Course reflection

The most memorable and meaningful aspects to me is that the class emphasized on understanding mathematical concepts deeply, using visual aids, and the importance of personalized learning experiences in teaching. I would use these ideas in my future teaching for sure. And for my future study, I would like to explore more about how to integrate technological tools in teaching. 

Lesson Plan

Here is my unit plan for Math 9 Linear relations: 

https://docs.google.com/document/d/1qE93c8g2VYmnWg7J6HTZTa78JXma_Xfu/edit?usp=sharing&ouid=110954195094836418945&rtpof=true&sd=true

Here is my supportive activities/worksheets materials: 

https://drive.google.com/file/d/1mklAcUlWFysfoTtMrNNz3FSgI1mZYD0h/view?usp=sharing

I only included the materials that needed for the three detailed lesson plans. I did not include much more about the other activities. I think the link I attached will be enough for the lessons plans I wrote. 

textbooks and how they position their readers

As a teacher, I'm sure I'd appreciate the article's in-depth analysis. It emphasizes the significance of comprehending how textbooks can position students in relation to mathematics, their peers, teachers, and their surroundings. This understanding can help teachers choose and use textbooks in ways that are consistent with their teaching philosophy and the needs of their students. The framework presented in the article encourages teachers to evaluate textbooks critically, not only for their mathematical content, but also for the language and ideologies they present.

As a former student, the article may elicit memories of my own interactions with mathematics textbooks. It led to an understanding of how textbooks influenced my perception of mathematics, its relevance in the real world, and the roles of students and teachers in the learning process. The focus of the article on textbook language and structure may also shed light on why certain aspects of mathematics felt more or less accessible or relevant during education.

Textbooks are used for a variety of reasons, including providing a structured curriculum and ensuring that essential topics are covered. However, the article's critique suggests that textbooks may not always account for students' diverse experiences and backgrounds, potentially leading to a disconnect between students' experiences and the content presented.

The call for more critical engagement with math textbooks reflects the changing role of these resources. There is a focus on the need for textbooks that recognize students' uniqueness, meaningfully connect mathematics to real-life contexts, and promote a more interactive and student-centered approach to learning. This viewpoint challenges the traditional view of textbooks as the primary source of knowledge, arguing that they should play a more facilitative role in the larger educational context.

Dave Hewitt & mathematical awareness

Use of Visual Aids: Hewitt's use of visual aids to represent fractions was striking. It made me stop and think about the importance of visual learning in mathematics. Visual representations can make abstract concepts more concrete, which leads to deeper understanding.

Emphasis on understanding over memorization: The focus on understanding the "why" behind fraction operations, rather than just memorizing procedures, was another point of reflection. This approach encourages critical thinking and a deeper understanding of mathematical concepts.

Simplifying complex concepts: Hewitt's ability to simplify complex fraction concepts into understandable segments was remarkable. It made me reflect on the importance of breaking down complex ideas into simpler parts for effective teaching.

Hewitt likely created these fraction problems by considering the common difficulties students have with fractions. His problems seem designed to challenge misconceptions and promote conceptual understanding of fractions. They are excellent examples of teacher-created math problems because they are rooted in real-world contexts and encourage students to think critically.

Takeaways:

From Hewitt's ideas, I take away the importance of: 

Conceptual understanding: Focusing on the "why" behind math concepts.

Visual Learning: Using visual aids to enhance understanding.

Simplifying Complex Ideas: Breaking down difficult topics into simpler, more digestible pieces.

These approaches can make math more accessible and engaging for students, fostering a deeper and more lasting understanding of the subject. 

Short blog post on 'Flow'

This state is a profound psychological experience in which an individual becomes completely immersed in an activity and experiences a deep sense of enjoyment and involvement.

Flow can indeed be experienced in mathematical contexts. When students or mathematicians engage deeply with a problem, especially one that is appropriate to their skill level and challenging, they can enter a state of flow. This is often seen when individuals work on complex problems that require their full attention and provide a sense of accomplishment when solved.

The key is to strike a balance between challenge and skill. If a math problem is too easy, it leads to boredom; if it's too hard, it leads to anxiety. The sweet spot in the middle promotes flow.

It's possible to facilitate a state of flow in secondary math classrooms. The challenge is to tailor the learning experience to students' varying levels of ability.

Personalized learning paths, where students are given problems that are neither too easy nor too hard for their current understanding, can help. This requires careful observation and understanding of each student's abilities and progress.

We can use engaging materials: Present mathematics in ways that connect to students' interests and real-world applications. This relevance can spark curiosity and engagement.

Arbitrary and necessary

The concepts of "arbitrary" and "necessary" are central to understanding how students engage with mathematics. Hewitt discusses how certain mathematical concepts can seem arbitrary to students if they do not see the inherent necessity behind them.

For my lesson, instead of presenting mathematics as a set of rules to be memorized, I would encourage exploration and discovery. This might involve presenting students with a problem before teaching them the method, so that they can see for themselves the necessity of the mathematical concept. This is also what one of my SAs does in her class. I really like this idea. 

The language I use can make a concept arbitrary or necessary. I would be mindful of how I introduce terms and symbols, making sure they have a clear purpose and are connected to students' prior knowledge. This is also important for a class with a large number of ELL students. 

I would encourage students to discuss their understanding of why certain procedures are used in mathematics. This might help them move from seeing concepts as arbitrary to seeing them as necessary.

The Giant soup can puzzle

Your task: work on this puzzle yourself, and let your 'teacher bird' and 'student bird' notice how you approach it, where you can use reasoning and where you need to research, where you get stuck and un-stuck.


Then work on either: (a) extending this puzzle, or (b) coming up with your own puzzle for secondary math students based on a real-life observation you have made (and include a photo or graphic to support it).

The teacher bird would start by analyzing the problem statement thoroughly, identifying the knowns and unknowns. Then the teacher would rely on prior knowledge about the proportions of a Campbell's Soup can and might even know the average height of a bike. Teacher would confidently make assumptions where needed, such as assuming the bike's height in relation to the tank.The teacher bird would set up proportions and equations to relate the knowns to the unknowns. Then consider the implications of the results. Then the teacher would think about how this problem could be extended or modified for different learning objectives, like introducing concepts of scale factor, volume calculations, real-world applications and so on. 

If we assume the height of the bike (from the ground to the handlebars) is approximately 42 inches (3.5 feet), we can use this as a reference to estimate the height of the tank. By visually estimating, it seems the bike's height is roughly a quarter (or less) of the tank's height.

Using these assumptions:

Tank Height = 4 x 42 inches = 168 inches 

Given the soup can's proportions: Can Diameter : Can Height = Tank Diameter : Tank Height 

4.06 inches : 4.6 inches = Tank Diameter : 168 inches

From this proportion, we can solve for the Tank Diameter: Tank Diameter = (4.06/4.6) x 168 inches ≈ 148.3 inches 

Volume = πr^2h Where r is half the diameter and h is the height. Volume ≈ π((148.3/2)^2)(168) ≈ 923701.38π cubic inches ≈ 2901893.46 cubic inches or 12562.31 gallons of water

Conclusion:

The tank can hold approximately 12562.31 gallons of water. This would be more than enough to put out multiple house fires.

The student bird might initially be overwhelmed or confused by the problem, unsure where to start. They might need to look up the dimensions of a Campbell's Soup can or ask for clarification about certain aspects of the problem. The student bird might try different approaches before finding one that works. For instance, they might struggle with setting up the correct proportions or making reasonable assumptions. They might ask a peer, teacher, or use resources like textbooks or the internet to help them understand or solve parts of the problem.

While solving this question, they may first get stuck on what the height of the bicycle is and how to construct the equation related to the can. After asking for help, they may have the equation similar to the teacher's solution. 

Extended puzzle: 

The Hornby Island Council wants to use the space around the water tanks to create a community art project. They plan to have local schools create large, painted tiles that will be laid out in a pattern around each tank. The tiles will be square and each side will be the same length as the diameter of the bicycle wheels.

If the bike from the provided image is used as a reference and is known to be 42 inches tall, and the height of the water tank is four times the height of the bike, calculate the following: Tile Calculation: If the diameter of the bicycle wheels is 26 inches, how many tiles will be needed to create a single row of tiles that extends from the base of the tank outwards by 10 feet?