Fractals, Algorithms, and Us: Building the syllabus

I will break this post into three parts!

Choosing the readings

Once I had the previously discussed teaching goals as a basis, the next step was to choose reading material to guide the journey. I knew that I wanted students to have at least one reading a week, with space in class for them to unpack. Having only six weeks at my disposal, I had time for only five selections—six if I allowed in-class reading in the first week and maybe a couple more by squeezing in some shorter stuff as one assignment or taking additional class time. As a man who craves structure—I'm an algebraic topologist, it's just who I am—I decided to go mostly chronologically.

Remember, what I needed to do was tell a six-week story of mathematics which created the environments necessary to accomplish my teaching goals along the way. The story arc I wanted to follow was roughly this:

• Math is part of the human effort to solve problems and therefore it is found in every culture. Since math arises in this way, we should spend time understanding how it connects to the cultures, societies, and physical environments of the people who do a particular form of math. 
• Now that the students have just spent time internalizing that math is deeply tied to the shared experiences of the people that develop and practice it, they are better prepared to understand the role of math in colonial violence. This includes the use of math as a tool to materially shape who had access to education and wealth, but also the use of math towards the eradication of indigenous knowledge in colonial societies and establishment of cultural hegemony.
• Moving into the postcolonial era, we can continue to talk about the connections between math and institutional power. There are more well-known and clear examples—weapons development, code-breaking, interest rates, and gerrymandering, to name a few—but also more furtive ones, such as artificial intelligence or big data.
• Students see by now that the act of doing math, science, and intellectual labor more broadly is hugely contextualized by the world around us. Therefore, as a sort of epilogue, I want them to see that STEM workers have a shared responsibility for how the knowledge we create is used. In other words, to build a more just society, a crucial element has to include scientists and engineers who understand power and share a notion of solidarity and justice in a way that protects and empowers the most marginalized among us.

I read a ton of books to get to a place where I felt like I could meet these tall requirements. So
my honorable mentions which didn't make the final list (though some students read excerpts on their own) were Decolonising the University, The Imperial University, Power in Numbers: The Rebel Women of Mathematics, A Mathematician's Lament, The Imagineers of War, How to Bake Pi: An Edible Exploration of the Mathematics of Mathematics, and Rehumanizing Mathematics for Black, Indigenous, and Latinx Students. I also have to mention this great repository on decolonizing science, put together by Professor Prescod-Weinstein, which I am only partway through myself. Remember that you can also see the old syllabus for Fractals, Algorithms, and Us here.

The final readings

• Week 1: Joy in the Struggle, by Bea Lumpkin. We read a section from the beginning of Chapter 22, in which she recounted uncovering evidence for the Egyptian notion of zero while attempting to meet the demands of students pushing for African American studies at Malcolm X College in 1968. 
• Week 1-2: African Fractals: Modern Computing and Indigenous Design, by Ron Eglash. This book was used by many students for their final projects, but the section we discussed in class was the cross-cultural survey of indigenous mathematics which, among other things, highlights the remarkable ubiquity of a fractal aesthetic across the entire African continent (but nearly nowhere else). 
• Week 1-2: Pedagogy of the Oppressed, by Paulo Freire. This whole book is great, but I asked the students to read the beginning of chapter 2 on the teacher-student relationship so that we could all think together about our roles with respect to one another. I wanted to create an environment where students could recognize and help each other to reject the so-called banking model of education, wherein "students are the depositories and the teacher is the depositor."
• Week 2-3: "Western Mathematics: The Secret Weapon of Cultural Imperialism," by Alan Bishop. Not a book, but a great read about math in colonial societies which critiques the widely held belief that math is culturally neutral. We also used Chanda Prescod-Weinstein’s “Making Meaning of ‘Decolonising’” to frame our discussion, which I personally think is an incredible post to truly think about (1) decolonization, not as a feel-good metaphor but a serious endeavor to dismantle settler colonialism, and (2) decolonizing science in particular. Especially after the first two weeks of internalizing math as a living and breathing embodiment of cultural heritage and shared struggle, the way in which students immediately recognized the violent colonialism inherent in the eradication of indigenous mathematics by European ideas was very moving to me.
• Week 3-4: "Geometry versus Gerrymandering," by Moon Duchin in the Scientific American. This is a really well-written article by an incredible mathematician about gerrymandering, which is the act of securing a political advantage for a particular party in power by manipulating district boundaries, and the use of math (Markov chain Monte Carlo methods) to fight it. This is a pretty overt use of mathematics for political gain, but still a good transition to the contemporary era—especially in a city as gerrymandered as Chicago.
• Week 4-5: "Racist in the Machine: The Disturbing Implications of Algorithmic Bias," by Megan Garcia. This article does a great job of breaking down the unconscious and institutional biases that affect modern algorithms, as well as some of the racist, sexist, and xenophobic consequences.
• Week 5-6: Scientists at War: The Ethics of Cold War Weapons Research, by Sarah Bridger. This book talks about, among other things, scientists who lobbied against the use of chemical defoliants, napalm, and the 1980s Star Wars program. It does a good job of looking at how individual science advisors, looking to wield influence from within the government, were ousted and how these advisory roles were diminished in the administrations of future presidents—this fits nicely into the importance of collective and organized power in the pursuit of justice, which I wanted to emphasize.
• Week 6: "The Dual Nature of Science," by Conor Dempsey. This is a neat essay in Science for the People which links militarism, reproductive justice, eugenics, and organized scientists fighting for a better world. I thought it would be a good epilogue to the course, with even more space to think about power. 

Interweaving the "math"

I say "math" here because, of course, we're already doing math in some sense. Apologies to old English teachers who insist that quotations aren't for emphasis—it's not my first language.

Anyways, now that I was armed with these really cool readings, all I needed to do was populate the course with the more recognizable "math" content and we were ready to go. My goal here was to pick mathematical topics which would pair nicely with what we were covering as we read.

All told, this part was actually pretty easy. The subfield of math which I specialize in, called topology, lends itself really well to picture-drawing (or, if you're me, interpretive dance) and is deeply connected to the age-old problem of solving polynomials. In fact, I once gave a talk called "The topology of the quadratic formula." In another life, in the far-off mountains of Socorro, New Mexico, I was both a computer scientist and physicist—that just means I had a very eclectic background of example material to draw from. Here's what we did in class:

• Week 1: We start by solving equations, as motivated by the Babylonians over 5,000 years ago, which naturally leads the students to manipulating complex numbers. This also led to our first example of an algorithm—the Babylonian square root-finding method—which the students played with for homework and we coded up together. To practice coming up with their own algorithms, but with the stated goal being to create fun art, students did a lot of in-class exercises with the Turtle module in Python. We also played around with sorting lists, where the students broke into groups and independently created the insertion and bubble methods—this was a really remarkable moment for me to witness, having offered no advice on the problem and thus being truly de-centered as the Source of Knowledge.
• Week 2: After the Babylonian method, we moved on to the so-called Newton root-finding method for higher-degree polynomials. Some of the students hadn't taken calculus yet so, rather than go into detail rigorously defining a notion of derivative, we stuck to heuristics: they'd draw pictures and do calculations with various examples to see how the algorithm worked. Once you do that, the Newton fractal pops out (this was such an incredible a-ha moment for the students)! 
• Week 3: We hopped back on the fractal train by exploring Mandelbrot/Julia sets and Buddhabrot renders. We also reviewed some of our old sorting problems and introduced the heap sort as an alternative to the bubble and insertion methods they had developed. Homework consisted of hypothesizing which numbers would escape the "Mandelbrot Mill," comparing how different sorting algorithms performed in different situations, and each student creating a portfolio of hand-drawn and computer-generated fractal art.  
• Week 4: We started talking about more advanced algorithmic ideas, including page-rank algorithms and neural networks. Both required students have a basic understanding of linear algebra—I had been surprised to know they all had seen matrices already, though they also unanimously expressed understandable anxiety and disgust when I mentioned the word. We started talking about the problem of recognizing digits written by hand, tried to gerrymander and prevent gerrymandering in a simulated city, and rediscovered RSA encryption.
• Week 5: With students preparing their final projects, we had to pull back before building a full-fledged neural network together from scratch. Instead, the students played with the idea of gradient descent and cost functions before we looked at a visual representation of a neural network I coded up to recognize handwriting. We had a blast trying to understand how different neurons fired when different shapes or patterns were present in the samples. 
• Week 6: Final projects! We also listened to some computer-generated Shakespeare.

Phew. Okay, that's enough for tonight. Stay tuned for more!