Math Wars!

Who would have thought the history of mathematics education in North America since 1900 would be so spicy? Obviously it should not come as that much of a surprise that viewpoints on math education have been and continue to be very polarizing, mimicking the political left vs. right dichotomy, but for some reason it still feels a bit shocking that something that has a reputation of being so objective (math) can still be used as a proxy for this seemingly never-ending battle. It truly is astonishing how much of society is pigeonholed into this binary. I thought Table M.1 did a fantastic job of summarizing the differences between these two main camps in math education - there really is nothing like a good table. 

I stopped when faced again with the proposition that one of the fundamental issues holding back mathematics education is the fact that so many adults, and especially educators, are so uncomfortable with math and do not understand most of its foundations. This creates a self-feeding loop that is incredibly hard to escape. I can't help but think that having math specialists in schools of all levels could help alleviate some of the negative consequences. I remember having a French specialist at my elementary school who taught every grade French. I'm sure most elementary teachers could blindly "teach" the fundamentals of French without speaking the language by focusing on spelling and grammar and watching videos, but the school recognized that learning a language from someone who does not speak it is a ridiculous thing. In this vain, it is preposterous that most kids are being taught math by people who don't speak it. Actually, I suspect most educators don't even recognize that math is a language.

The ties of the conservative school of thought in math education to fascism, neoliberalism, and religious extremism were not super surprising, but still exceptionally refreshing to see laid out in such an explicit and concise manner. I am in no way shocked that Thatcher & Reagan were (at least partially) behind the "back-to-basics" curriculum movement. I have a general rule that it is usually a fairly safe bet that when I find some policy from the last 50 years that is egregiously evil and/or stupid, it was probably one or both of them behind it.

Another thing that made me stop, and this time out of shock, was the Bourbaki group's banning of diagrams and call to stop teaching geometry. As someone who is generally more comfortable with algebra than geometry, I guess I selfishly understand the sentiment, but teaching math should be about providing as many avenues to understanding as possible, and a physical, geometric approach is often one of the more successful paths for students.

Overall I really enjoyed this reading. It is so important to learn the history of things like this so that we know how to avoid the traps of the past and trace a good path forward. It is also always good to be reminded that pretty much everything can be political. Really enjoying reading some of your work Susan :)

Dooo dododoo

Binary Tree Math Art Project - Collaborative Account

Brandon, Henderson, Jacklyn: Georgina Ryan’s Binary Tree Georgina Ryan used cotton fabric and cotton embroidery thread on a 21.5 x 20.0 x 1.0cm plastic embroidery loop

Georgina Ryan: Binary Tree (Original Version)

When remaking this project we first decided to approach the binary tree through the lens of fractals instead of combinatorics. This meant we had to simulate the appearance of many more iterations than the 8 that Georgina Ryan did.

The first step in making a binary tree is deciding on an angle between branches and a ratio with which the branches shorten at each iteration. In an attempt to mimic the proportions of a “classic” tree we chose an angle of 30° and a ratio of 1/2. We found the branches got too short too fast, making it very difficult to do many iterations and resulting in no interesting fractal geometry. We then decided to increase the ratio to 2/3. With this change, we found we could create more iterations and got some overlap in our later iterations because the branches got smaller slower. Choosing the length of the initial iteration was also a problem we faced. We tried sketching out a couple versions with different lengths, but struggled with finding a satisfactory length. We ended up committing to a length that resulted in the ends of the tree being a little off the canvas, but it gave it an interesting aesthetic so we are not upset. All this exploration was done through a combination of drawing fractal trees on paper, sketching out our tree on the canvas, and using an online binary tree generator (https://homo-deus.com/lab/mathematics/fractal-tree).

Computer generated tree with our parameters

The original art piece was embroidered using thread. In order to make this project our own, we decided to make a binary tree out of trees themselves! We looked around the garden and found sticks of varying thickness. We then cut them to the proper proportions, attempting to pick thinner sticks for each iteration of the binary tree. We then hot glued the sticks onto a canvas (30.05 x 40.05 cm)  in order to recreate an actual binary tree! In order to simulate the appearance of the chaotic ends of a binary tree with many iterations, we used moss. Adding the moss also had the effect of making the art piece look even more like a tree found in the wild 🌳

Exploring binary trees in the sun ☀️

For our interactive activity, we decided we will split the class into groups (preferably those they are sitting with), and assign them each different angles to explore with their own binary trees. Within each group, each person can choose a different ratio and everyone can draw a binary tree. Afterwards each group can show their drawing to the rest of the class. The goal is to get a visual representation of how binary trees change depending on different ratios and angles, along with instilling a general confidence in how binary trees work. 

The final product 🌳

Critical Curricula

Elliot Eisner's "The Three Curricula That All Schools Teach" was very thought provoking. Eisner took a very critical view of the education system, an approach I often share. When I think of "curriculum" I think of anything that is taught in schools, but I believe most people think of it as the things that have been explicitly mandated to be taught in schools. This explicitly mandated curriculum is one of the three curricula that Eisner discusses, and probably the least interesting of the three. I was particularly intrigued by Eisner's analysis of the "implicit" curriculum. This curriculum appears to be closely tied to the expectations of capitalism, teaching kids the skills and behaviours needed to be good little proletarians. This sort of analysis is often my default lens so in some ways while reading it felt like Eisner was simply preaching to the choir (I'm the choir). Some arguments that made me pause were the arguments steeped in luddism. In particular, the long quote from Mumford (1938) read like something that could have been written this year, just with the technologies swapped out for modern ones like AI. I am quite sympathetic to the luddite cause, but in the anti-capitalist sense, not the more propagated and over-simplified "technology bad" sense. Another moment that made me pause was the reference to the "back-to-basics" movement; this made me feel like the cycles of mandated curriculum is a lot like the cycles of fashion, repeating themselves every few decades. The ideas of the "null" curriculum were very interesting and the biggest concept in the paper that I hadn't grappled with much before. I do agree with the underlying argument that students should be given as diverse an education as possible, but also fear that introducing too many mandatory courses would only serve to remove the freedom of choice from students, furthering the effects of the implicit curriculum. Regardless, I look back at my schooling and do wish I had explored more diversity in fields instead of committing myself so heavily to the maths and sciences so intensely and so early.

I believe this more nuanced understanding of curriculum that Eisner lays out is essential to understanding what is taught in schools and what our roles as teachers are. With the general vagueness of the BC curriculum, I believe there is more room to implement some of the good things like incorporating elements of the null curriculum into class, and to resist some of the bad things like attempting to remove elements of the implicit curriculum. Furthermore, the generality of the core competencies feel related to some of the ideas that Eisner proposed, especially with how they focus on teaching skills that are mostly independent of the specific topics being taught.

My favourite quote from the paper was "Schools are educational churches, and our gods, judging from the altars we build, are economy and efficiency. Hardly a nod is given to the spirit" (p. 97).

I wish I wasn't so tired when I read this paper so I could have engaged even deeper with it.

so.many.lockers.

Due to tiredness and a general difficulty with visualization (I am 99% sure I have aphantasia), I skipped the "classic" 1st step of trying it out in my head and went straight to brute-forcing the n=10 case and looking for patterns:

Through this process, I found that if the number of the locker has an even number of factors it will be closed at the end, and if it has an odd number of factors it will be open at the end (excluding 1 as a factor in both cases for some reason 😅).

I proceeded to ask how many lockers will be open at the end of the process and tried to build a series using the pattern that each "group" of open lockers is double the size of the previous. I struggled to find a good way to iterate & bound the series though and eventually gave up.

I then challenged my partner (also a math teacher) to try the problem:

She found that the number of each closed locker is a perfect square, something I just didn't see 😮.

From this, determining the number of open and closed lockers became easy because I just needed to find the largest perfect square that is less than 1000.

Starting from 30^2=900, I went up and found 31^2=961 & 32^2=1024 (was too lazy to get a calculator so I did it by hand). This told me that there were 31 closed lockers and therefore 969 open lockers.

The teachers that shaped me

First off some context. Ironically, my favourite math teacher gave me my lowest mark in university and my least favourite math teacher gave me my lowest mark in high school. This is indictive of the fact that marks are not always tied to the quality of a learning experience.

Now I'm a save the best for last kind of person so let's start with my least favourite math teacher.

My least favourite math teacher was my grade 11 teacher. They were set to retire the next year, and though they weren't quite completely checked out like some workers in that position are, they were clearly tired and stuck in their ways. The class was composed of dry lectures and endless textbook problems. The policing of homework completion (usually like 20 textbook problems a night) even if I quickly understood the concept drove me particularly mad. At a time when I was trying to figure out what I wanted to do in university and with life, this class almost ruined math for me, but luckily I can be quite stubborn.

My favourite math teacher was my Calculus 2 professor in university. They were always passionate and excited, handled a blackboard like a wizard, and really cared about kids not falling behind. They started every class with a review of the previous class, integrated history and application into most lessons, and loved showing us weird and beautiful little things like i^i or the derivation of Euler's identity. Unfortunately I mis-studied for the final exam and ended up getting a 50 in the course, meaning I had to redo it due to prerequisite requirements for some 2nd year courses. Regardless, I still look back on that course incredibly fondly. The icing on the cake is that this professor ended up becoming my masters supervisor, actively affirming to me that marks are dumb and not representative of a person's potential.

To this day I use a lot of the techniques that my favourite teacher used in my own classes, especially the starting every class with a review of the previous class. In contrast, remembering what I disliked about my grade 11 class motivates what type of practice I give and expectations I have for my class, trying to avoid putting students through the meat grinder I had to endure.

My favourite "celebrity" educator Carl Sagan

Don't Skemp on the Details

It is always an interesting experience when I find myself reading a formalization of an idea that has been floating around in my head for some time, but that I never got around to properly researching. Skemp's conception of 2 types of mathematics, largely differentiated by motivation and process, made me stop because it is something that I noticed early on in my post-secondary studies. In particular, it was through discussions with people in applied fields, such as engineering, that I learned how many people value mathematics simply as a tool and do not engage with the deeper, more artistic elements of the field.

I also stopped when Skemp first introduced his definitions of "relational" vs. "instrumental" understanding because my thoughts immediately went to the current discourse in the math world around AI proofs. Sparked by OpenAI's ethically questionable brute forcing of the Navier-Stokes smoothness problem, the STEM community has been abuzz the last week with discussions incredibly similar to the arguments made in Skemp's paper. All this drama has culminated in Terence Tao penning a letter signed by 25 Fields Medalists titled "A Severe Misalignment of AI in Mathematics" in which he details the dangers and travesty of pursuing mathematics simply for the result, and not for the process (and what is learned along the way). This coming exactly 50 years after Skemp wrote "Relational Understanding and Instrumental Understanding" is quite amazing and speaks to how much Skemp's paper is still relevant.

Finally, when Skemp was discussing the attributes and prevalence of instrumental understanding I stopped because 3 big things came to mind:
1. The fields that incentivize this form of understanding (engineering, economics, programming, etc.) are some of the most socially "valuable" fields because they create tangibly useful things in people's lives (I'm pretending the type of money economists work with is tangible for the sake of the argument). There are many consequences of this, some obvious and some less so.
2. Having worked in Quebec the last 4 years helping students pass the grade 10 standardized math exam there, I was ostensibly forced to teach instrumental understanding due to the structures of their education system. This realization has helped me put words to an element of my dislike of standardized exams that was mostly just intuitive up to this point.
3. Most teachers who lay the foundational mathematical knowledge in children on which everything is built only have an instrumental understanding of math themselves, so how can they be expected to teach relational understanding? This serves to deepen my belief that there should be math specialists in elementary schools.

By this point it's probably pretty clear where I stand on the issue raised by Skemp, but just to make it abundantly clear:
Relational Understanding > Instrumental Understanding
Sure I see the value of instrumental understanding as a stepping stone in certain contexts, but ultimately if you only know how to swing a hammer, eventually you're going to break something and not know how to fix it.

P.S. Sorry this wasn't very brief; I really liked this paper.

hello world

heyheyy welcome to my blog