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?

ProD day blog post

I attended Supporting Students with Exceptionalities in K-12 Schools Pro-D Day Event. 

This session talked about the importance of supporting students with disabilities in the BC education system and how we should do it. It introduced the importance of terminology, with an emphasis on using "students with disabilities" instead of "disabled students". There was a brief introduction to the evolution of legislation from the 1996 BC School Act to the 2021 Access BC Act. This session also introduced some supports we can have to help students with disabilities and what we should do to help students. 

This session provided a lot of useful information. I learned about the Inclusive ABC's from Kristin Wiens. The funding scheme is not what I thought it should be. Sometimes the funding goes to another school, another class, even if the student is in your class, which surprises me a lot. The emphasis on not using dehumanizing terms and the importance of a holistic approach to students with special needs stood out to me. The Individual Education Plan (IEP) focuses on teaching students accordingly, having specific materials and assessment methods to meet individual needs. This resonated with me. I would like to follow up on the Inclusive ABCs website. This resource is a valuable tool and can help me gain a deeper understanding of inclusion. I am also interested in learning more about Universal Design for Learning (UDL) and how it can be effectively implemented in diverse classrooms. 

#2 micro-teaching reflection

We planned this group lesson beforehand and agreed on how we would introduce the content, how long each section should take, etc. However, we did not practice thoroughly before actually giving the lesson. There are still some improvements we can make for next time. We had a mistake in our practice questions. This can be fixed if we have time to practice on our own in advance. When giving the lesson, we used the method of asking students if they knew the definition of the terms we gave, rather than giving the definitions directly. This is more like a review lesson. It should be a new lesson. From the peer assessment sheet we got a suggestion from Jacky that we could reduce the number of times we say "you guys", which is good to know that we always say it unconsciously during the lesson. 

I will change my teaching methods for next time. Maybe I will say the definitions of the terms first and then let the students try to use the definition to work on the example questions. This method will work for teaching students instead of peers. They won't be confused because they all know the terms for sure. 
 

#1 micro-teaching reflection

My first mini-lesson is about how to eat chips without getting your hands dirty. This lesson should take 10 minutes. While teaching this lesson, I did not use all 10 minutes and did not plan enough information to teach the remaining few minutes. Also, based on the feedback I received from my peers, I could speak louder so that everyone could hear my instruction clearly. I would work on the lesson plan design and make sure I have enough content to cover for my future lessons and be confident while teaching to let people hear me clearly. And do more preparation and practice before actually giving the lesson. 

What is meant by "curriculum"

Two stops: 

I really like the idea about explicit curricula and implicit curricula. The article emphasizes that children spend a large portion of their childhood in school, and during this time they are exposed to both explicit and implicit curricula. The explicit curriculum includes the teaching of reading, writing, arithmetic, and history. The implicit curriculum is the set of expectations and rules that define schooling as a cultural system. These definitions are different from traditional understanding of curriculum. 

The document emphasizes that schools have consequences not only because of what they teach, but also because of what they fail to teach. Ignorance isn't a neutral void; it affects the options one can consider and the perspectives one can adopt. Thus, what students are not taught can limit their understanding, options, and perspectives, leading to a biased or simplistic view of the world.

If the BC curriculum follows traditional curriculum models and ignores alternative subjects or perspectives, it may not be fully aligned with Eisner's ideas. To ensure alignment, the curriculum would need to consider the broader implications of the explicit, implicit, and null curricula and ensure a comprehensive and diverse educational experience for students.

Revised: 

The explicit curriculum, which encompasses the direct teaching of subjects like reading, writing, arithmetic, and history, aligns well with the traditional approach I've encountered in schools. I recall a focus on these subjects, with an emphasis on standardized testing and measurable outcomes. However, Eisner's concept of the implicit curriculum, the unspoken norms, values, and expectations embedded in the schooling environment, made me reflect on the less obvious yet impactful aspects of my education.

From my observations, schools not only impart academic knowledge but also socialize students into particular ways of thinking and behaving. For instance, the implicit curriculum in my school experience involved learning respect for authority, adherence to time management, and developing a competitive spirit through grading systems. These aspects, though not part of the formal curriculum, played a significant role in shaping my approach to learning and interacting with others.

Comparing these ideas with the BC Provincial Curriculum, I realize there might be a disconnect. If the BC curriculum primarily focuses on traditional academic subjects and overlooks the implicit and null curricula, it might not fully align with Eisner's broader vision of education. For a truly comprehensive educational experience, it's crucial to consider not just the explicit content taught but also the underlying values, skills, and perspectives that are being implicitly communicated to students. 

In conclusion, Eisner's ideas have prompted me to reevaluate my understanding of curriculum and its impact. It's clear that a well-rounded education should encompass not just the explicit knowledge imparted in classrooms but also the implicit and null curricula that significantly influence a student's development.

#2 micro-teaching lesson plan

Here is the link for our group lesson plan: 

https://docs.google.com/document/d/16xvDyFZra4Q5vhyGsahxjM35cfoCSvJydHw_t9MTKH8/edit?usp=sharing

TPI reflection

 

It is surprising to see the difference between my strong "Nurturing" beliefs and intentions and the higher "Actions" score in the "Transmission" perspective. This could mean that I am leaning toward more traditional teaching methods. My Beliefs score in the Transmission section is low and this surprises me. I see myself as a content-based teacher because I believe that the content of every subject is important and fundamental for building future deeper study. 

I am wondering that how might the scores change as I gain more experience during my teaching career? And how would I balance transmission and nurturing perspectives in my teaching approach? The TPI test lets me know which perspective I need to balance or pay more attention to in my teaching career. 

Microteaching lesson plan

 Updated lesson plan: 

Battleground Schools "stops"

1. This reading identifies several negative perspectives that students and even adults may have about mathematics subjects. This is quite common, even for now. Educators should think of some specific ideas to deal with these fixed thoughts and find some more engaging ways to let students enjoy mathematics. 

2. The core of mathematics remains the same despite the various movements and "wars" in mathematics education. In the future, it will be important to adapt to the needs of the times and to draw on the best aspects of both progressive and traditional views. 

3. The article discusses the argument between progressive and traditionalist views. While it's important to prepare students for real-world problem solving and critical thinking (progressive approaches), foundational skills (traditionalist views) are also crucial. 

Update: 

1. Math has been taught in a way that emphasizes memorization and procedure over understanding. This can lead to a lack of confidence and a fear of making mistakes. In addition, social norms and stereotypes that only certain people are "math people" can contribute to these negative views. Many countries and people face similar challenges in mathematics. This is universal. The traditional way of teaching math can lead to anxiety and fear in students because it emphasizes rote learning and does not focus on different learning styles. Also, from the video in EDCP442, there was a person who talked about her fear of math and her father was a math teacher. That kind of family pressure can also lead to fear of math. Some people may want to avoid showing their weak understanding of math by stopping learning math. I believe her father did not want her to feel stressed while learning math. 

2. I would start with foundational skills using traditional methods, to ensure that students have a solid understanding of basic concepts. Then, progressive methods can be introduced, where students apply these concepts to real-world problems, engage in group discussions, and explore topics through doing activities. 

3. I would start classes with traditional methods to introduce and practice new concepts. Once students are comfortable, shift to progressive techniques. For example, after traditionally teaching the Pythagorean Theorem, students can be grouped to measure some right-triangle objects in the classroom and verify the theorem. This combination ensures that students have foundational knowledge while developing critical thinking and problem-solving skills.

Microteaching topic

 I am going to teach: How to eat chips without getting your hands dirty?

Penrose Tiling project

 

What we have done for this project is a Penrose Tiling artwork that shows periodic and aperiodic patterns. It has stars, pentagons, rhombuses and several different shapes in different colors and makes a wonderful piece of art.

The challenge for me is to understand the non-repetitive nature of the tiles and their arrangement. We have to make sure that they are perfectly connected and that the vertices are touching. But it is satisfying because you can draw different shapes and cut them to prepare for class activities and let students try on their own to feel the magic behind tiling. And we found out that there are several different ways to complete the tiling with the rules. There are an almost infinite number of solutions. 

Now for the teaching purpose: I would like to use Penrose tiling as a project to introduce students to the connection between art and mathematics and try to explore some concepts such as geometry and symmetry. However, the challenge part is that it will be hard to make sure that the students grasp the principles we emphasize and don't get lost just in the artwork.

The Dishes Problem

1. One way to solve this problem is to make a reasonable guess: We can guess a number that is the common multiple of 2, 3, and 4. By testing, we will be able to find the correct answer to this problem.

2. Yes. This method can somehow broaden students' perspective and understand the universal use of mathematics. Also, it can promote a great sense of inclusion and engagement by practicing some of the problems from their country or the countries they are interested in.

3. Yes. Students may find the problems easier to understand if they read a story or see the pictures of the word problem. And problems with related pictures can get students engaged in the content.

Lockhart’s Lament

I agree with the emphasis on the importance of real understanding: Just as Lockhart emphasizes the beauty and art of true mathematical understanding. Many educators emphasize the need for deep understanding in every subject. Teaching for memorization or simply to pass exams misses the essence of true learning.  

Both articles emphasize the importance of real understanding. While Lockhart focuses on the art and beauty of mathematics, Skemp discusses the difference between instrumental and relational understanding.

Edited: Lockhart emphasizes that mathematics is an art form and should be treated as such. He argues that true understanding in mathematics comes not from rote memorization but from engaging with and enjoying the subject. This aligns with my previous agreement on the importance of real understanding in education. From my personal experience, I never understood the meaning behind Pythagorean Theorem when I was in high school. What my math teacher told me is to memorize the formula and learn how to apply it for questions. There is no engagement in this content. And I really appreciate that now teachers are more focus on teaching the meaning behind the math contents. 

While Lockhart is highly critical of the way mathematics is taught in schools, seeing it as destructive and overly rigid, I would like to say that some structure and guidance are necessary for learning. Introducing foundational concepts in a structured way can provide a framework upon which students can build their understanding. I do agree that creativity is crucial but in my opinion, it can be fostered within a structured environment.

Letters from future students

Love: 

I hope this finds you well. Although it has been several years since I took your math class, I still wanted to contact you to express my gratitude for playing such an important role in my academic career. I never realized that I had a passion for math like you did. You are the reason I decided to study math in college. I was able to develop a deepening love for numbers because of the difficulties you presented and the support you provided. Thank you!

Not love: 

I was in your math class several years ago. I would like to share some thoughts with you.  I struggled a lot in your class. Math has never been my strong suit, and I often felt overwhelmed. I wished there was more understanding of the pace at which I grasped concepts. I hope that future students will find a better balance in your class, making math more approachable for all.

Reflection: 

I hope to inspire a love of mathematics in my students and lead them to a successful future. But I'm worried about how to balance giving each type of student my full attention. Some students may feel like they are being left behind. It's important to keep in mind that every student is different, and one single strategy may not always work. I may continue to develop myself by being mindful of these possible outcomes. I should be more adaptable and look for additional tools or strategies to meet the needs of different learners. This awareness makes sure that I keep trying to be a kind and successful teacher who is constantly improving to best serve the students.