Battlegrounds Schools - Response

I was amazed to see how math education has never been just about numbers, it has also been connected to huge political battleground for over a century. 

One part that really stood out to me was the events that followed Sputnik in 1957. The United States attempted to instantly change the way math was taught in schools to transform children into rocket scientists. Reformers, influenced by a French math group called “Nicolas Bourbaki,” very abstract ideas like outlaws geometry diagrams and university concepts like set theory into elementary schools. These ideas were taught to young students even though teachers were not fully prepared for them. Since parents were also unable to assist with homework and children were simply perplexed. It was a complete failure. To me, it showed you cannot create a curriculum that ignores the students and teachers in favour of lofty academic ideals. 

Another thing that made me pause was learning about “teacher-proof” textbooks. Some educational leaders wanted textbooks to be written in materials so rigid that they assumed literally anyone could teach maths just by following the book. Teaching is not just reading instruction from a page. Because of teachers’ shortage, non-specialists were often assigned to lower-grade math classes because people assume basic math is just reading rules out of a textbook. Treating math like a script really degrades what teachers do and leaves kids doing repetitive drills without really understanding what they are learning. It also made me think how this can contribute to math anxiety. 

The last major idea that really caught my attention was the ongoing “Math Wars” between traditional math and progressive math. Traditional math focuses more on memorization, algorithms, speed and getting the correct answer, while progressive math focuses more on problem-solving, exploration, and understanding why something works. 

These arguments are often more than just math. Traditional math gets tied to ideas of discipline, order, and obedience, while progressive math gets tied to open inquiry and democratic thinking. This may be the one reason people are projecting their own political views and personal memories of school onto the curriculum. 

After learning about all this history, I noticed that math education keeps moving back and forth between two extremes. But as a future teacher, basic skills and deep conceptual understanding don’t need to fight each other. Knowing this history helps us to avoid getting caught up in these debates and instead focus on classrooms where students actually understand math and feel comfortable learn. 

What is Meat by curriculum? - Response

The first impression of curriculum that I had is I viewed "curriculum" simply as the prescribed syllabus—the specific topics and learning outcomes mandated by the BC Ministry of Education for high school courses. Eisner broadened this definition by showing that curriculum is a lived experience made up of three distinct dimensions: the explicit, the implicit, and the null.

My 1st stop: How Schools Teach Compliance Instead of Thinking - When Eisner described how schools unintentionally lead students to focus getting good marks over learning, I stopped to think that it is somewhat true as per my own experience. When I was in high school, every single student of my class was so focused on getting good grades that they memorized steps and formulas to just to find the correct answer. Instead of taking the time to comprehend how the maths works, they learn to follow the teacher's instructions.

My 2nd Stop: How the Bell Schedule Interrupts Learning - The constant interruption of momentum created by fixed 50- or 80-minute block scheduling at the high school level sends students a clear message: don't invest yourself too much into anything. When I was a student, I remember finally starting to understand a hard math concept right as class ended and we had to move on. It made me think about the importance of giving students enough time to explore and understand mathematical ideas.

My 3rd Stop: The Null Curriculum (what is not taught) - I agree with Eisner's view that the amount of things that schools fail to teach is just as important as the things they do teach. In my own experience in high school math classes we did a great deal of practice on worksheets, but we were never shown how formulas were created and what the use of them is in real life. 

Connecting Eisner to the BC Secondary Math Curriculum

The BC math curriculum determines the explicit curriculum by identifying concepts (i.e. algebra) and skills (i.e. Logical reasoning) that must be incorporated. The implicit curriculum will largely depend on the classroom culture. For instance, if I reward students for providing the correct answer quickly, the culture will be compliance. To counter this, I will ignore students who are first to provide the correct answer. To counter the null curriculum, I will incorporate what has been traditionally excluded from math. The First People’s Principles of Learning emphasize learning through experience. I therefore, incorporate experiences that will enable students to learn math by engaging and sharing in various ways through a sense of responsibility and accountability. I will further integrate what traditional math has excluded and engage students to help identify real-world situations.

As a future teacher, Eisner’s work reminds me that my job is not just to cover textbook topics. I need to build a classroom environment where high school students feel safe to ask questions, learn from their mistakes, and see how math connects to the real world.

Experiences with Math Teachers!!

Looking back on my math learning, the one teacher who stands out is the person who made me feel comfortable in her classroom. I recall that when she taught me something I didn’t understand the first time, she never made me feel like I was inferior or that I should already know it. She would slow down, give another illustration, or clarify the same idea in a different way until it began to make sense. So I never felt afraid to put my hand up or answer questions even when I didn’t know. In the end, I know I learned more in her class because I wasn’t afraid to be wrong.

My least favorite math teacher was almost completely opposite. He was more concerned with methods, formulas, and achieving the right answer. I remember sometimes being able to complete the homework because I had memorized the steps, but if the question was presented differently, I would get stuck on it. I also did not feel as comfortable asking him to explain something again. Over time, I stopped thinking about the arithmetic and started concentrating more on whether my response was correct or incorrect. I learned from that experience how easily a student can seem to be doing well while really not understanding the content.

I'm thinking about each of them differently as I'm getting ready to become a teacher. I want every student in my classroom to be at the ease of speaking up, asking questions and making mistakes as they are here to learn. At the same time, my experience with my least favourite teacher reminds me not to assume that a correct answer always means a student understands. I want to ask students how they got their answers, encourage different approaches, and make mistakes feel like a normal part of learning. 

Our First Math Puzzle!!!

 I first misunderstood the question and thought there were only three students taking turns. Because of that, I started with the first 15 lockers and worked out what would happen to each one. When I read the question again, I realized that there were actually 1,000 students. I decided to increase my example to 30 lockers and made an Excel sheet so I could keep track of what was happening and hopefully see a pattern.

While working through the lockers, I noticed that a student only changes a locker when their student number is a factor of that locker number. For example, locker 12 is changed by students 1, 2, 3, 4, 6, and 12. That means it changes six times and ends up open again. Locker 16 is changed by students 1, 2, 4, 8, and 16, so it changes five times and ends up closed.

This is where the pattern started to make sense to me. Most numbers have factors that come in pairs, so those lockers are changed an even number of times and end up open. Perfect squares are different because they have an odd number of factors. For example, with 16, one of the factor pairs is $4\times 4$, so 4 is only counted once.

The perfect squares go 1, 4, 9, 16, 25, ... all the way to 961. Since ${31}^2=961$ and ${32}^2=1024$, there are 31 perfect squares between 1 and 1,000. This means 31 lockers will be closed and 969 lockers will be open.

My Mathematical Thinking

For this problem, I started with a smaller example, organized my work in Excel, and looked for patterns. I also tested different lockers to see if the pattern continued. Even though I misunderstood the question at first, working through that mistake actually helped me understand the problem better. Once I saw the connection between the locker numbers and their factors, I was able to use that pattern to figure out what would happen with all 1,000 lockers.

Response on Skemp's Reading

Three things really made me stop and think while reading Skemp’s article. The first one was the difference between knowing how to do it and knowing why it works. I’ve learned that a lot of the time in math, I’ve memorized a formula or approach to get the right answer without really knowing what was going on. The second was the example where students got the area question wrong because the measurements were in different units.They knew the formula, but as soon as the question changed a little, the rule they memorized wasn’t enough. The third thing that stood out to me was Skemp’s example of knowing a few routes around a town versus actually having a mental map of the whole town. That made his argument much easier for me to understand because if you know the map, you can figure out where to go even when you haven’t taken that exact route before.

I agree with Skemp for the most part. “I think it’s better in the long run to know why something works rather than just memorizing rules. At the same time, I don’t think memorization is necessarily a bad thing. Skemp also admits that sometimes it’s best to learn a simple rule first, so you can get started and build confidence. I think the best learning method is a mix of the two: learn how to do something first, but don’t stop there, learn why it works eventually too.

 

Hello World!!!