Final Reflection

This course helped me shape my teaching philosophy in the math curriculum. The math art project helped me understand the strong correlation between math and art. I did not know that art exists in mathematics to this extent; seeing the beauty of this relationship made me want to implement art in my lesson plans. I believe that teaching mathematics through art could be a fun and engaging activity that will even help students be less fearful of the subject. Further, microteaching and group curricular lesson plans were very practical assignments. They helped me understand what I needed to be aware of and what I needed to work on as a teacher. Learning about the BC curriculum also helped me understand what I needed to focus on for my lesson plans. The hidden curriculum was eye-opening for me; I learned the importance of focusing on a student's development as an individual. I also felt privileged working in BC as it focuses on student's skills and helps them in personal, social, educational, and workplace contexts. The difference between arbitrary and necessary is now something I value when planning my lesson plans. I want my students to develop their awareness, creativity, and logic in mathematics and problem-solving. Finally, the unit plan project was a practical and resourceful guide for my extended practicum. As I was creating the unit plan, I focused on my experiences from my short practicum from what was effective. It was also a helpful experience because I got to think about diversifying my lesson plans, which was something I needed to work on after my short practicum. 

Unit Plan

Unit Plan 

Textbooks

As a teacher, I learned that I need to be aware of the textbooks I use in class and their linguistic choices. For example, I was not able to realize the difference between exclusive and inclusive imperatives before reading this article. It was interesting to think that these linguistic choices will not only help students reflect on their independent work, but think about their mathematics in the community. This will positively affect their involvement in class and enhance their mathematical thinking. Further, I learned that I need to be aware of the language I use when I give out my own instructions as well. As a student, I believe I will find myself reflecting on my thinking if I am given instructions such as "describe the pattern" or "suppose you use" instead of "make a table" or "graph your data." The verbs such as make and graph are very straightforward and require no thinking. However, describe and suppose are verbs that allow me to expand my thinking process. 

I personally do like textbooks as I believe they offer a guideline for teachers. However, I don't think I will be relying on them too much. Many textbooks that I encountered contain a lot of exclusive imperatives. Further, I think I will be able to allow students to reflect on their mathematics in the community if I give them personalized instructions that contain inclusive imperatives. Finally, I also need to be aware of what textbook I choose to use in class. I would prefer to use textbooks that have a separate section for problem-solving and applications (ex. MathQuest - I examined this textbook last class) as they allow students to make connections and understand relationships between math and the world.

Flow

There were many times when I experienced a state of flow. The model of flow was very relatable and made sense to me right away. When I encountered math problems, I felt all the different emotions that were listed in the model, so I knew what it felt like to be in a state of flow. This was possible when my challenges and abilities were balanced out. It always felt good to be in a state of flow as I was completely involved in what I was doing and I was able to experience a meaningful study session. Hence, I believe that it is possible to achieve a state of flow in secondary math classes. Teachers should implement challenging but not too challenging as well as fun activities. Deciding the difficulty level is challenging because everyone has different strengths and weaknesses. Hence, teachers should also prepare extending or easier questions just in case. Teachers should also know who their learners are in order to implement an appropriate level of difficulty. Lastly, teachers must make the assignments/activities engaging. They must be engaging enough to grab students' attention. Thus, when students are interested and are challenged enough, they will be able to get into the state of flow. 

Dave Hewitt

In the first video, I liked Dave's idea of using the entire classroom as a learning tool. His introduction to algebra and number line lesson was also taught by allowing students to speak in unison. This gave every student an opportunity to reflect on their learning. This made me stop and reflect on my role as a teacher candidate. However, as I was watching the video, I felt like the lesson was dragged on for too long. For example, I would implement more activities and foster group collaboration after a quick interactive lesson. Otherwise, if my lessons are too long, I believe my students will lose focus and interest. Another stop that was interesting was the concept of powers of the mind that was mentioned in the second video. He mentioned that there are different powers of the mind; one of them being memory. As teachers, we need to make sure that memory isn't the only skill that is being taught to the students. This is so common, especially in a mathematics classroom. It was also interesting to learn that we can learn a lot from little. His example of fractions made sense. We need a common name to add fractions and this can be easily explained with the concept of addition. I was able to learn that educating awareness is necessary instead of making students memorize information. I think Hewitt created the fraction problems to support awareness. This problem helped me understand how fractions work and that fractions can be equivalent even when they don't have the same denominator. These examples are crucial for students as they are able to build awareness in them. It will allow them to apply their knowledge instead of using their memory to solve it. Hence, I will need to be definitely thoughtful of the examples I give to my students in my teaching. 

I was able to find fractions between 5/7 and 3/4. Here is the work:

Arbitrary and Necessary

I was always aware of the difference between arbitrary and necessary realms in mathematics. As a student, I never understood why my teacher made me memorize formulas in a math classroom. It did not enhance my learning journey and made me spend less time practicing my logical and critical skills when solving a problem. As a math teacher, I need to make sure I am helping my students to practice their own awareness as this is a crucial skill in mathematics and problem solving. The article talked about a teacher informing their students that the angles of a triangle add up to 180 degrees. This example made me think about one of my lessons during my short practicum. I had to teach my students the quadratic transformations: stretch, vertical translation, horizontal translation, and reflection. However, my SA and I agreed that we did not want to tell our students what a, p, and q does in y=a(x-p)^2+q. Instead, we created an activity where students were able to investigate and play with Desmos, numbers, and tables of values to observe what each variable represents. Hence, this will drive students to use their awareness to come to their own conclusions. This also can be easily done through whiteboards. The day after I taught my students each of the transformations, I made them work in pairs and use the whiteboards to draw the graph of y=a(x-p)^2+q. The process of playing and struggling first before I gave them the lesson enhanced my student's understanding of this specific chapter. When I become a math teacher, I want to implement activities that will allow students to use their intuitive thinking to play and struggle before I give them my lesson. I believe that giving a lesson at the start of a class will allow students to practice developing their awareness instead of memorizing information.

Unit Plan Assignment

 I will be teaching Pre-Calculus 12 Trigonometry for my long practicum

The Giant Soup Can of Hornby Island

We need 1291 liters of water to control a fire, hence there is enough water to put out an average house fire because the water tank can hold around 17,861 L of water. 

This problem allows students to use proportions, estimating, and reasoning. Students may get stuck when they need to find the dimensions of the actual Campbell Soup can and the height of the bike in the photo. They also might get stuck figuring out the diameter of the tank from the height of the bike just by looking at the picture. However, this is a good problem as it allows them to connect to the real world. The problem is realistic and practical. Being able to estimate is a crucial skill because we use it all the time in our daily lives when we go to the grocery store. Further, allowing students to find the information they need to solve word problems is also an important skill that is needed when they become adults. 

One way I would extend the problem is by allowing students to look at the length of the bike in the photo instead of the height. In this case, they would need to figure out how high the tank is when given the length of the bike. This will help them realize that they only need one dimension of the bike to solve the problem when they are given the dimensions of a normal-sized soup can. 

Update November 5, 2023

Here is another extension problem. About one person can fit in between two hedges. What would be the total area of the six hedges at the front not including the hedges at the back. This will allow students to estimate the width and length of the hedges using their knowledge of human measurements. Further, they will need to consider the gap between the hedges when calculating the area.

Update November 7, 2023

This photo was taken at Jeju Island in South Korea when I traveled this summer. This place is called Osulloc Tea Musem and it is a place that shares the history and culture of Korean tea. Osulloc cultivated a farm to create an organic tea farm in Korea. However, it is considered one of the greatest tourist attractions because it was nearly impossible to grow crops on this land due to the rocks. Since it is challenging to see the hedges at the back, only consider the six hedges at the front of the photo and find the total area!

Province Wide Pro D Day

On Friday, I will be attending the BCAMT Conference virtually!

Added Friday, October 20, 2023

I attended all the sessions at the conference online; the first session by Sean Chorney, "Going to Math Class vs. Belonging to Math Class: Elevating Classroom Communities" by Timothy Spray, "Quizzes Where Wrong Answers Don't Count Against You" by Patrick Nelson, "Culturally Responsive Teachingthrough a Thinking Classroom Approach" by Aleda Klassen and the AGM. 

The first session focused on allowing students to be active participants in communities. He highlights that students need to be wanting to be a part of math class. I really enjoyed this presentation because he offered practical tips to implement on the first day of class. He provided a website (flippity.net) and emphasized that randomizing seating arrangements is beneficial and will help students work with their classmates.  Even though some students may be not comfortable in the beginning, they need to practice working through the awkwardness of all relationships even for their future. He created an environment where all students knew each other names and took risks when learning. He also emphasized the benefits of using whiteboards and allowing students to work in pairs. I enjoyed this seminar the most because there were many practical tips that I would utilize in my own classroom. 

The second session was about assessing content and how he uses quizzes for standards-based assessment. He showed us an example of how he marks quizzes. Instead of using the proficiency scale, he either marks the question right or wrong. He also requires the students to do the quiz twice for two demonstrations. His grade book is categorized into skills that he is looking for and puts a checkmark if the student acquired the skill and leaves it blank if the student did not. I liked this method because it focuses on students' curricular competencies rather than driving them to chase for grades. I would definitely try to create a classroom environment where assessment is authentic, but I think I do like the proficiency scale better because it has clear and detailed expectations and skills that are required for the students to accomplish.

The last session was focused on the thinking classrooms framework. She emphasizes that thinking classrooms allow students to be more independent. They can develop agency and independence through language and conversation. Hence, we need to implement group work so that students can connect to their classmates. She offers practical strategies such as VRG (visibly random groups). I need to make sure that I switch groups for every task and explicitly teach and give feedback on collaboration so that there is an increase in knowledge mobility and a decrease in reliance on the teacher. I also learned that we need to be culturally responsive and this can be done through instructional conversation. I would love to know more about the thinking classroom as I want to foster an environment that includes critical thinking, collaborative learning, and reflection. 

Group Curricular Teaching Reflection

Our group did a good job of integrating Desmos into our lesson. It was one way to help our learners be engaged and interested in the topic. Many students did enjoy using Desmos as it taught them an important skill for future math classes. However, some feedback talked about how they wished we shared our screens on a projector for visual aid. I initially thought it would be okay just to write down the instructions on the whiteboard but I realized I really need to think about my learner's perspective. Further, some feedback talked about our group's need to work on voice projection and body language. Using whiteboards and teaching the class is hard to do simultaneously, so our group was speaking to the board many times. This is something I need to be aware of in the future. Also, our group didn't have a proper closure. I need to be flexible as a teacher because I realized our lesson did not exactly go the way we planned it. I really learned the importance of lesson preparation and walking through the lesson in my head beforehand. Further, co-teaching was also very challenging to do. I think this is because we needed to spend more time collaborating and working on our lesson plan as a group. However, trying to combine all three of our ideas into one lesson plan was difficult. I also found that allowing students to use whiteboards would've been beneficial for group work instead of giving them handouts. Even if there were no vertical surfaces in my future classroom, I would buy portable ones so that students can freely write down their thought processes and answers. Ultimately, it was a great experience to co-teach and I found many aspects that I could work on as a teacher candidate.

What is meant by curriculum?

The author mentions that what schools teach is largely unintentional. I only carried negative perspectives on the hidden curriculum. I first thought about whether the hidden curriculum was also unintended by teachers. Teachers tend to teach based on their experiences as a student, thus their learning experience from society is a natural process. Hence, teachers tend to teach implicitly with values that are closely aligned with societal norms. If our values are aligned with societal norms, then there will be exclusion and a lack of diversity in the classroom because we are not considering our diverse learners. Nevertheless, as I read through the article, it mentioned another aspect of the implicit curriculum that I had not thought of; the implicit curriculum does not entirely have a negative impact. The reasons were that the school "can teach a host of intellectual and social virtues: punctuality, a willingness to work hard on tasks that are not immediately enjoyable, and the ability to defer immediate gratification" (Eisner). Ultimately, we need to be aware of the hidden aspects of education because it has a significant impact on students' development including academically, socially, and emotionally. It was interesting to learn that "what they teach may be among the most important lessons a child learns" (Eisner). Hence, educators must intentionally create a positive school environment because what we teach should focus on the student's development as an individual.

The BC Provincial Curriculum focuses on three big competencies including communication, thinking, and personal and social. I believe that the three competencies focus on students as an individual and provides them the need to thrive as members of their community and society. Likewise, Eisner highlights that the explicit curriculum is not all for students' learning. The BC curriculum allows students to acquire certain skills that will help them even in personal, social, educational, and workplace contexts. Hence, this is closely aligned with Eisner's ideas of the implicit curriculum. 

Group Curricular Microteaching

This is our group curricular lesson plan with Asiya and Chernie! 

Group Lesson Plan Link 

Link to Worksheet 

Microteaching Reflection

Overall, my microteaching session went well. My group members seemed to be engaged and interested in my lesson. My peer evaluations had a similar comment as well. However, as I was going through my lesson, the time went faster than I expected. Hence, I did not go by schedule at all. I don't know if this was because I was rushing thinking that 10 minutes was too short, or if it was due to my lack of experience. Thus, 10 minutes was longer than I thought and I now have a good grasp of how much I can teach in that time frame.  Two of my peer evaluations also complimented my slides. The slides consisted of playable combinations with pictures and had quick check-up questions at the end to enhance their understanding. I also think this was a good strategy because the card combinations were confusing to understand at first. But, as I was teaching, I thought having a separate piece of paper that had the combinations or rules would've been helpful during the game. My peer evaluations had the same comment. The combinations and rules were too hard to memorize, so they had to ask questions and write out the rules themselves. If I had more time, I think it would've been fun to give them a quiz on the basic rules and card combinations at the end. Lastly, because I had a minute or two left at the end, I could've shared strategies or techniques that could've been useful for all the members. Even though I walked through the lesson inside my head before my microteaching session, there were many adjustments that had to be made. Hence, I learned that I need to be flexible during my lessons and I need to be more meticulous in the future. Ultimately, it was a great experience and I am looking forward to more of these sessions in the future!

Lesson Plan for Microteaching

Presentation for Lesson

Lesson Plan (for better view) 

*updated October 11, 2023 after microteaching*

Battleground Schools

I didn't know that mathematical education was highly conservative. But it made more sense as the article talked about mathematicians who thrived in traditional mathematics classrooms. I personally did well in a traditional math classroom, so I wasn't able to think about other students who may not have enjoyed it. Memorizing formulas drives students to lose motivation and interest ultimately giving them math phobia. I also learned that the United States was worried about a lack of college students who could become mathematicians or scientists during the period of the Cold War. Even in history, the teaching of mathematics was an issue and concerned many people as it was not relevant to their lives. Lastly, the NCTM Standards and the new California Mathematics Framework emphasized skills that are needed in today's curriculum; these included problem-solving skills, understanding mathematical relationships, the use of technologies, and mathematical communication. All these skills promote relational understanding and learning in mathematics. I was able to learn that these skills should be valued and taught in the present day because it was significant even in history.

Updated October 12, 2023

I did well in a traditional math classroom because I liked teaching myself the content and I loved to solve various questions. However, after taking several education courses, I slowly started realizing the importance of diversity in assessments and assignments. I became aware that everyone learns differently. As a teacher candidate, I must remember to implement both traditional and nontraditional teaching so that everyone gets a chance to thrive in their own learning experience. Through this article, I learned that mathematics in education concerned many people as the way it was being taught was not relevant to the students' lives even in history. This is still an issue in the modern world. Hence, teachers must be flexible in their teaching so that their learners can relate and be engaged with mathematics. If we teach traditional and non-relevant mathematics to our students, then they will find no meaning in learning. We need to help students become individuals who can do good for the world and be new leaders for future generations. However, I want to be a teacher who can fully support their learning experience by being flexible and giving autonomy to my students.

The Teaching Perspectives Inventory

The highest perspective I have is transmission. My sub-score for beliefs tells me I value a substantial commitment to the content. The best way to teach is to have a strong understanding of the subject because this is the only way I can implement diverse unique assignments and assessments. Further, I can explain the content in a variety of ways only if I know the content. My social reform perspective was the lowest and I was not surprised by this result. It is difficult for me to connect mathematics to how they can do 'good' for the world. I don't know if this is because I have no experience myself or if it is because I am not aware of social reforms myself. I was surprised to see that my sub-score for actions is higher than my beliefs for developmental and nurturing perspectives. It was interesting to see that I was already trying to accomplish certain skills in my teaching without being aware of my own beliefs! Something I want to work on in the future is my developmental perspective because I believe it is crucial to always think about the learner's point of view. I can develop this skill by asking them questions and providing them practical examples. However, it is also significant to get to know who my learners are in order to provide them with the right guidance and solutions. The TPI results helped me understand where I am as a teacher candidate. Some questions I have are, "What would be the best way to enhance my perspective on social reforms? Do I need to have a stronger belief or do I need to demonstrate it as I am teaching? Where do I start?"

Individual Microteaching

 Topic: Big 2 (Card Game)

Math Art Projects

1. We decided to analyze and extend upon the artwork "Between the Devil and the Deep Blue Sea" by Gauthier Cerf, a series of hexagons embedded within each other, where their side lengths correspond to the Fibonacci numbers. Different tasks were broken down and we discussed ways in which such an artwork may be used as an active way to engage students in understanding how the Fibonacci numbers are related to each other through its recursive definition. We extended the Fibonacci pattern using the square, which generated the golden spiral (which is seen in many natural objects) and created a colorful aesthetically pleasing butterfly. Linking patterns to other mathematical ideas visually allows students to see how some formulas came to be/make more sense, such as explaining an infinite geometric series with equilateral triangles.

 2. There were many challenges when creating this project. We struggled to find a way teachers could connect Fibonacci sequences and golden ratios to the math curriculum. Although the Fibonacci sequence may be a simple sequence of the sum of two preceding numbers, there is much more students can learn about, including the golden ratio and the proofs. However, we thought proofs would be too complex at their level, so we focused on students and their ability to observe and identify patterns. Having two different shapes that represent the Fibonacci sequence allows students to identify patterns and visualize the sequence. Further, the golden ratio exists in many of our surrounding artworks and nature. So we thought it would be a good activity for students to understand the Fibonacci sequence and connect it to their lives. Finally, I personally got to develop a better understanding of the topic and it helped me understand the importance of implementing math art projects in a classroom. Not only did I learn something new, but I had fun and was engaged!

 3. As our group was presenting, we realized it would be challenging for students to create the hexagon artwork as it took up so much time and lots of explanation was needed. Further, we thought it was important for our classmates to play with the hexagons because it could be difficult to visualize the Fibonacci sequence right away. However, it would have been better if our classmates were given pre-cut hexagons to play with and identify the pattern themselves. Further, providing them with the three triangles would've helped them realize that the diagonals of the hexagon are twice the side length. I realized that I need to be flexible in the future and be aware that my lesson plan could not go the way I intended. Further, I need to be aware of seating arrangements as cutting and tracing paper in a group did not go well in the way my classmates were seated. Nevertheless, I would keep the golden ratio artwork because it extends the idea of the Fibonacci sequence in a different shape. I could implement this activity/assignment in many ways. I can allow students to go outside in the garden to see if they can find the golden ratio themselves and allow them to create an artwork. Finally, I can even connect this topic to other subjects such as the history of art because the golden ratio exists in many famous artworks. Overall, this project helped me reflect on my teaching behavior and I was able to learn how to implement engaging projects in a math curriculum.