Fractals, Algorithms, and Us: Discussion and reflection

Hey all, sorry for the long break! Life has been a bit hectic but I'm back and excited to share more.

In Fractals, Algorithms, and Us, students were regularly assigned readings that we would discuss and journal on in class. This post is about the way I facilitated these activities and is split into three sections: an introduction on why we did it, the actual process of doing it, and final thoughts on feelings in math. Of course, you are free to skip around to the bits you are interested in—I tried to keep sections pretty self-contained.

Motivation

There is a great deal of talk in math circles about the so-called math-phobia that grips our world, in which most everyday people excitedly distance themselves from this practice we call mathematics. Many mathematicians talk about their goals of eradicating this social ill, together with some analysis for the current state of affairs.

It is my humble opinion that math-phobia isn't just some superficial problem with the way we teach or talk about math or even the more structure issues of who teaches it—these are symptoms rather than the root cause. Instead, I believe that this phenomenon is deeply connected to the profound disconnect most people feel between mathematics and their lives; the many forms of oppression associated to the institutions of mathematics, often compounded upon people who identify across multiple axes of marginalization; and the resulting collective trauma we nearly all share around this thing called math. 

I am, of course, not the first person to present such an analysis—check out Professor Peter McLaren's Life in Schools: an Introduction to Critical Pedagogy in the Foundations of Education, for example, to see an explicit discussion of education through the lenses of class, labor, culture, ideology, and hegemony. Moreover, these deeply institutionalized problems are inextricable from neoliberalism, colonialism, and the mass disenfranchisement of black, brown, and indigenous peoples in the U.S. and across the world.

Quick aside: If you are unfamiliar with the concept of neoliberalism, by the way, here's a great video by Irami Osei-Frimpong at the University of Georgia on what it is and how it relates to education. Something he points out which I really took to heart is that when a student comes into my office to argue for 5% back on a homework set that is worth 3% of their grade (rather than searching for deeper insight on the math they may have missed out on) it’s because they’ve internalized that the answer to this question "What is education?" is "What my test scores tell me it is." Education has become fundamentally about students' assessments, not to equip them for self-determination as workers, partners, citizens, human beings, and maybe even as mathematicians. 

All that to say: understanding power is deeply critical to building a better world. That was the founding premise of this course. As such, I wanted to facilitate a space in which students felt safe to be honest about the feelings associated with these oppressive institutions and to think actively about what it could look like to dismantle them. These ideas weren't based on my experience as a trained pedagogue so much as they were based on my experiences as an organizer. 

PS. If education is supposed to be about self-actualization, then dammit part of what we learn in school should be how to talk about our feelings!

Methodology

I'd like to explain the facilitation practice through example, by running through the first hour of the six-week class. But, before that, a quick two-paragraph anecdote.

On January 17 of 2012, I attended the first lecture of MATH 352: Basic Concepts of Mathematics taught by Professor Ivan Avramidi at the New Mexico Institute of Mining and Technology. I was 19 years old, in my first year of college, and believed I knew practically everything about math. Little did I suspect the question that would kick off the class: "What is an integer?" It was a frustrating question that stumped the room, where we could only answer by saying things like, "Uh... positive and negative whole numbers?" to which he would demand, "But what is a whole number!" We realized very quickly that we couldn't define the real essence of an integer, not without using other words that already carried that essence.

The solution he finally gave, when we had no more guesses to offer, was simply: "An integer is an element of the set of integers!" which was both technically correct and apparently useless. His point, we'd later realize, was that abstraction via the language of sets, functions, and so on is a powerful tool for exploring and making rigorous the ideas we encounter every day.

This was an experience that really stuck with me. I liked the idea of giving students pause by posing a very fundamental question, one that we could unpack and think more about over the course of the six weeks we had together. So, after handing out the syllabus and putting up the day's agenda on the whiteboard, I asked the students to take out a piece of paper and write out a sentence or two answering a question:

"What is math?"

They didn't hand the papers in yet, but I'll spoil some of their answers:

• "A system of numbers used to explain the world"
• "The representation of real-world problems with abstract ideas and symbols"
• "Ideas and observations that are quantified""
• "The science and logic of shapes and arrangement"
• "Techniques for calculations"
• "Really really hard"
• "A precise way to communicate"

Next I passed out an excerpt from chapter 22 of Bea Lumpkin's Joy in the Struggle, but before reading I told the students that throughout the course we would be covering material that could be difficult, triggering, or otherwise make different people uncomfortable; my goal was to facilitate an accepting and honest space to do our work. Then we wrote down a list of ground rules for our discussions, to make sure that we respected each other and ourselves. They came up with:

• Don't be afraid to ask questions
• Take space, make space (this means that students who are normally quiet should feel safe to take up more space, while those who speak a lot should be mindful of relinquishing it) 
• Respect others with your use of technology (each student had access to a desktop computer, along with their cell phones)
• Disagree with ideas, not people
• Don't talk over each other 
• Be respectful and open-minded
• No hateful language
• Challenge your own perceptions

I added:

• Engage in good faith
• Don’t invalidate experiences

Before proceeding, the students voted unanimously to uphold these principles in our discussions.
We also decided to use the practice of a "parking lot," wherein a corner of the whiteboard is devoted to points we want to revisit later so we can stay on topic.

We then read aloud, taking turns in different sections, on the re-discovery of the Egyptian notion of zero and Eurocentrism in mathematical history—in all subsequent sessions this bit would be done at home, but this was the first day of class. Afterward, we returned to the board and engaged in an exercise roughly adapted from what I’ve learned as an organizer. I asked the students: "Tell me some feeling words. How did reading this make you feel?" After some of the girls spoke up, I urged the boys to offer some feelings too.

• Angry & disappointed 
• Disbelief
• Surprised
• Proud
• Excitement
• Curious

Then I asked the students to unpack those feelings. I tried my best to record faithfully on the board:

• Angry & disappointed
• racism/bias/etc. reflected in the things we create
• didn't think about how racists change/write history
• omnipresence of these attitudes in all our institutions
• Disbelief
• what is the truth?
• what else do we not know about our histories?
• Surprised
• human nature and values across millennia
• how can science be racist?
• Proud
• Egyptian, Incan, Aztec, Mayan, Indian, Chinese, Islamic, Polynesian—math isn't just from Europe
• math started in Africa!
• Excitement
• cool to see how math was connected to Egyptians' lives
• liked seeing how similar ancient people are to us
• Curious
• when did people start doing math?
• how can we learn more about ancient math?

When people would unpack a feeling, often someone else would want to add to it; this was allowed with the permission of the original feeling-giver or, otherwise, split into a separate point. This led to open discussion and naming tensions—I reminded the students that tensions are not necessarily bad since they help us grow—and added more to our feeling words. We used a stack, wherein I kept track of people waiting to speak; the stack was progressive in that I prioritized those who had spoken less. I spoke very little, less than 20% of the time according to my TA's measure, only to ask guiding questions or check if a particular rephrasing was acceptable while I copied thoughts onto the board.

Forty-five minutes into class, we wrapped up our discussions. I asked the students to write down a summary of their thoughts on the reading, only a couple paragraphs, for the next ten minutes before stretching their legs and taking a quick break. I also asked that they write a new answer to the first question, "What is math?" next to their original answer. A few examples:

• "A universal understanding of the world around us that started with humans in Africa"
• "Math is a shared human effort to solve problems and it doesn't belong to anyone"
• "The combination of lived experiences across human history used to solve common problems"
• "Fundamental to who we are as humans"

When we dismissed the first day of class, one of the students asked: "So—what is math?" I mention this as a point of caution because I don't like the idea of reducing the whole exercise into a riddle. After all, even if I knew a succinct or clever answer, it wouldn't make a whole lot of sense to suddenly center myself as the Source of Knowledge. I simply replied that we could figure out together, which turned out to be profoundly true.

In future classes, aside from giving out readings a week ahead of time, the principles of discussion and reflection were similar:

1. Feelings—listing single words expressing the emotions evoked while reading.
2. Unpacking—unpacking the aforementioned feelings and discussing together where they came from. 
3. Tensions—reflecting together on things, either in the reading or in the discussion (careful to avoid accusatory language), which created tension.
4. Journaling—give students time to write down final reflections on the reading, using guiding questions.

Sometimes we would vote to extend discussion time on certain hot-button matters. As the class progressed, I'd see students come in with their readings covered in highlighter marks with feeling words written throughout the text. That was pretty cool.

Final thoughts

I decided to move this section up in the blog lineup after an illuminating discussion with Professor Dagan Karp at Harvey Mudd who, among other things, pointed out some of the tensions in this class and suggested that I spend time unpacking them. I've also been really excited to write about this section since it is one of the more uncommon pedagogical features which sets this class apart from most other math courses (well, aside from nearly all the content) and one I'm very keen to continue exploring in the future. We talked about feelings and power in math class!

One of the most trademark aspects of mathematics is its powers of abstraction. Ranging from the simple use of numbers to the lofty realms of category and model theory, practitioners of math can extract the underlying essence of a particular concept in order to remove troublesome real-world details that inspired the problem in the first place. As espoused by many mathematicians (Dr. Eugenia Cheng, Scientist in Residence at the School of the Art Institute of Chicago, comes to mind) and evidenced by countless marvels of human invention, abstraction is an incredibly useful tool for solving problems. And solving our problems is why we’ve done mathematics for over forty thousand years! Indeed, a fascination with abstraction is why I am a mathematician (in particular, a topologist) rather than staying the course of physics or computer science from my undergrad career.

So why, when nearly every math course is at least implicitly predicated on the idea that abstraction is the premier feature of mathematics, did we devote two whole weeks of class time to exploring math as the amalgamation of the lived experiences of humans solving problems? Why did we spend so much time talking about feelings, power, colonialism, and so on, making the point that math is contextualized by the people who do it when, at the end of the day, 2+2 is always 4?

Because it can be both, of course, and we do ourselves an incredible disservice by pretending otherwise! In the words of Peter McLaren, we challenged the dichotomy of "technical knowledge" (the kind you measure with SATs) and "practical knowledge" (gleaned by describing and analyzing); instead, we replaced this false choice by embracing "emancipatory knowledge." According to McLaren:

"Emancipatory knowledge helps us understand how social relationships are distorted and manipulated by relationships of power and privilege. It also aims at creating the conditions under which irrationality, domination, and oppression can be overcome and transformed through deliberate, collective action."

Mathematics is simultaneously human and abstract, creative and logical, invented and discovered, beautiful and terrible, liberating and oppressive, indigenous and post-colonial, and there is no contradiction.