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!)

September 6 - Richard Skemp on instrumental and relational ways of understanding mathematics

 After reading this article, I found myself relating to Skemp in a way that as an educator, I too have been guilty of prioritizing and having a bias in favour of relational understanding in mathematics. It wasn't until I started my job as an educator at an institution in Vancouver that I was made aware of the two types of understandings that Skemp mentions. I remember being called into my manager's office to discuss the goals of a new student. Upon looking into the student's file, we decided that not focusing much at all on deeper understanding of concepts would be the best way to go in his case and in all honesty, I felt frustrated and wanted to push my student's limits. It took some self control on my part not to do that and the results were outstanding. The student made it through math with a decent enough grade and was able to prioritize other things that truly interested him and he was happy!

 As a product of the Indian high school system though, where grades and mindless memorization are given priority over relational understanding of any material, I was not surprised that I needed to pause and think about the concepts Skemp mentions in his article that I too was made to understand in an instrumental way. As for where I stand on the issue Skemp raises, I believe that as an educator, you really have to talk to your student about their goals and find the balance between instrumental and relational understanding. They may be interested in learning about fractions in a relational way but not Areas. Engaging with the student and being proactive in setting goals together would make this easier.

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