Final Blog Post

Looking back at my blog posts this semester, I realize how much I’ve grown in my understanding of math education. The articles I was assigned to read challenged me in the best possible way and gave me the language to articulate the changes I aspire to bring into my teaching. These readings didn’t just inform me—they transformed how I think about teaching and learning mathematics.

One area where I’ve made significant progress is in understanding the purpose of assessment and what makes assessment authentic. This has reshaped how I plan my lessons and evaluate student learning. Delving into the history of the curriculum was another eye-opening experience. It deepened my understanding of the BC curriculum and gave me insight into its structure and intent.

The concept of Arbitrary and Necessary teaching in mathematics has been especially impactful. It’s something I actively consider while designing lessons, and it’s making me a more intentional and effective teacher than I imagined I could be.

I’m grateful for everything I’ve learned in this course—from the guidance of Susan and Malihe to the inspiring and supportive community of future math educators I’ve found in my classmates. This journey has been nothing short of transformative.

Unit Plan Assignment

 Here is the link to my Unit Plan Assignment:

https://docs.google.com/document/d/1anfJM17FNjYWhc1bJHoN9kI5fihK5fKf/edit?usp=sharing&ouid=100957161943925271254&rtpof=true&sd=true

Here are the links to the 3 lesson plans to accompany this unit plan:

Lesson Plan 1: Introduction to Rational Numbers

https://docs.google.com/document/d/1QvRdqM0Oy5SclInQLI7mDd2YATQjlN30O9a4U8qYN8/edit?usp=sharing

Lesson Plan 2: Converting Fractions to Decimals

https://docs.google.com/document/d/1NM9vlUKQJyQKky0Opq965Y7IHchiYT28TILWfNhdqXs/edit?usp=sharing

Lesson Plan 3: Converting Decimals to Fractions

https://docs.google.com/document/d/1g9HrMmhS1t8Gb8Q1bGdJLtAaqpZjGPr3_bxn5sILWiY/edit?usp=sharing

Nov 20 - Textbooks and How They Position Their Readers

 Math textbooks have always played a key role in education. They provide structure, practice problems, and a roadmap for learning. But I have never stopped think about the effectiveness of specific textbooks until I started teaching a few years back. I have also noticed that I read the textbooks in more details now as an educator than when I was a student. Reflecting on the why it is that students skip to the exercises on textbooks without reading the Introduction of the chapter or even the intro to the book itself is something I have thought about often. As a tutor (and an avid reader), I deliver my lessons to students, focusing on what the essence of the chapter is, wishing my students took the time to read the "extra" sections the textbook has to help them make more sense of what they are learning. 

Textbooks are one of my go-to tools for structuring lessons and ensuring my students meet learning goals. They’re reliable and standardized, which helps me stay on track. But after reading the article, I realize textbooks do more than just teach math—they influence how students relate to the subject, their peers, and even their teachers. For instance, the language in many textbooks can feel impersonal. Phrases like, “The equation shows you...” or “You must calculate...” make math seem rigid and detached from human involvement. This can unintentionally alienate students and reinforce the idea that math is about memorization rather than creativity or understanding. This can potentially be addressed by adding context to the problem because story telling is a powerful tool that helps students connect math to the real world. 

Considering my student bird, I can see how textbooks have shaped my academic career and life both positively and negatively. As a student who was naturally good at following instructions, textbooks were great for me as they provided me a clear path to follow and lots of practice problems to build my skills. That being said, textbooks did also feel dry and unrelatable at times.

These days, the role of textbook is not as rigid as it used to be. In class, we looked at some online open source textbooks developed by UBC professors and with the integration of technology in classrooms, there are more options for teachers to decide. I think this is a good thing because one textbook can only offer so many ways to approach and solve a math problem but when you have the entire world wide web at your disposal, the possibilities are endless. Teachers can modify their handouts to incorporate the different methods they want to teach depending on the collective and individual abilities of the class.

Nov 13 - Flow

 It was intriguing to delve into Mihaly Csikszentmihalyi's TED talk on 'flow' and happiness, where he describes flow as a state of complete immersion and enjoyment in what one is doing. Reflecting on this concept, I realized that I've experienced flow at various moments, particularly when I'm solving geometry problems. I view geometry as precise, yet creative in nature. I find the process of visualizing shapes, figuring out properties, and solving geometrical problems quite immersive. This same deep engagement occurs when cleaning; it’s a process where I can see immediate results and feel a sense of order and accomplishment.

However, transferring this state of flow into a secondary math classroom is a challenge, especially with 30 students who each may find flow in different activities. This is not to say that it is impossible; it's just tricky. By setting up problem-solving tasks for those who thrive on challenges, integrating creative math projects for those who think more artistically, and applying math to real-life scenarios for those who benefit from practical applications. We could also give students the choice to work alone or collaboratively.

By tailoring tasks to match individual interests and skill levels, we aim to make learning experiences both enjoyable and engaging. This approach could help each student find their flow in math class, making the learning process as captivating as their favorite activities. I'm looking forward to testing out these strategies and hopefully seeing our classroom transform into a place where everyone can discover their best way to learn.

Nov 13 - Hornby Island Puzzle

 When I first saw the problem, I immediately thought about using proportions and I was able to link it to Math 9 BC curriculum. I knew that since the water tank was in the same proportion as the soup can, scaling up from the can’s dimensions would let me estimate the tank’s dimensions. Right away, I could recognize the problem as a blend of practical scaling and real-life math applications, which made it feel engaging and approachable.

I decided to start by determining the dimensions of the soup can. A quick google search revealed that it’s about 10 cm tall with a 6.5 cm diameter. Next. I needed the dimensions of the bike. Since, I'm not very well versed in the types of bikes, I went with the average 27 inches or 68 cm diameter of the wheel of the bike. Letting go of perfection was a little challenging for mw in this step as I have always focused on accuracy in math. Once I had dimensions for both the soup can and an estimated bike height, I found the scale factor by dividing the tank height by the soup can height. Applying the scale factor to the soup can’s radius and height gave the water tank’s dimensions. At this stage, I realized that I needed to be careful with the units to avoid simple but easy-to-make mistakes, like mixing up diameter and radius. Then, to calculate the volume of the tank, I used the formula V = πr2h. The calculations were straightforward since I already had both the radius and height for the tank.

Reflecting on this puzzle from Teacher's Perspective, I was immediately excited to see this problem because I could connect to the curriculum and I could see how it has the scope to engage students. To extend this problem, surface area could be introduced. Students could work on how many buckets of paint would be needed to paint the tank.

Another puzzle like this that would engage the students would be to use a picture of themselves next to an object of interest and use their own height and their height in the image to come up with a scale factor, then using that scale factor, students can figure out the height of the object of interest. Some students have pictures with athletes and other celebrities that they could use to find their true height.

November 6 - Arbitrary and Necessary

 In the article Arbitrary and Necessary Part 1: A Way of Viewing the Mathematics Curriculum, Dave Hewitt argues that a lot of the time in a mathematics classrooms is being dedicated to teaching conventions rather than doing actual mathematics. I have always been aware of this and my approach so far has been letting my students come to their conclusions where conventions are concerned and then I tell them the agreed upon conventions by society. This makes the students realize that their method was perfectly acceptable and a lot less time is spent on memorizing the arbitrary.

One thing in the article that caught my attention was when the author says that by simply telling the students that the sum of interior angles of a triangle is equal to a half turn, we communicate to them that this is arbitrary information. They may or may not remember this in the future. I have had students struggling to remember this piece of information, quite possibly because of this same reason that Hewitt mentions. When I plan my lessons, I focus a lot on the contents covered in that specific unit that sometimes I do not pay attention to  what arbitrary memory informations students are bringing with themselves and if it even belongs in the realm or arbitrary. In my future lessons, I am determined to let students work out on their own the concepts that belong in their realm of the necessary, so as to avoid accidentally letting students simply receive information and potentially forget it right after my class.

Oct 25 - Pro D Day Reflection

 For my Professional Development Day,  I went to the BCAMT Conference in Whistler. My experience as a new teacher was very positive. I attended Peter Liljedahl's session on the importance of closing a lesson effectively. I learned that a third of your class time should be dedicated to closing. I had never thought of how much time must be dedicated to closing prior to this, and I like this idea because Peter argues that when students learn something new, their thoughts are unorganized and chaotic and with effective closing practices, we can help organize those thoughts so the students understand and retain more information.

I attended another session in the math Lounge where I learned about the importance of organizing practice questions in the categories of mild, medium and spicy. In this session, the speaker got us to list the things in the Linear Equations chapter that we want our students to know and then organize them in a way that the next topic would build on the previous one. We then looked at some qualities that make these questions mild, medium or spicy. My understanding was that mild questions are supposed to be approachable for students, medium need to be the same as mild with one more tier of difficulty and spicy is supposed to be 2 or more tiers of difficulty.

I also learned that with the four categories in the Proficiency Scale, students tend to think of the fourth category as a secret garden where only the elite students are granted entry and so calling the questions "mild, medium or spicy" will get more engagement with students and encourage them to try more!

Oh, and I also got a lot of freebies and merch! I got a few math textbooks and some stickers and other things for free!

Oct 16 - Curricular micro teaching Refelction

 I think the micro teaching this time around went really well. I was confident throughout and the group members were engaged. I wrote a script for the lesson this time and even though I ended up not needing it, I was relieved that I had it if needed it. In terms of my voice, I was trying to be mindful of the other lesson going on in the classroom at the same time as ours but perhaps it would be a good idea to check with the students if they can all hear me so I can adjust in the moment.

October 16 - Curriculum Micro Teaching Lesson Plan

 Jasmine, Sahl and I chose "Binomial Distribution" as our micro teaching topic. Here is our lesson plan:

https://docs.google.com/document/d/1plxr1qunmgXorAwWT3Sm6ixAnhiQ-C6Q/edit?usp=sharing&ouid=100957161943925271254&rtpof=true&sd=true

Here is the thinking task questions that we are planning to use:

https://docs.google.com/document/d/1Ij0x7cCchVxxvsjILGsrwKsgLLL73aI8/edit?usp=sharing&ouid=100957161943925271254&rtpof=true&sd=true

Thinking Mathematically Puzzle

 Here is my mathematical work and thoughts while solving the Cistern Filling problem:

Oct 9 - Thinking Mathematically

 I found this reading to be very insightful. I have been teaching math for 2 years now and I have seen my students being stuck on a problem at least once a week. While I believe I did my best to convey the same ideas as mentioned in this chapter about "slowing down when in full flight" and "write what you know and want," I appreciate now the well phrased reasoning as to why it is important to do these things.

However, there were a couple of words in this chapter I did not quite understand in context, like RUBRIC writing and specialization. Perhaps it is because I have not read the previous chapters that I do not understand what Mason means by RUBRIC writing, but I would love to talk to my peers about it and know their thoughts. "Specialization" did not make sense to me because the word itself means having a mastery of something and the context was being stuck. Therefore, I had to remind myself of the context while reading this chapter.

Oct 7 - Non-Curricular Micro-teaching Reflection

I found this to be a very enlightening and reassuring activity for me. I do not believe I have ever had by teaching observed, let alone evaluated. I very much appreciated the words of encouragement and feedback from my peers.

What went well: People were engaging with the content, my materials were all prepared and ready to be used and I believe I organized the lesson appropriately.

What could be improved: I found myself in the middle of the guided practice time realizing that I forgot to mention that I encourage multilingual people to contribute words from their language. I also think I should have given members an outline of what were going to be doing before beginning the activity. 

What I might do differently next time: I knew what I was going to say to introduce my lesson but I felt as though I was nervous until I got in the rhythm of teaching, so I would like to explore writing a script for my lesson that I can fall back on in such cases.

 

October 7 - Non-Curricular Micro-teaching Lesson Plan

 

 Here is my lesson plan that I will be following for our non-curricular micro-teaching activity in class on Monday!

I will be filling out the reflection section of this lesson plan after delivering the lesson.

October 2 - Battleground Schools

 

 Upon reading this article, I realized that we have been fighting for the same things in mathematics education for a century. I was surprised to read that John Dewey's movement came before the New Math movement and it made me stop and think about how Dewey's recommendations were never accepted in a wholesale fashion across North America but it did not take as much effort for people, not just in North America, but worldwide, adopted the New Math curricula in fear of being left behind. This was especially surprising to me as the article mentions that "Far too often, the jet-setting curriculum developers showed little regard for local conditions, cultures, or educational traditions".

My second stop while reading this article was when math phobia was brought up and learnt that not much effort was made to combat this in students', teachers' and parents, lives. Instead, parents and teachers continued on to pass that trauma on to the new generations in the name of advancement of science. As someone who grew up not having much understanding of the "why" of mathematics in her early years, I realized how important it is for our students to learn from teachers who are themselves passionate about mathematics and can recognize the beauty of it. This also reminds me of Lockhart's point that we must do math for the sake of simply doing math.

My third and final point of wonder and discovery was throughout this reading when I realized how much control one political party has over the education of millions of children. I do not believe it is fair for politics to have such a huge influence in education. Of course, education is and will always be political but as a future teacher, I hope that we can have more say than we do now in how we learn and teach in schools.

Math Art Project Write Up

Original Artwork By Holly Laws

Tricolorability of Trefoil Knot

 

Our group's rendition of Trefoil Knot made with clay

 

Our group - Jasmine, Caris, Krystal, and I - recreated Holly Laws's "Trefoil Knot" using clay and discussed Knot Theory as a branch of topology in our presentation. We introduced key concepts like knot invariants and tricolorability. Knot invariants remain topologically equivalent through simple transformations such as stretching, bending, twisting, or shrinking. Tricolorability, a tool for identifying these invariants, refers to the ability of a knot to be coloured with three distinct colours, where each uninterrupted chain is one colour, and each intersection has either all three or just one colour. In our model, one side demonstrated tricolorability, while the other side reflected Holly Laws’s original choice of colours and materials. Additionally, we highlighted the cultural, scientific, and everyday significance of knots, noting how knot theory models DNA replication, helping visualize and predict the complex topological structures of DNA. As part of our activity, we provided clay and strings for classmates to create their own trefoil knots, challenging them to explore transformations that preserve topological equivalence. This hands-on approach deepened our understanding and reinforced the value of incorporating interactive activities into classroom teaching to help students grasp abstract concepts.Moreover, to pique further interest, we talked about the importance of Knot Theory in understanding the behaviours of various diseases related to protein aggregation such as Alzheimers, as well as, knots in higher dimensions.

It was really fascinating to discover the major real life applications knot theory had in scientific research. It was during this process that I found out that the basis of research for understanding Alzheimers was knot theory! I will definitely be doing a lot more research on this. The frustrating part with this project for me was that there were not any applications of this that coincided with the high school BC math curriculum. 

While I found this project to be a fun activity that would interest students to further research this topic, I do not think I would be able to indulge in this as a teacher if covering the core competencies and big ideas is to be given priority. However, working on this project and seeing my classmates present their art projects, I learned that projects like this one can be done just for the sake of art and connecting a few dots with mathematics. In my future classroom, perhaps math art projects can be a little break from "harder" topics.

Here is the link to the slides: https://docs.google.com/presentation/d/1wYTgETin8K9mm2xTXP2MH_n5jskdQLL1ENTD1sR11NU/edit?usp=sharing

Sept 25 - Lockhart's Lament

 

The article "A Mathematician's Lament" by Paul Lockhart is a very powerful read in my opinion. I found myself agreeing with him that there is such beauty in mathematics much like music or art. It is unfortunate that this beauty is stripped away by the school systems and what remains is a depressing gray classroom where formulas are memorized. I like the analogy that Lockhart uses to argue that math teachers must know and be able to recognize its beauty in order to 'teach' it and share the appreciation of mathematics with their students. I resonate with Lockhart's point that students need autonomy and freedom to figure out their own solutions and proofs and I wonder how much of students' perspectives of math will change if we stopped focusing so much on the correct notation and rather give more weight to notion, as G.H. Hardy said. I think that notation is the very last think we should be teaching our kids after they figure out a math idea for and by themselves. In my experience, students are the most confused when they have to learn set notation and interval notation, for example. Perhaps taking this approach towards 'doing' mathematics in classrooms would encourage Relational Understating that Richard Skemp talks about in his article.

In this article, Paul Lockhart focuses majorly on appreciating the beauty of math and 'doing' it rather than teaching it. In my opinion, Lockhart fails to take into consideration the people that do not necessarily want to appreciate art. Building on his art class analogy, I am one of the (probably few) people that do not enjoy making art. I have given it my best and despite all my efforts, I simply do not have a knack for art and this is okay. Similarly, there are people that do not want to do math and that is completely fine. For them, Lockhart's proposed approach might be even more frustrating. Fr this reason, I believe that Skemp's Instrumental Understanding is also just as important. Hence, I do not agree with Lockhart that his 'new' artistic approach will solve the problem. It might improve the condition of math classrooms a little, but we need to consider all the implications it will have.

Sept 16 - Locker Problem

 I have come across the locker problem in the past, kind of in a funny situation - it was when all my colleagues and I from the education institute I worked at were enjoying the Christmas party in 2023! When my colleague first showed me this problem, my initial thoughts were to think about the factors of every locker number. We started writing out the first 20 numbers and their factors to conclude whether they would be same or different from their initial state. We quickly noticed that all the numbers that were perfect squares (1, 4, 9, 16...) ended up being the opposite state. This is because they have odd number of factors because you would need to pair the factors in two to return to the original state.

Sept 16 - Favourite and Least Favourite Math Teacher

My experience learning math in elementary and high school in India has been a wild ride. A few of my teachers that I encountered in my learning journey were incompetent, unmotivated and unsupportive. I remember my grade 3 teacher comparing me to other students and calling me out in front of the whole class for not understanding simple single digit addition. She did not create any sense of belonging for me in her classroom. Another one of my teacher was physically abusive and instilled so much fear in me that I memorized all the steps to solve questions rather than really understanding any of it. I had to work really hard to overcome the trauma caused by my teachers.

I was fortunate enough to encounter my favourite math teacher in grade 8. He was patient, kind and cared for his students. He himself was so passionate about math that he changed careers at 60 years old to pursue teaching. He praised every effort that I made and every little bit of progress I showed. He inspired me to try to learn math and this soon made me feel as though I belong in a room where people discuss math. When I think about the kind of teacher I want to be, his clear image pops up in my head. As I take up the 'mantel of the teacher,' I aspire to to be patient, inclusive and to create a safe space for my students where mistakes are allowed and questions are encouraged.

Sept 10 - Eisner on Three Curricula That All Schools Teach

 Reading this article by Elliot Eisner was a very interesting experience most of what Eisner talks about is what I find myself discussing with my peers since the beginning of this program. I thought it was interesting that for every point he raises against the school system, he adds that it need not be a totally negative things. For example, he mentions competitiveness being a the thing that drives most students, and yet it has an ugly side to it. He also mentions schools fostering complaint behaviour. This got me wondering how can I push the limits of the school system. How can I promote free thinking, strong opinions and perhaps some behaviours that are deemed "not appropriate classroom behaviour" in my classroom? What would happen if I allow my students to eat in class or take a 10 minute nap if they need it? Perhaps there is a way to do this so the students can recognize the general professional expectations and still be able to deliver on that front.

I also thought the concept of the Null Curriculum was very interesting and very important. I agreed with Eisner that not teaching something has consequences and they are not ones that we look for in the real world. I went to a high school that did not teach how a bank worked and I experienced embarrassing consequences because of that. The only classes that were taught any economics were the "Commerce" classes and I unfortunately was part of the "Non-Medical" stream. This relates back to the reward system that Eisner mentions. It left me wondering why must I be left out of important lessons like that simply because I chose to learn more about science instead of accounting?

Reading this article has definitely got me thinking and planning about how I want my classroom to look like in terms of the explicit, implicit and the null curricula I will eventually be responsible to teach. I am excited to learn some ways I can incorporate the important lessons that are usually left out in schools.

Group discussion on Skemp Article

 Following is what my group talked about in relation to the guiding questions on the Skemp article:

1. Are the two kinds of understandings distinct/separable?

In response to this question, we almost instantly and unanimously decided that yes the two kinds of understandings are indeed different. Instrumental and Relational understandings target different parts of students' brains. Instrumental's focus is on memorization, whereas relational encourages students to explore and come to their own conclusions. 

2. Is there a "best" order to teach them?

We thought that there is no right order to get a student to master a topic. This is a very case by case situation. It depends on the student's learning methods and interest in the subject matter. In a classroom, it would depend on the unit being covered. Sometimes, it is a good idea to provide the formula to get the ball rolling and then make relational connections at the end and witness students' surprise!

3. What kind of activities promote one or the other?

Instrumental understanding is mostly promoted my institutions such as Kumon where the priority is given to filling the pages. I once worked at a reward based math learning centre where students would get rewards for every five pages completed. Activities like this promote instrumental understanding. Relational understanding, on the other hand, is promoted by class group activities where students are required to research, class discussions, the teacher explaining the deeper connections of the material, and of course, real life examples.

4. How to assess understanding?

My group came up with some really cool ideas to assess understanding like - having a low stakes quiz, throwing them a curveball or asking trick questions and adding layers to the questions but the one that I brought up and relate to the most is having the students teach part of the lecture. I say this because my grandpa once said to me that "If you want to make sure to fully understand something, teach it to me." This is a practice that I utilized throughout my life and it has always worked for me (and everyone I recommend this to!)