PBL: Peoples’ Math and Not-Knowing Math

Yesterday, I had my first meeting with colleagues preparing for the new school year. I am still happily entrenched in the pace of the summer (the only tolerable pace right now, as NYC sits in the middle of a 5-day heat advisory), so while a meeting like this has that potential to upend one’s vibe, this did nothing of the sort. Just a change of scenery and a nice little ramble through the Village for a burrito afterwards.

My colleagues and I are coming off of a wonderful summit for PBL-style mathematics, where we got LOTS of great ideas from far-flung teachers doing exciting and inspiring work. It took me a while to recognize it, but Problem-Based Learning answers a number of difficult questions I have always had about the meaning and purpose of math education, every since my earliest days as a public school teacher and grad student. It was only after tossing about for two trimesters at my new placement at a progressive school this past year, that I finally started to pay attention to some of my fellow math and science teachers who were engaged in this approach, and recognized its potential to help student actively engage and create mathematics.

What is PBL? Here’s the definition that Carmel Schettino, the summit organizer offers:

“An approach to curriculum and pedagogy where student learning and content material are (co)-constructed by students and teachers through mostly contextually-based problems in a discussion-based classroom where student voice, experience, and prior knowledge are valued in a non-hierarchical environment..” (Schettino, 2013)

The big words for me here are “co-constructed” and “non-hierarchical”. The first addresses the idea that to be actively engaged in math is to create math, and that this is accomplished mostly in community. The second addresses the idea that math is not separate from us, and rather than being accepted and practiced on the basis of external authority, it is developed by (and in turn develops) our own insight, autonomy, and capacity to make sense of the world. Creating this experience for students is exactly what math teaching is about, what I’ve talked about on this blog often, and what I’ve created in the most fleeting moments you can imagine in my classrooms.

What does this look like? Students work on problem sets in groups and present their solutions to one another. These problem sets are the “text” of the class, but they differ from a typical math textbook. The problems are not organized by type of content and no procedures or theorems or samples are offered to address the problems. Instead, these problems are meant to call upon what the students know already, and are authored and arranged so as to guide the students, step by step, through the interconnected web of mathematics. The teacher’s role is purely as facilitator; significantly, her or she does not provide correct answers. Process is emphasized over product in homework and in writing about what they’re doing. There are tests, but they are one of many different forms of assessment — and they are also (ideally) recognized as opportunities to learn (imagine that!).

There is a lot of struggle, but also a lot of student ownership, when this kind of thing works. In my experience, students will come up with approaches to solutions that I did not anticipate, and they grow more comfortable seeing my approach as one way to come at a problem, rather than the “right” way. This kind of problem-based work — open-ended but with a definite end goal, group-oriented but generally relying on individual difference and participation — most closely resembles the kind of work our students will be engaged in in any field they choose in their professional lives, not too mention their postsecondary training to get there. I speak often of my time in engineering school, where it was clear that group work was necessary to stay afloat, but everyone still had to pull their own weight, working on problems that a step-by-step procedure simply could not resolve. It was jarring and disorienting to find this paradigm dominant in my college, after a very different experience in high school. Obviously college is more challenging than high school, but there is nothing about the pedagogical approach that should be saved for college. Rather, it is best to foster this method of study early, giving students autonomy and building the competency in a more authentic way. This is the basis for the entire progressive education movement.

And indeed as I got a closer look at PBL this year in my junior Precalculus class, I began to call it People’s Math. There is a little irony to this, since I work in a independent school which by its nature is elite. But the school comes out of the progressive political tradition of the last century and is committed to a progressive pedagogy. That means different things to different teachers, but for me it means empowering students to own, create, and critique mathematical thinking for themselves. I call it Peoples’ Math, because it draws on a common mathematical capacity that can be cultivated in all persons — and because the capacity it cultivates will benefit the broadest set of persons for everyday mathematical thinking, an essential component for a functioning democratic society. While People’s Math recognizes the presence of experts, and gives their analyses due respect, it does not cede the mathematical lens to them.

This is math for the People, yes.

Universal education is an unprecedented development in the history of the world, but its purpose is not fulfilled just by bringing bodies into buildings. It’s by actually educating universally. In this context, abstraction and specialized language are not necessarily the goal of math education — they are tools, to be applied skillfully, when the time is right, in order to best produce mathematical thinkers who are both self-sufficient and also comfortable in community. The long goal is to produce a society full of people who can reason mathematically together. This is the promise of universal education, and it is also crucial that at this time in our history, we engineer a society that is ready to take on unprecedented challenges.

In fact, the whole class experience that I intend to create is an immersion in small-scale but still unprecedented challenge — I will ask my students to tolerate a level of not-knowing that most adults would not tolerate if they had a choice. Most students and their families understand math as an objective body of content that can be communicated clearly, memorized, and then presented in some rote manner that is open to easy, quantifiable measure. PBL, with its talk of co-creating math in a non-hierarchical environment, throws a wrench in that whole scheme. If you’ve been really good at math and have good strategies in place for a different kind of math class, suddenly you’re standing on unstable ground: there are no textbook examples, no answers in the back of the book, and no procedure to memorize. If you’ve been really bad at math, you suddenly find yourself called upon to engage with this unpleasant content in a whole different way (though these types of students have the most to gain from this approach, in my estimation). Either way, it’s a significant upending of what students usually expect — and the harder they hold on to what they’ve come to expect, the harder a time they have (not unlike life…).

In Zen, the term not-knowing is used to convey a quality of openness and presence that is free from preconceived ideas. It calls upon an authentic response, rather that a reflexive reaction. It is not the absence of knowledge per se, but rather a flexible stance and freedom of motion and expression that comes before the application of knowledge. It is not an idea, but it is the ground from which ideas can grow. It is more given to heuristic rather than procedure and to listening rather than talking. In a Western cast, it might be called wisdom — a quality rarely mentioned in education policy and debate, but a worthy goal of our schooling.

Above, I used the term not-knowing as something to be tolerated, and that sense still holds somewhat true — because not-knowing is a little scary. It is challenging to dispense with one’s pre-existing assumptions, because they are how we navigate the world and protect ourselves from danger. And while it might seem ridiculous to consider the possibilities of danger in a math class, I invite you to see if you can remember a time that you failed to do something expected of you in a math class at any level. What feelings were attached to that? I bet they were pretty strong. It is not easy to open ourselves to strong feelings.

However, when not-knowing is skillfully and persistently cultivated, students can start to let go of their fixed ideas about how math should go and how good or bad they should be at math. If that happens, even for a moment, space is made for insight. When cultivated carefully and encouraged, these moments become more frequent. And suddenly you don’t have students doing math anymore, you have students mathing math. Or, to achieve complete Zen saturation, you have: math mathing math! I imagine this as the goal of a PBL class — creating a space for the creation and expression of math that flows naturally and authentically from the students, as if they were writing poems or singing songs. And it starts with the ability to tolerate, protect, and sustain an open stance of not-knowing.


In this way I think PBL, People’s Math, can also be called Not-Knowing Math. This inches into the realm of wisdom disciplines, akin to philosophy, the humanities, and even to theology — where it should be! Math knowledge and content, so often taken to be the whole of math, become accompanied by a mathematical stance, a pattern-seeking predisposition that playfully and pointedly bounces around a group. Letting go of who’s good and who’s bad at math, and in an environment that supports and encourages persistent effort — beauty, truth, and the good can all be created spontaneously.

Here is Bertrand Russell from Education and the Good Life. What I call the mathematical stance he calls the “scientific temper” — and he suggests a middle way utilizing both mind (“intellectual culture”) and heart (the opposite of “emotional atrophy” which I would call aliveness) that is none other than the attentive openness of not-knowing. He says:

Neither acquiescence in skepticism nor acquiescence in dogma is what education should produce.  What is should produce is a belief that knowledge is attainable in a measure, though with difficulty; that much of what passes for knowledge at any given time is likely to be more or less mistaken, but that the mistakes can be rectified by care and industry.  In acting upon our beliefs, we should be very cautious where a small error would mean disaster; nevertheless it is upon our beliefs that we must act.  This state of mind is rather difficult: it requires a high degree of intellectual culture without emotional atrophy.  But though difficult it is not impossible; it is in fact the scientific temper.  Knowledge, like other good things, is difficult, but not impossible; the dogmatists forget the difficulty, the skeptic denies the possibility.  Both are mistaken, and their errors, when wide-spread, produce social disaster.

Speaking of social disaster, I also call this approach Math for the End of the World — an end precipitated by our societal inability to recognize patterns and work together effectively. (I suppose Math for the Anthropocene, or Math for the Climate Crisis would work as well, but it’s best not to be coy about the end of the world.) I’ll have to write more about that another time.

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