locker problem

 

For this problem I learned that my roommate loves math puzzles, so he insisted on trying it as well, and I included his writeups too (the second page is mostly his writing, while the first and third are mostly mine). 

When solving this problem, I first tried a simpler problem, where there are five lockers and five students, and wrote out what happens explicitly. I was still confused, and my roommate tried a similar approach, but was also confused (and misinterpreted the problem at first, until I clarified it for him). 

Eventually I noticed a pattern where, given a locker, every time a student of number dividing the locker number walks in, the state of the locker flips. From this, I concluded that the locker state at the end is dependent on whether the locker has an odd or even number of divisors.

I thought we were done, but as I was looking around at mathematical equivalences to a number having an odd or even number of divisors, I stumbled on the elementary number theory fact that a number has an odd number of factors iff it is a perfect square. This can be justified by the heuristic argument -- that each time a number has a factor pair, like a = b times c where b,c < a, then this factor pair adds TWO divisors to the list of all the divisors of "a", EXCEPT if a is a perfect square, in which case there will be b = c for ONLY ONE factor pair, so the number of factors would be odd. 

Overall, some strategies me and my roommate used --

- trying a simpler example (n=5 instead of n=1000)

-drawing a picture (kiinda what my roommate did with his worked-out example on page 2)

- fix one aspect of the problem constant (trying to figure out the state of ONE locker instead of all 1000 at once)

- and lastly, research based on a hunch (me thinking "there's gotta be a simpler way to check if a number has an odd or even number of factors, let me google this")

OH! I guess it's not the point of this post, but my answer is that a locker n is closed iff n is a perfect square.

ALSO! I was writing the euler totient function phi(t) sometimes, because I thought there must be a prime number connection, but nothing really came of that thought.

Math art individual writeup

Although art often uses math in interesting ways, I haven't done an art project relating to math before. It was an interesting experience as a student, and as a teacher, I may want to do something like it in the future. 

At first, it was difficult to understand the original artwork, especially with the opaque shapes involved. Chris helped a lot in piecing together the shape. I suggested making the shape from a hard metal wire but we ended up using pipe cleaners. It was satisfying creating more of the building blocks for the shape with the class, even if we ran out of time to talk a bit more about the implications of the shape, and the history of "non-euclidean" polyhedra like this one. 

I think the artwork my group chose is perhaps not as directly relevant to the BC curriculum as some other groups' projects, but the assignment as a whole is relevant to my teaching practice. There may be a balance needed between letting the students pick their art pieces freely, and making sure the project still teaches them something that's in the curriculum. 

Another point to consider is varying degrees of access to art supplies, techniques, and extensive time and patience (like with origami). Perhaps suggesting some simpler, cheaper art supplies, and maybe providing some could mitigate this issue. Overall, I think high school students could make many fascinating math art pieces! 

Math art group writeup

Group: Chris, Jason, Maxine

Artwork: 7 triangles meet at a point

The original artwork piece was made out of black MDF wood, but we decided to make it out of pipe cleaners instead since it was cheaper and more accessible to us. At first, it was difficult to know how to put together the polyhedron, but the structure made more sense once we learned that the shape was made by connecting square antiprisms together. It took some experimentation, but we learned that each square antiprism could be made by connecting 4 parallelograms together where each parallelogram was made using a single pipe cleaner. Working with pipe cleaners was easy in that the shape was very malleable and we could make adjustments where needed, but also fragile and prone to getting squished for the same reasons, so we had to be gentle with it.

The main piece that we recreated was mostly 1:1 in terms of its geometry, but we also colour-coded each antiprism in the object to highlight the underlying structure. The piece being a wireframe instead of filled in also makes it a bit easier to see how the antiprisms are connected. In addition to the main piece, we also made a supplementary tetrahedron, octahedron, and icosahedron to provide motivation for why having 7 triangles around a vertex is interesting.

One aspect of the shape that felt slightly unsatisfying was that it included both triangular and square faces. We wondered if it was possible to make a similar shape that also featured vertices with 7 triangles around them which had only triangular faces. We realized this was possible by connecting square-based pyramids to the exposed square faces. So, as an extension to the project, we also made a simplified version of the shape (with just 3 square antiprisms connected together) that still contained 7-triangle vertices but which featured exclusively triangular faces (by adding square-based pyramids.) We don’t have a photo of this object, but it’s currently sitting in the presentation room beside the main piece.

For our interactive activity, it felt natural to focus on the square antiprisms since those were the building blocks of the original art piece and our recreation. Each antiprism needs four parallelograms, so we thought it would be best to assign groups of three or four so that each person can at least make a parallelogram and get the experience of building up the shape from the smallest unit. After the groups put their parallelograms together, we used their antiprisms to make a common structure seen in both works: three antiprisms joined together such that their adjacent edges form a triangular prism. We were inspired by a previous group (Doreen, Minami, Yikang: Neel Shrestha's artwork) that had each person make a small part of a larger origami artwork, since we enjoyed the collaborative aspect of their activity.

Eisner Three Curriculum

 Three stops:

My first stop is near the end of page three. Eisner mentions Vallance's research on the covert messages of texts, and Eisner notes that some argue that [these messages] are profoundly more powerful and longer lasting than what is intentionally taught. With this I absolutely agree. This is the center of the apparent paradox that schools in BC, in 2025, teach explicitly that students should not be racist, should respect gay and trans people, should not abuse disabled people, but anyone who has spent much time in a school and paid attention knows that the social reality of schools are intensely racist, intensely homophobic, and intensely ableist. This is because of implicit messaging. The teacher may tell students not to say gay as a slur, but it is obvious to the students that the teacher is uncomfortable with the topic, and all their cool peers do it, so the message is clear. 

My next stop is near the end of page 4. The phrase "the school seeks to modify the child's behavior to comply with goals that the child had no hand in formulating and that might not have any intrinsic meaning." This really resonates with my experience in school, most of the time! I hope to change this with whatever agency I have.

My third stop is near the end of page 6, where Eisner mentions the Calvinist tradition of associating failure with sin and success with goodness. I'm always very interested in the ways that Christianity influences majority culture in North America, particularly the "radical" protestant traditions that the pilgrims brought, as well as the imbrication and negotiation vis-a-vis Catholicism. I always appreciate when people point out that "common-sense" things are actually Christian cultural values in disguise. 

The BC curriculum is a very complex text, as are most texts that are designed by multiple authors with conflicting views and aims. That said, I think Eisner's article has helped clarify for me the (in my view increasing) tension between implicit and explicit curriculum in BC schools. The implicit values schools teach, in my view, have not changed much. Be complacent, fit in, don't stand out, be competitive, succeed and don't fail. The explicit curriculum has increasingly incorporated concepts about social justice as well as nonstandard pedagogical approaches and assessment approaches. But despite this, many students still need to take the SAT. How to work with this tension and use my limited agency in a meaningful way? This will be a question I will ask myself again and again. 

Favourite and Least Favourite Math teachers

 There are a few characters in this story, they go by the pseudonyms Mr. S, Ms. L, Mr. T, and my dad. 

My least favourite teacher was an old man I had for pre-calculus 10. He had the most monotone voice of anyone I'd met. He didn't seem inspired about the material in the slightest. His class was one of the most boring I ever took (up there with Socials 11). Even when we did stuff beyond rote questions, like a little project, he somehow still made it boring!

My dad is unfortunately one of my least favourite math teachers. He's not that awful, but he is pretty bad at explaining things, and he tends to get upset if I'm not swayed by his explanation. It usually ends up being me doing emotional labour for him while he tries to remember stuff from when he was in high school. Overall, he is overconfident in his knowledge. 

Mr. T was not always great at explaining things, but he had a method for grading tests that really resonated with me. If you failed a test, he'd make you re-take it at the end of the year, and you would need to re-take the tests as many times as it took to get a passing (50%) grade. There were so many of my classmates who told me how much they appreciated this policy. I talked about this policy in my admissions essay to the program, because I think it's so valuable that students experience failure without making it the end of the world. 

My favourite math teacher was also for pre-calculus 10 (I had two teachers that year). Her name was Ms. L, and she was very kind, and she took her job seriously, but not too seriously -- she let us watch movies sometime, and I was introduced to Howl's Moving Castle, one of my favourites. I'm not even sure if she was that effective at explaining the math, but her attitude was really great and I actually enjoyed math class with her. 

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