Announcing: How Chance Changes the World

Hello everyone! It's been a long time and I have a lot to update you all on.

Firstly, I graduated! It wasn't something I expected to do this year, but at the last minute (literally mid-October, after some really tough heart-to-hearts) my advisor urged me to apply for jobs. In retrospect, I really wish that I had been blogging in that time because it was truly a revolting experience to be on the market™. After some really important application advice from Jen Taback, I decided to really own my history as an anti-racist union organizer and how this has guided my trajectory as an academic and an educator. That definitely scared a lot of places away, not to mention that fundamentally applying for academic jobs is truly a morbid (and racist, and sexist, and transphobic, and in many other ways violently exclusionary) game of chance that has nothing to do with what we as people and mathematicians deserve. At the end of the day, I am very lucky to be able to accept a year-long postdoc/VAP at the University of California, Irvine, where I will work with Jesse Wolfson and Nathan Kaplan. In the next two weeks, I will leave Chicago (my home of six years) to live in Albuquerque (my hometown) and from there to Irvine (pandemic permitting).

There is literally so much more to talk about (truly I can't believe it's been a whole year, yet at the same time it feels like a decade) but I'd like to get to the point of this post. The Collegiate Scholars Program at the University of Chicago has asked me to teach one final summer class, and this time I'd like to start blogging before we've begun (rather than trying to remember how things went). Because of COVID, the class will be done entirely via Zoom; we also have been given half the usual instruction time, with three hours per week over four weeks (instead of four hours per week over six weeks). I'll be co-teaching with Chloe Avery, a fellow graduate worker at the University of Chicago and also a Benson Farb student (she's also an incredible organizer if anyone's curious). We decided to teach a class on probability: How Chance Changes the World.

I've been hunting for the perfect textbook for a while and (as you'll see if you keep reading) at this point I feel pretty hopeless. Yesterday I chatted with Chloe about writing our own course notes and we decided to create a shared TeX file to start on the project. Last night I sat down to start working on said notes but found myself unable to focus. My thoughts kept drifting to the world outside, to all the suffering and violence and resistance and reimagining, and how all that connects to our classrooms. So, instead, I found myself writing a preface to the notes, trying to express exactly why the way we teach math now is inadequate and the struggles that I'm having with reimagining it. I can't imagine sharing what I've written with students in this current state—if for no other reason than it's all about me and my feelings, falling for the classic trap where education is somehow centered about the teacher rather than the students—so maybe it's more of a postface or something else entirely. Regardless, I'd like to share it here, for those who are interested:

Content warning: pandemic, sexual violence, lynching, racialized trans murder.
Preface (postface? something else?) to course notes

I'm hoping to do a good job of blogging about this course, sharing what Chloe and the students and I come up with during the pursuit of making probability class into a liberatory endeavor. Of course, we are also supported by so many other incredibly talented pedagogues. Hopefully, you all will be excited to join us on the journey of How Chance Changes the World.

Inclusive pedagogy and neoliberalism

At this moment in Chicago, there are 35,000 workers from the Chicago Teachers Union (CTU) and the Service Employees International Union (SEIU) Local 73 out on strike. These everyday people—teachers, clinicians, support staff, special ed classroom assistants, custodians, security officers, bus aides, park supervisors, attendants, landscape laborers, and more—are picketing not only for themselves but for the common good. Their demands include affordable housing for students, sufficient teacher prep time, and class size caps; proper staffing of school librarians, nurses, social workers, clinicians, and counselors; fair pay and benefits for Chicago Park District workers; and, ultimately, the dignity and resources they need to do their jobs. My own union of graduate employees at the University of Chicago, Graduate Students United (GSU), has been zooming across Hyde Park to both help bolster the lines and keep sharp our own picketing skills, as has the University of Chicago Labor Council.

At the same time, the coalition of faculty at the University of New Mexico (United Academics of UNM) have won their union election by a huge margin (almost identical to GSU's from two years ago). This is rad in and of itself, but especially wonderful to me because Albuquerque is my hometown. In the words of Professor Nick Estes: "with a union we can better serve our students and thus better serve ourselves as workers of UNM." and "with a union we can increase hiring and sustaining Native faculty."

These folks are truly inspiring, as are the people of Chicago and Albuquerque coming to their aid. To those of you who insist unions should stay in their lane and only fight for wages... well, frankly, you're racist.

As any of the amazing educators currently out on Chicago picket lines or in New Mexican classrooms will tell you, teaching is hard. Furthermore, going on strike and witnessing true collective power in the face of oppression profoundly changes the ways you teach, as evidenced by these blog entries.

With all this in mind, I've talked a lot about neoliberal education (remember Irami Osei-Frimpong's video?) and I want to make a note here on what is called "inclusive pedagogy," just because it's come up a lot for me the last couple of weeks. Inclusive pedagogy refers to a paradigm of teaching which acknowledges the varied backgrounds, learning styles, and abilities of all the learners in a particular classroom, which is of course very important and cool. At the same time, "inclusivity" has become a buzzword in a lot of liberal institutions and, as a result, you see a lot of teaching workshops centered around it. That's good, right?

My big beef with the rise of ~inclusivity~ is that institution-sanctioned spaces rarely do their due diligence in making white, wealthy, male, and otherwise privileged people uncomfortable with the current state of affairs. That means these workshops fail to confront the reality of racial capitalism in perpetuating a stratified society, where simply making our ivory towers more inclusive passes as a radical act. To borrow the words of Professor Chanda Prescod-Weinstein: "I harp on liberalism here because I think too often liberals who believe we should seek to grow the status quo ('America is good, it just needs to be better!') co-opt the language of those trying to destroy the status quo ('America is an imperial project built on genocide!')"

We academics should be working to remake the institutions of higher education—as political, economic, and social engines—to serve working people, not to just promote a handful of black and brown and female faces while perpetuating the ongoing underlying system of stratified exploitation.

Now, this quarter I've been attending a Fundamentals of Teaching in the Mathematical Sciences seminar run by two graduate workers, Karl and MurphyKate. These two are both brilliant mathematicians, dear friends, and incredible instructors who constantly strive to be better and I love them very much. They are running these workshops under the close supervision of the Chicago Center for Teaching at UChicago and this week was specifically on inclusive math pedagogy.

We opened with an exercise that I think can be useful to warm up academics who come from a great deal of privilege and haven't thought a lot about what it's like to not be themselves in educational contexts: we broke into small groups, were given a bag of supplies, and told to build mobiles. The secret was that different groups had different items in their bags—some had wire, glue, markers, glitter, scissors, and other materials, but my group only had a clothes hanger, paper, and some tape. After we created our mobiles, Karl and MurphyKate graded them—the groups that had received plentiful resources were praised but the rest of us, held to the same standard, were critiqued harshly despite creative efforts. At the conclusion of the exercise, we were asked when we realized the distribution of resources was unequal, what our reaction was to these asymmetries, and how it felt to be judged in such an unjust way. As you might expect, the groups with lots of resources hoarded them while those with very little tried to find ways to cooperate.

Even though I think this exercise was fairly tame—no one had even brought up words like “race” explicitly yet—I could see a few folks take what was being discussed very personally as they started connecting dots. The room, by the way, was overwhelmingly white and male. As we moved on to think about axes of identity which could lead people to feel excluded in a math classroom, a friend of mine brought up gender and was immediately met with a retort that "2/3 of my calculus students are regularly female, so I disagree." While we were discussing methods we use to encourage collaboration, an attendee became visibly agitated when I said that I regularly pause class to remind my students that math is not an individual endeavor and we ought to be working collectively to create spaces where we can be wrong and learn together.

I'd rather not dwell on those really unpleasant interactions; props to Karl and MK for mediating them. Instead, I'd like to focus on the takeaways that have been bouncing around my head this week after speaking with other BIPOC and radical educators, in particular during a course on Critical Pedagogy with Dr. Cheryl Richardson.

A few people in the math educator workshop had trouble thinking about ways they could be inclusive with respect to race and ethnicity in their classrooms: "Instead of dividing up a cake, do I pick... I don't know, something else from another culture?" Yeah, someone actually said that. The underlying sentiment they were expressing match those I encounter a lot in pedagogy spaces, where sometimes even the facilitators make excuses for me (usually the only STEM person in the room) by saying "Oh but in math, the answer is right or wrong—so all this doesn't apply."

How do you make your class culturally relevant when all you’re doing is computing integrals? We lecture, we have homework, and we have tests—how do you make that inclusive? Don’t our students just care about their grades anyways? There’s no way to be inclusive when all that matters is doing calculations!

These ideas are all connected. Also, to be clear: If you are worried that your students only care about their grades, rather than the content of your course, then you should reflect carefully on yourself and what the environment you create in class says about your priorities. Do you only spend class time doing things that will be graded, either by homework or quiz or exam? After all, how can we expect students to value the collective struggle for knowledge if we ourselves do not? This last bit is a reference to a passage from bell hooks' Teaching to Transgress, flagged by a friend from Dr. Richardson's class (who would like to remain anonymous):

"On another day, I might ask students to ponder what we want to make happen in the class, to name what we hope to know, what might be most useful. I ask them what standpoint is a personal experience. Then there are times when personal experience keeps us from reaching the mountaintop and so we let it go because the weight of it is too heavy. And sometimes the mountaintop is difficult to reach with all our resources, factual and confessional, so we are just there collectively grasping,

feeling the limitations of knowledge, longing together, yearning for a way to reach that highest point. Even this yearning is a way to know."

My friend pointed out, I think truthfully, that this idea of yearning as a way to know "would freak out a UChicago student."

Remember, the hallmark of modern education is that it relies on a market-type analysis to evaluate both teachers and students. As said on neoliberalism by Irami Osei-Frimpong: "the principles of our analysis lead us to believe that it is irrational to invest in people... unless those people show themselves to be poised to increase in value in terms of quantifiable assessments." There’s no time in today’s classroom for wonder—just facts and the cold evaluation of regurgitated facts, so we can decide who is worth investing in and divesting from! This is why inclusivity, in the context of modern higher education, often just gets reduced to cosmetic changes in metrics rather than truly transforming our pedagogy. After all, how the heck are we supposed to quantify a student's yearning??

I try to make my classroom more inclusive by learning about how math as a discipline is tied to political power and human experiences, both historically and today, and connecting these ideas to the lives of my students. I do that by giving weekly readings, by making time in class to discuss these readings, and by asking students to journal about their thoughts in an ungraded format; I intentionally provide space that is separate from the economized endeavor of technical knowledge, where "the answer is right or wrong" and comes with an all-important grade attached. But this isn't the only way to be inclusive and I'm excited to continue learning for the rest of my life about the many ways that I can be better.

Teaching is hard. So is inclusivity. If you want to make your classrooms inclusive, workshops are great but just going to one and expecting to learn the secret two-second fix isn’t enough. Justice takes work, from learning the political history of mathematics to making intentional spaces for discomfort, from organizing a democratic union to trusting our students' capacity for yearning. As teachers, our battle with neoliberalism is eternal—both in New Mexican universities and in Chicago streets.

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.

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!

Fractals, Algorithms, and Us: Teaching objectives

The content of the course was designed around the following broad goals, assembled from my own experiences as a queer New Mexican Chicano mathematician and labor organizer at the University of Chicago:

• Establish mathematics as a living, breathing subject, part of the lived experiences of the student, their communities, and their ancestors. In particular, students should feel emotions throughout the course, reflect upon them, and then connect them to the practice of mathematics. Instead of the cold and sterile presentation of a typical math class, where knowledge is "about information only" (as said beautifully by bell hooks) and colored only by half-hearted attempts at connecting to reality through contrived word problems, we should immerse ourselves in the context of what problems people were actually trying to solve in their daily lives as they developed these mathematical ideas.
• Explore math, and access to mathematical knowledge, as a dynamic political force that changes the world rather than existing separately from it. This goal is both contemporary and historic in its focus. I also intentionally wanted to challenge the usually individualistic and thus ironically disempowered saying that “knowledge is power.” Instead, I wanted students to reckon with how math as a social construct, together with its perceived neutrality, has contributed and continues to contribute to mass disenfranchisement, colonial violence, and the modern surveillance state.
• Challenge the Eurocentric bias of mathematical history. Because math didn’t begin with the Ancient Greeks! Instead, the goal should be to center indigenous people throughout history to understand how their lives played into the mathematics they developed and used. Especially taken with the goal described in the first bullet point, wherein math is thought of as an inseparable part of the human experience rather than an abstract set of rules dreamt up by some boring and long-dead European men, I hoped that students would begin to see for themselves the colonialism inherent in modern mathematical practice.
• De-center the Teacher as the Source of Knowledge. In most modern math classes the Teacher enters the space of waiting students, projects ignorance upon them, and then proclaims Knowledge upon the chalkboard. The students are measured by how dutifully they are able to act as empty vessels to be filled by said Knowledge, signaling whether they should be invested in or divested from by future Teachers and institutions of Knowledge. All told, there can be no learning without the Teacher and their Knowledge; the Teacher is the subject, rather than the students. Instead, I aimed to facilitate an environment where students enjoyed a high degree of agency over their education, especially through student-led projects and breakout groups. Check out Pedagogy of the Oppressed for more of what inspired me here.
• Develop a working knowledge of programming and some basic algorithms. This goal is very personal to me, deeply connected to my own quest for agency and self-actualization. Despite growing up with a mother who taught me to love math before I could speak, I hated the subject for most of my childhood because of my experiences with math education in school. The thing which allowed me to fall back in love with the thing I have spent my professional life studying was a teacher, Mr. Maier, who gave me a book called Processing: A Programming Handbook for Visual Designers and Artists. Armed with this book, I'd steal away whenever I could to our high school's ancient computer lab and create. Programming allowed me to explore gravity, magnetism, fractals, art, predator-prey behavior, music, and so much more (some old code is here—please remember no one had taught me about commenting or style yet) in a completely new, self-driven way. So, with this class, I wanted to teach students to code with the aim of developing a heuristic approach to both mathematics and problem solving so that they could feel empowered to investigate things on their own.
• Expose the students to mathematical topics and material resources they would ordinarily not encounter in high school. This goal was geared especially toward contemporary topics, like machine learning and chaotic systems, whose applications range from predicting the weather to scheduling employees' shifts. So the idea would be that we could black-box some computationally complicated prerequisites for the sake of giving students an intuitive understanding of deeply pervasive modern ideas. Also, the University of Chicago has 3D printers, laser etchers, and video games labs—I wanted students to have access to these amazing materials and resources.

These guiding principles led to the following learning objectives. Students will:

• Explore math as a social and political concept through weekly readings and reflections. This objective includes both following a narrative throughout human history which expresses math as a collective endeavor to solve problems and also understanding the colonial context of math research and education.
• Write algorithms to solve polynomials, sort lists, and render fractals. The first is a five-thousand-year-old problem, the second is both instructive and delightful, and the latter unlocks infinite creativity.
• Be comfortable manipulating complex numbers. This particular set of numbers is not only necessary for understanding many concrete examples of chaotic dynamics and the emergent fractals, but also deeply related to the problem of equation solving. Since this was an endeavor necessary to the lives of countless humans (millennia before Greek societies came to be), the complex numbers feel like an appropriate avenue toward humanizing mathematics. 
• Develop a working knowledge of basic programming concepts, including logic, loops, conditionals, and functions, via the Python language. Programming is an increasingly essential skill set in the modern world, but learning how to code also equips students to question on their own in the way that I once did in high school.
• Collectively design and train a neural network to recognize handwritten numbers. Neural networks are an increasingly ubiquitous tool of modern data science and are a concrete example of a mathematical idea that is currently changing the world in a deeply politicized way.

In the next entry, we'll take these objectives and actually synthesize a reading list and collection of abstract mathematical topics into a real class. After that, we'll talk about the day-to-day structure of the classroom and how exactly it looked to facilitate student-led discussions and learning.

Fractals, Algorithms, and Us: Preface

I've been thinking a lot about how to share my experiences from this summer. A handful of friends had suggested that I describe the process of developing my syllabus, framed around some of my teaching goals, then use the actual teaching of the class as a sort of data gathering to either support or refute my hypotheses. I took studious notes and kept a journal of the whole experience, in addition to keeping hold of all the students’ work, so this seemed right—the idea was to submit the final result for publication in a variety of pedagogy journals and it seemed like the right thing to do as I prepare for the job market.

As a good scientist, I tried to outline a report structured in the same format as the lab reports that my incredible AP Chemistry teacher (shout out to Ms. Varoz) taught us to write in high school—complete with an abstract, introduction, methods, results, discussion of theory, and conclusion. I even gave it a grand title:

"Fractals, algorithms, and us: a case in practicing critical pedagogy in math education."

This was me overcompensating for my general lack of formal pedagogical knowledge, a fear that everyone would immediately recognize that I'm just some math grad student who has spent his professional life working in national laboratories, has only taken a single non-STEM course since high school, and has no place occupying the same space as trained pedagogy experts. The resulting write-up, of course, was something that felt entirely cold and dehumanized, which is completely antithetical to what I set out to do with this class in the first place.

In my day-to-day life as a working mathematician, I have to use big words all the time: homotopy equivalence, ℚ-acylic, semi-locally simply connected, and quasifibration, to name a few. While of course these words have their uses in conveying very precise meaning with respect to one another in the contexts of research, they have the added bonus of acting as a sort of linguistic veneer which signals to the people around me that, despite ongoing imposter feelings, I'm qualified to exist as a graduate student in the UChicago math department.

But, when writing about my teaching experiences, suddenly I don't have big words to hide behind anymore!

Appropriately enough, it was by reflecting on an experience from this summer that I was able to move past these imposter feelings (or at least some of them—as anyone with imposter syndrome will tell you, it's an ongoing battle). While discussing the political opinions and activities of Albert Einstein and other scientists in the context of nuclear weapons research, one of my students said something very astute:

"I feel like if he said that stuff today, people would tell him to stay in his lane. Like, that he's just a physicist and it's not his place to talk about it."

The name Einstein is synonymous with genius. As we go through school and generally exist in society, we often hear his name in the context of physics—relativity, the photoelectric effect, E=mc², and so on. Occasionally other details enter the popular zeitgeist: we might hear pleasant anecdotes of an aged Einstein pausing on his daily walk to help fix a child's bicycle or sharing a can of beans with someone seeking help with her homework, or tales of an Einstein speaking out against American racism, or occasionally more scandalous rumors of an Einstein who might've stole work from his first wife Mileva Marić and might've failed out of math classes in his youth (he definitely didn't do the latter, at least). But we rarely, if ever, hear about the Einstein who vocally opposed war, the Einstein who proudly held union membership in AFT Local 552, the Einstein who was staunchly anti-capitalist, or even just the Einstein whose diary contained jarring examples of Orientalism.

This passing comment in my class inspired a wonderful student-led conversation about intellectual labor, shared responsibility, and exactly who wins in a society where workers are defined by their jobs and simultaneously told that what they toil to produce doesn't belong to them. But the students in my classroom also made an explicit point: a lack of formal education in a thing should not invalidate the feelings and wisdom you glean from directly experiencing said thing. These narratives of "stay in your lane," when used by those in power, cow people from exercising agency over their lives. To my students, this conclusion came from the tale of a physicist—not a statesman or political scientist or military general "qualified" to have an opinion on war. But, upon reflection, they realized these were the same sort of feelings that many of them have associated with math, science, and education for much of their lives—we are made to feel stupid and powerless by people who ostensibly know more than us. After all, to see evidence of the apparently universal trauma we all share around math, all it takes is telling a stranger in nearly any social gathering my field of study: almost without fail, the response is either “Oh I hate math” or maybe “I’m so bad at math!”

In the same way, the practices of critical pedagogy did not begin when the modern giants like Friere and McLaren started using big words to describe it. Instead, as noted by bell hooks in the introduction of Teaching to Transgress: Education as the Practice of Freedom: "We learned early that our devotion to learning, to a life of the mind, was a counter-hegemonic act, a fundamental way to resist every strategy of white racist colonization. Though they did not define or articulate these practices in theoretical terms, my teachers were enacting a revolutionary pedagogy of resistance that was profoundly anticolonial." Like everything else, these ideas were part of the lived experiences of everyday people doing work long before they were studied by the modern-day academy.

So before we begin, let me be forthright: I am a mere mathematician and teacher of mathematicians, completely untrained in the more abstract and jargon-heavy aspects of pedagogy but equipped with almost a decade of experience in actually teaching. I'm not pretending to be all-knowing, especially when it comes to the millennia of shared human experience in passing on knowledge which I am only just learning about. I simply love my work and I love my students: my life has been hugely shaped by my experiences with both. I am also not a lone pioneer of these ideas by any means and do my best to attribute ideas to those who have come before me whenever possible.

As such, I present the following of my 2019 summer teaching both humbly and excitedly. Check back regularly as I will be using this blog to tell the stories of creating and conducting this course. I hope this medium will facilitate a more genuine sharing of the journey, free from the fears of my journal editors, their scathing emails, and wildly inaccessible paywalls.

If you are a teacher, STEM or otherwise, I hope that some of these anecdotes might inspire you to try something new, share your own experiences too, or maybe both and more. If you once were or still are a student, I hope that these stories can help to give you a new perspective on math in which you are empowered to create, rather than being inundated by formulas and shared trauma.

I have only my experiences to share. Join me, won't you?

Fractals, Algorithms, and Us

This summer I taught a six-week class, which I had designed over the spring, called Fractals, Algorithms, and Us. The students are currently CPS high schoolers—most are black and Latinx, roughly 80% are low-income, and about 60% will be first-generation college students. The original syllabus is here. Things had to be flexible because of the short timeline and lack of experience with the students, so we deviated from it a bit, but the course description remains true:

This course is designed around developing mathematical skills, with a focus on programming, while also taking seriously the cultural and social implications of mathematical work. We will design algorithms to solve equations, create and explore a variety of different fractals, and use neural networks to recognize images and generate art; simultaneously, we will discuss how mathematics combined with various forms of power have shaped the world, consider the ethical implications of artificial intelligence on human lives, and embark on a cross-cultural survey to reckon with the richly varied lived experiences of people doing math. 

This course will have a considerable component of in-class discussion and writing, as well as homework mostly conducted in Python, and will include a substantial final project.

I've had a lot of friends and fellow educators ask me to share the experiences of creating and facilitating this course, which I will do via this blog here and possibly by formal write-up sometime in the future. The planned outline is as follows:

• Preface: why the blog format?
• Teaching objectives: agency, power, and shared trauma in learning math.
• Building the syllabus: choosing readings and math that fit together.
• Discussion and reflection: talking about our feelings in a math class!
• Solving polynomials*: complex numbers, ancient algorithms, and wicked fractals.
• Intro to coding*: tying shoes, fun with Turtles, and sorting lists.
• Graphics and dynamics*: finding the beauty hiding in complex numbers with code.
• Indigenous math and colonialism*: the deeply human origins of math contrasted with post-colonial Eurocentrism, imperialism, and the myth of cultural neutrality.
• Neural networks*: cool math problems while peeling back the curtain on computers' assumed objectivity.
• Contemporary STEM*: gerrymandering, nuclear weapons, and marching for science.
• Final projects*: revisiting agency in math education.
• Postface*: evaluations, next steps, and math according to the students.

* = coming soon!