The word polymath describes someone whose knowledge spans a wide array of subjects. The word comes from the Greek roots πολύς (polús, or many) and μανθάνω (manthanein, or knowledge). The word's appearance in English is quite curious when you think about it. Despite referring to someone who has a diverse set of knowledge, we use the word "math", a specific field with an often esoteric reputation.
So will it go with this project. Despite having "math" in the title, I will not limit myself to an investigation of just math pedagogy. Rather, I will use it as a starting point with which to frame my investigation of how to thoughtfully teach undergraduates and learn as an undergraduate in our modern era.
I will specifically ground my investigation in a math course I took in Fall 2024, titled Discrete Math (MA293) and taught by Debra Borkovitz. The course has two sections of 35 students each, has no prerequisites, and there are very few courses that have it as a prerequisite. In terms of the collegiate mathematical sequence, where courses build themselves off of calculus or differential equations or linear algebra, discrete math is kind of in its own separate boat.
Despite its relative independence, however, I think it is one of the courses that I have learned the most from so far in BU. The reason is pedagogy; I don't know exactly what I was expecting when I walked into class on the first day, but I certainly wasn't expecting my professor to come in with a big box of tiles and to challenge us to beat her in what she called the "tile game", a two-player competitive game that seemed like it had nothing to do with math.
Many of Prof. Borkovitz's prior students rave about her unique, tactile teaching style, saying that they've never taken a class where that kind of experiential learning is prioritized. When asked by one of her students when we'd actually start doing math, she simply held up a tile and said "this is math."
I was kind of used to experiential learning from my humanities classes, where little games or other exercises are often prioritized in the small class environment to get a point across more strongly.
Two things in particular surprised me, though.
The first was the unique way Prof. Borkovitz's teaching style made us students work. The math class environment I was used to was very passive; we were simply given the relevant formulas and taught how to apply them. In Prof. Borkovitz's class, we had to essentially figure out the formulas ourselves, wrapped in these more accessible tactile games.
The second was how equalizing the class environment was, especially at the start. Through the lens of the tile, or some of the other mathematical objects she brought to class which grounded the concepts we learned about, everyone could think about math separated from both the literal notation and the intimidating aspect of what math means to each student.
I also just really admired Prof. Borkovitz's willingness to welcome both people and what most would consider non-mathematical things into math. In math, we usually separate ourselves from others, more often talking about how this mathematical concept can be used "in practice" as part of some other field. Rarely are others actually brought into math.
But as I'm writing this, something else is bringing itself into math—I'm using the general definition of "any scholarly field" now—and not with permission. Of course, we cannot discuss pedagogy in the year 2025 without discussing artificial intelligence. In fact, for a while, I was thinking of naming my whole project "On AI and Pedagogy". However, much like AI's current role in most classrooms, I didn't want to center my investigation on AI. Just like an AI disclaimer or policy at the bottom of a syllabus or when a professor gives "the talk" when grades on the last exam were the lowest they've ever been, I want AI to appear naturally throughout the course of my investigation. I don't need to draw attention to it; I'm sure it'll come up plenty.
If you think about it, AI is a type of equalizer too. It literally takes text from across the internet and merges it into a sort of homogenous paste it draws from to complete prompts, choosing the next word as it probabilistically decides on a response. However, unlike the tile, AI strips the work out of the tasks students need to do in class.
But what is the importance of "actually doing the work" in school? What does that even mean? Why can't students just sit in lectures learning formulas? Do other approaches have a meaningful impact, or do they just feel more satisfying in the end? What does it mean to have a meaningful impact?
These are some of the foundational questions I want to start from in my investigation of math and pedagogy. Consider my narrative above a sort of preliminary excerpt to preface my final work, which will be a multi-chapter text strongly reminiscent of Descartes' MeditationsCC201Core Humanities 3: Renaissance, Rediscovery, and Reformation. I envision this text being at the intersection of journalism, philosophy, and narrative. I hope to interview both professors and students who have had many different pedagogical experiences, particularly looking closely at how modern-day shake-ups like AI have affected their experiences. Ultimately, I want these interviews as well as close reflections of my own experiences to serve as a place where I can start from as close to no assumptions as possible and build up a sort of axiomatic set of ideas relating to teaching and learning, much in the vein of Descartes.
As for other Core texts, there is a lot that I plan to draw from. Of course, as a sort of philosophical text as the final product, I will likely draw from all of the philosophers that we read in the humanities sequence. In particular, I hope to write short dialogues and narratives that accentuate the argument that I build throughout my text. However, I am also very interested in depictions of both teachers and learners in some of the texts we read. I think these depictions will both give us a better understanding of some of the historical elements of pedagogy as well as give us archetypes to draw from and make comparisons with.
For example, one of my favorite learners in any piece of literature I've read is Sansón Carrasco from Don QuixoteCC201Core Humanities 3: Renaissance, Rediscovery, and Reformation, since he is perhaps the embodiment of a surface-level analyst. In Part Two, he happily describes the passive experience of reading Part One of Don Quixote due to its strict conformance to Catholic norms. I've thought for a while that Carrasco's behavior reminds me a lot of a LLM. He's certainly not stupid, and he ends up being really useful in stopping Quixote's delusions (to me, the way Cervantes distinguishes intelligence and effectiveness is one of the brilliant things about his character writing), but he seems to think one word at a time.
Overall, I think there is a lot of potential in this project, and I think it's really ambitious. The major effort I'll make to temper my ambitions is that I won't place a certain expectation on the number of chapters or number of people I interview. Much like the MeditationsCC201Core Humanities 3: Renaissance, Rediscovery, and Reformation, I will end my work when it feels right.
I'm honestly very excited at the prospect of this project. In many ways, I see myself as a polymath: I participate in both the teaching and learning aspects of collegiate education, I am a math & CS student but not too heavily involved in AI, and I take a wide and even range of different courses, always balancing two STEM and two humanities courses a semester.
I hope my project will end up as not just a personal reflection of these aspects of my personality but as stronger food for thought about what it means to teach and learn in an era when everything we know about pedagogy feels like it's changing.