Response to The Giant Soup Can of Hornby Island

I tried to solve this puzzle by using only what I could see in the photo. The bike became my measuring tool since a medium hybrid bike usually has a wheel diameter of about 0.70 m and a handlebar height around 1.05 m. When I compared the bike to the tank, the diameter looked like it covered about 5 bike wheels, so I estimated the diameter to be around 3.5 m . Campbell's cans usually have a height that is about one and a half times their diameter, so I estimated the tank's height to be close to 5.25 m.

Once I treated the tank as a cylinder, the volume followed naturally. I used V= 𝝅 r^2 h 

with r ≈ 1.75 m and h ≈ 5.25 m. This gave me V ≈ 𝝅 × (1.75)^2 × 5.25 ≈ 5.2  × 10^1 m^3. That is roughly 52000 litres, or about 13700 gallons. This amount of water would still help with the early stages of a house fire.

What I liked about this puzzle was how much I could figure out from a single picture. It reminded me that a simple detail like a bicycle can give you enough structure to build a reasonable estimate without relying on outside numbers. If I extended this for students, I would ask them to compare two different reference points from the bike to see how their estimates change, or to calculate how much paint is needed to cover the red and white sections of the tank. 

Arbitrary and Necessary

Hewitt's idea of arbitrary and necessary made me rethink how I usually plan math lessons and even whole units. I realized I often try to "explain everything well," but I do not always separate what students actually need to remember from what they should be able to figure out on their own. This distinction changes the way I look at my planning. If something is arbitrary, like notation or vocabulary, then I can simply introduce it clearly and move on. But if something is necessary, then I need to create moments where students can actually see why it must be true. That means I cannot jump straight to formulas or polished steps just because it feels efficient. I need to slow down at the right places and design experiences that let the idea appear naturally.

This also reshapes how I plan a unit. Instead of listing pages to cover or examples to finish, I start thinking about the "awareness moments" I want students to have. For example, in geometry, I want students to notice angle relationships by cutting and rearranging shapes. In algebra, I want them to understand why certain operations preserve equality instead of memorizing slogans like "two negatives make a positive." During my short practicum, I often heard my SA say that he really did not like students using the phrase cross multiplication. What exactly is cross multiplication? We are not doing it because we need to memorize a rule. We do it because we are multiplying or dividing both sides of the equation by the same number. These moments require time and space, so my pacing needs to be flexible rather than packed with every exercise. I also find myself paying more attention to which parts of a lesson might overload students with things they can't reason out yet. If I ask them to memorize too many arbitrary details before giving them something meaningful to reason through, they lose motivation.

Overall, Hewitt's framework pushes me to design lessons where understanding is not handed to students but discovered step by step. It reminds me that math is not just a collection of rules but something that makes sense when students have the chance to look for structure. As I keep planning future lessons, I want to build a classroom where students remember the arbitrary because they have to, but they understand the necessary because they saw it happen.

Microteaching Lesson Plan

                                                              

Response to Battleground Schools

I found it interesting that debates about how math should be taught in North America have been repeating for over a century. The constant shift between focusing on basic skills and emphasizing understanding made me think that education reform is often more about ideology than actual learning. It feels like schools keep changing direction without really solving the core problem of how to make math meaningful for students.

I was also struck by how the New Math movement, which aimed to modernize math education, ended up failing because teachers and parents weren't ready for it. It reminded me that even a well-intentioned reform can't work if people don't fully understand or believe in it. Real change in education probably depends more on teacher support and communication than on new theories or curricula.

Lastly, I noticed how politics and media shaped public opinion during the Math Wars. What surprised me was that something as neutral as math became tied to ideas about authority, competition, and national identity. It made me realize that discussions about curriculum are never just about knowledge, they also reflect the values and power structures of the time.

Post Microteaching Reflection

To be honest, I started facing challenges with this microteaching from the very beginning when I was choosing the topic. I am actually more comfortable with things like table tennis or photography, but since those are not really classroom activities, I decided to choose magic instead. That choice brought two main difficulties. One was figuring out what kind of activities to do after teaching the trick. The other was how to tell if everyone had really learned it, because once the secret is revealed, any kind of test no longer makes sense.

Another question I kept thinking about was what it actually means to teach magic. Is it just about showing techniques, or is there something more meaningful behind it? I once read a book that explained the difference between teaching magic and simply exposing secrets. What makes a magic performance memorable is the mystery of not knowing how it works, and the most enjoyable part is the process of trying to figure it out. When someone tells you the answer right away, the magic suddenly loses its charm. That is the hardest part of teaching magic for me. I did not want to focus on tricks that rely on hand skills, but rather on the methods of deception in magic, such as how magicians lead people to make certain choices or shift their attention. However, I think I did not manage the timing and emphasis very well. Some people mentioned in their feedback that it was hard to follow the goal of the lesson, and I understand why. I wanted everyone to experience the idea first instead of being told directly at the beginning, but I probably did not highlight those key points clearly enough during the lesson.

Aside from that, most parts of my lesson went as I expected. I think this experience was very valuable because it helped me clearly see the gap between lesson planning and actual teaching. I really enjoyed the whole process, and I am very thankful to my group members. They gave me a lot of support and feedback while I was teaching, which made me feel encouraged and confident.

Response to Lockhat's Lament

After reading Lockhart's Lament, I really agree with his point that math should be about curiosity and creativity instead of just following rules. I like how he describes mathematics as a kind of art, and that part really stood out to me. I also don't agree with the idea that students should only memorize formulas or follow the exact steps shown in practice exams. When students only plug numbers into formulas, they lose the chance to actually think and see patterns. I think giving students space to explore and even make mistakes is what makes math meaningful.

But I don't fully agree with how strongly Lockhart dismisses the practical side of math. I understand his frustration with "usefulness" being overemphasized, but I still think connecting math to real-life examples or applications is beneficial for students. In my own experience, my professors and classmates often say that math is the foundation of all sciences. I actually developed my mathematical thinking by exploring how math connects to the fields I am interested in. I think that a physics student, a statistics student, and a computer science student would each look at the same math problem differently, and that to me is also part of the beauty of mathematics.

This also reminded me of Skemp's idea of relational and instrumental understanding. Lockhart clearly values relational understanding, knowing why something works instead of just how to do it. Both authors care about genuine understanding, but Skemp seems more balanced. He admits that instrumental understanding can sometimes help students build confidence or make progress, while Lockhart mostly rejects it. From my perspective, I find myself agreeing a bit more with Skemp's view.

Micro-teaching Lesson Plan