever hearing the sound reminded me of my own math learning. It was lots of symbols but not much meaning. I can still remember teachers saying, "Even if you don't get it now, just memorize it, you'll understand later." That way of learning shaped a lot of my early math and it is exactly what Skemp is pointing out as a problem.

Finally, I agree with Skemp's call for relational understanding. For me, math makes more sense when I can connect rules with reasons, see how topics fit together, and try methods on new problems. His article made me think about my own experiences and also about my future as a teacher. If I only pass on rules without reasons, I might help students get through exams, but I will not give them the confidence, flexibility, or joy that comes from really understanding math.