photo by Phil Roeder

The Real Schedule and the Shadow Schedule
People often ask me “how do kids learn math?” But they’re missing the better question of how children learn at all…

When people ask me what I do for work I say I work at a school where we let children do whatever they want all day. Typically I get a lot of follow-up questions.

It’s on purpose because I’m a fan of questions, which is necessary when the work you do is at a school for self-directed learners. ALC-NYC, where I am officially a facilitator and librarian, is an independent K-12 school in East Harlem, New York. There, anywhere from one-to-three-dozen young people shape their own learning, in community with 3-4 Agile Learning Facilitators and a handful of volunteers, in a creative, cooperative space.

It’s a place where magic happens. I’ve been working at ALC-NYC for more than half a decade, after a lifetime of passionate rage against the educational machine, and in my professional experience, I’ve learned that the kinds of questions people ask reveal more about their values and assumptions than their statements do. Which is why I offer new folks the most flippant description of my job, and let them show me their cards in surprise.

The question at the root of most of the questions I get about my job is How do we educate children so they grow to be capable and well-adjusted adults? It is itself a valid, complicated, contentious subject throughout much of human history.

It’s a question of endless constructions and endless possible answers, because “capable and well-adjusted adults” is a broad and largely meaningless phrase, stripped of context. Capable of what? Well-adjusted to which conditions? Do you mean “adults who know their place in the hierarchy of capitalism and are willing and able to fill that role”? Or do you mean “adults who are prepared to be in good relations with their own selves, others, and the Earth, despite the current conditions, so we might build a better world”?

Pedagogically and politically, I occupy the latter position. It’s led me to a lot more interesting questions about what it means to be a person in a time of upheaval and uncovering. And yet, despite the incredible breadth of possible queries about my job, there’s one that somehow, always comes up; my sworn enemy, which I am here today to put a rest to:

But how do they learn math?

Math on the Real Schedule

At ALC-NYC, I co-facilitate info nights for prospective parents, where but how do they learn math is often paired with because I know my kid, and they’ll just play all day as if their kid is unique and that is not the whole point – that learning is play, and play is learning.

Not everyone comes to this conclusion, though, and few of us start here, and some parents never arrive at all, which is also okay. As far as opening questions go, this one makes sense, given how much focus US schools put on teaching and testing mathematical literacy.

Our school’s population is a diverse mix of families, most of whom do not come to Agile Learning Center because they’re longtime self-directed learners. Most families that find us do so because they have a child who is actively suffering in the school system, and they’re looking for a way out. Whether or not they already believe that play should be at the center of a good education, like I do, many of them are brave enough to take the leap of faith that has them sitting on the mismatched couches in the back room at ALC-NYC, looking around at all the weird art and piles of books and kid detritus, and wondering if they’re doing the right thing for their young person’s future.

In this context, I let my co-facilitator Ryan answer, because he’s the one who runs our Maths offering. We call our classes “offerings” because that’s exactly what they are: an (always-opt-out-able) offer to do a thing, learn something together, or otherwise connect at a scheduled time. They can be run by young people, facilitators, or folks from our wider community, and can happen just once, or repeat as frequently as there’s interest. Maths w/ Ryan, as it goes on the schedule, happens twice a week for an hour and has for the entire time I’ve worked at ALC-NYC. At parent interest nights, Ryan tells a compelling, true story about its providence:

I started the offering because I never really liked math but, in the early days of the school, I decided I wanted to model making a new relationship with an academic subject that I didn’t like. So, because I was playing fantasy baseball, I started working through the statistics unit on the online math curriculum that we recommend – Khan Academy – and I started to like it a lot more as I got better, learning at my own pace alongside the kids doing their own work. Over the years I’ve finished stats and something like 30% of Kahn’s total math curriculum – though these days a lot more kids have been coming to Maths and so I spend most of my time supporting them, or doing the logic puzzles one of our teens has been bringing in.

Ryan has endless patience with the young people who come to his Maths offering: working through problems with them, delighting in logic puzzles, and encouraging them to teach each other. He helps them get comfortable with clicking the “submit” button when they’re not certain the answer is right, and he helps them work out where they went wrong — even if he doesn’t yet know the answer himself. It’s the stuff that Ryan always does, with children, because he’s a skilled and compassionate facilitator. Kids go to Maths with Ryan, and they like it because kids like Ryan; he respects and supports them.

This reassuring answer to how do kids learn math becomes even more soothing, accompanied by the offer Ryan makes with it: If you want, I’m happy to help parents and young people to create family agreements that make everyone feel comfortable with the degree of freedom available, here at ALC-NYC.

A lot of parents take him up on it; we usually have at least one student with a family agreement that they will attend Maths x number of times a week, in exchange for the privilege of self-directing their education at our school.

We won’t chase down your child, the facilitators remind parents in these situations, and we won’t police your child’s choices if they choose to break their agreement with you. But, we will remind them that breaking an agreement has consequences. I, personally, believe more learning happens around power dynamics than mathematics in this kind of situation, but that’s not necessarily a bad thing. Understanding learning as something separate from schooling is a process that takes time. I’m lucky to work on a facilitation team with such clear boundaries, and I’m glad the kids have access to Maths with Ryan.

All of this is a nice, tidy answer to this question haunting me, but it is ultimately unsatisfying. Plenty of children attend ALC and never once go to Maths and yet they still learn math. What’s that about?

Math on the Grand Scale of Things

It fascinates me how many adults assume that if mathematics wasn’t required, no person would choose to learn it – as though there is no inherent satisfaction to uncovering this language of the universe? As though space and time and weight and measure and prediction and force and observation are not elemental? As though no musician, no engineer, no witness to the fractal beauty of nature ever wondered how the universe figures?

Why does mathematics get such a short shift? I’ll let you in on a secret: it has nothing to do with the content of the discipline and everything to do with how we teach it.

The Common Core State Standards, which are designed to “provide clear and consistent learning goals to help prepare students for college, career, and life” recognize two learning domains for K-8th grade students: Mathematics and English Language Arts/Literacy. According to the standards written by the US federal government and adopted by forty of fifty US states, any other area of study – the sciences, the fine arts, history, personal or interpersonal knowledge, just to name a few – is considered less important than math and literacy. Here, we have clear evidence of an academic hierarchy.

I want to be clear: I am not opposed to math and language literacy. I merely want to point out the existence of this hierarchy – that different subjects are considered more or less valuable not because they have more or less utility for the person who is trying to succeed at their specific college, career, or life goals. They’re considered more or less valuable objectively, regardless of the needs, skills, goals, or desires of the person whose life is spent learning them.

Mathematical literacy is one of many skills I want the children I work with to have available to them, so they can navigate the world when they are adults. But I don’t for a second believe that knowing how to solve a quadratic equation is an indicator of how prepared a young person is to survive this adult world – where I am writing this essay in the middle of a pandemic, as we approach a million deaths and the US empire shows its true eugenicist colors and threatens war, as climate change threatens the very islands I live and work on.

Regardless of the real, material conditions of the year 2022 – not to mention what’s to come in the 2030s, 40s and 50s, when these children will be living as adults and, presumably, you and I will be, too – contemporary academic systems, like ours in the US, prioritize abstract mathematics at the top of an intellectual hierarchy. It’s not because they’re the most useful; it’s because they draw on the forces of white, western patriarchy which associate math with the masculine, and the masculine with the superior.

People who are good at math, the thinking goes, will be good with money, and have access to “better” jobs, and will therefore be more poised to succeed in capitalism. Which means when we’re asking “but how do they learn math?” we’re by-proxy asking “but how will they get access to a place at the top of the ladder?” Simultaneously, we are punishing those who don’t succeed at math, often by taking away free choice and unscheduled time, in order to drill more rote memorization. We must punish children, the logic goes, because the world will punish adults who fail to acceptably acquire these skills. A world these young people haven’t even had the autonomy to interact with, beyond the bubble of the institution. It’s no wonder that so many kids who struggle with math class think they’re stupid – a terrible tragedy for the many brains and brillances unmeasurable by any standardized test.

As we teach math with hierarchy and punishment, hierarchy and punishment is what young people are learning. We can see the results of those systems all around us. Invariably, some young people are rising to the top of that hierarchy, and succeeding in demonstrating the skills our institutions deem most valuable. Also invariably, some curious and brilliant humans are having that curiosity crushed out of them. When it comes to our survival – our ability to build a better world than this one – I fear the continued repercussions of both.

Math is not inherently linear, rigid, or hierarchical. Our schooling system has made it thus, with skill at math sorting children into their place in a hierarchy that mirrors the hierarchy of the culture at large. Math becomes the bludgeon of the system, and it corrals us into place. As long as we teach it this way, it will be an unpleasant subject to learn, for most people.

It happened to me – school beat me down, with math, and even after all these years of deschooling I’m still grappling with it.

My struggle started nearly two decades ago, in middle school, when I got a concussion playing with my brother. After hitting my head, I struggled with numbers, and during the months of figuring out what was happening with me, I got demoted from advanced math to a regular, grade-standard class.

Shouldn’t be a big deal, right? Except that changing math classes meant that I had to change all my classes, and lose the support of my friends there, and that I was left to explain to a new group of kids why my schedule had changed and no one else’s had, and surrounded by new teachers in the middle of the semester, who didn’t know me either. It was the first time I’d ever struggled academically, and I felt a lot of shame. Making new friends was hard. I didn’t trust my new teachers. I felt stupid. It was a bad time. So I quit.

From that point in my academic career on, every time I was introduced to a new math topic I didn’t immediately understand, I shut down. I stopped even trying to understand and started trying to prove that I didn’t care about it. I understood enough to skate by, and I was content with that. It was a project that lasted years, into high school, when I finally got my hands on the graduation requirements and did the calculation I was most excited about: how soon could I drop math class?

The answer was much, much sooner than the adults around me had suggested – after taking the final state test required for my diploma, my second year of high school, I could be free of it. After I took that test, I never took another math class – not once, in six years of high school and college. I didn’t have to. Instead, I took computer science, which fulfilled the lingering credit requirements, and was marginally more useful and interesting to me.

The first Agile root is learning is natural and happening all the time. It’s at play in my conventional-school-story, at ALC, and everywhere young people are interacting with the world.

Just because there was a class called “math” that I was scheduled to attend daily in middle and high school, doesn’t mean that’s what I learned. Far from it. My teen self may not have been interested in calculus, but they did learn how to read an institution’s fine print and advocate for themselves – useful skills for any young person. They’ve served me well, navigating bureaucracies. I have no regrets. I may still learn calculus. Time is long, and I don’t know what I’ll be called to do.

Learning is natural and happening all the time, even when it’s not the learning the schedule says should be happening now. Content and container are not separable. It’s as true at ALC as it was in my conventional high school, because it’s true in all spaces where young people are: learning to be, in relationship with other humans, is always happening, and it is the foundation on which all other kinds of learning happens.

When those relationships are deeply hierarchical, full of comparison to peers and scrutiny from authority, dictated by a schedule you have no control over, evaluated by impersonal standardized tests, and enforced by punitive and carceral punishment, you learn a specific set of lessons. It’s not just my experience of math; all those parents asking but how do they learn math? are leaving out the implied because I was forced to, and it was a bad experience for me, and I don’t know how to reconcile this woundedness with the assertion that my child will feel differently, given the space and support.

But when you learn in community with people of all different skills, roles, ages, knowledges, and curiosities, with a calendar that is adaptable and emergent, with accountability practices rooted in transformative justice, learning just looks different. We learn because we are human, and curiosity is our condition. How we support young people in their learning will determine what kind of future they have the skills to shape.

Math on the Shadow Schedule

There is, of course, a real schedule at ALC-NYC which we create together, every Monday morning at a mandatory, all-school meeting called Set-the-Week. This is where Ryan’s Maths offering resides, and plenty others. It’s abundantly true that scheduled learning happens in ALC-land. But the magic – the learning that takes over spontaneously, in the relationship space of organic, self-directed community – happens on the shadow schedule.

I got the term from a friend and fellow facilitator, Amber, of Rivers and Roads ALC in Oklahoma City, on a call that started as a general support call and, over the years, became a call where we, and one more trusted ALF friend, just supported each other. She was talking about the making of fortune tellers – all the folded paper geometry and fine-motor glory of that hallowed childhood activity – which swept the entire community, despite not being on the schedule at all. When you work in Self-Directed Education for long enough, you come to realize that the shadow schedule is where we’re doing the most learning. Which is why when people ask me how do they learn math? I don’t talk about Ryan’s offering. Instead, I tell a story.

There is a craft/fabric store around the corner from ALC-NYC that I frequent with children – locally owned, a little dusty, and full of treasures. My first year facilitating, I often went with a trio of young people who loved the beads and baubles section. They were elementary-school-aged – between six and eight – and, on this particular day, they had a lot of coins between the three of them. I forget what they wanted.

Do we have enough money? they asked me.

I’m not sure, I said. Let’s see.

We crouched down in the back corner of the fabric store and dumped out all their change on the floor. I helped them sort the coins by type, and name them, and figure out how much each was worth. Then, I watched as they helped each other count. I didn’t rush them and I didn’t step in to intervene when they lost count and had to restart. I put my body between them and any adults who might interrupt, so they could use all their energy to focus on this learning. I created the scaffold and held the space for them to figure it out, together.

Back in college, I worked in a conventional elementary school; I remember math lessons with plastic coins, doled out so carefully, and the workbook problems associated with them.

This was the same, I realized with a jolt, crouched by the yarn display. This was a developmentally appropriate math lesson. You can look it up in the Common Core Grade 2 Math standards, under Measurement and Data: “Work with time and money.”

Except we were doing it now, not because the curriculum mandated it, but because these young people asked me for this lesson. They desired this knowledge in order to navigate the world.

I’ll never forget how triumphant they were when they figured out they had enough.

The question of how children learn math mistakes the forest for the trees. We all need the whole forest, and yet we are all drawn to different trees, at different moments, as we have different needs. Learning is natural and math –the study of quantity, quality, structure, shape, space, and change; a powerful way of understanding our universe – is not inherently more important or complicated than any other subject; it’s just one tree in the forest of knowledge.

Hierarchical math does not help people develop the curious, wondrous, flexible, systems-thinking that makes mathematics such a valuable discipline to begin with. But math, in relationship? Math on the shadow schedule? That math opens up new, fractal worlds – for young people and for grown-ups, too.

I learned about fractals my first year at ALC-NYC when a mischievous 9-year-old fellow Virgo, who reminded me strongly of my younger self, called me over to the grey couch in the back room to watch Vi Hart on Youtube with her.

“Say you’re in math class,” begins Vi in the popular, three-part series Doodling in Math: Spirals, Fibonacci, and Being a Plant, “and your teacher is talking about... well who knows what your teacher is talking about. Probably a good time to start doodling.” (What a mood.)

Vi doodles spirals and finds a fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34... With a pinecone or a pineapple or a sunflower, you can see over and over again how nature plays with this pattern; Vi traces them with glitter glue.

“It seems pretty cool and wondrous, but the cool thing about the Fibonnaci series and spirals is not that it’s this big complicated mystical magical super math thing beyond comprehension of our puny human minds that shows up mysteriously everywhere,” explains Vi. “We’ll find that these numbers aren’t weird at all. In fact, it would be weird if they weren’t there. The cool thing about it is that these incredibly intricate patterns can result from utterly simple beginnings.”

I won’t go beat-for-beat through the whole thing, because the videos are not that long, and free on the internet and worth watching (the end leaves me teary-eyed every time, even now, years after I first cried, on the couch, in the back room, with Mischievous Virgo). Through three videos – not even 45 minutes, shorter than my middle school math class – Vi traces how sequences like Fibonnaci are related to the most irrational number – phi – and dips into biology and physics to explain how hormone distribution results in the abundance of spirals all around us.

The point? All the fractals we can see are the result of a natural, irrational pattern of growth, each new thing immeasurable except in relation to what’s already there.

“Once it gets started, it’s a self-perpetuating cycle,” explains Vi. “All that these flower bits are doing is growing where there’s the most room for them. The rest happens auto-mathically.”

I met Mischievous Virgo when she was nine and we learned together for five years. In that time, she attended Ryan’s Maths offering weekly and studied with a tutor, and worked through Khan Academy, and completed standardized test-prep work on her own – the visible parts, on the real schedule, not the most enthusiastically but steadily, because she had an agreement with her parents and she was the kind of young person who kept her agreements.

But on the shadow schedule? She did math all the time. She taught me about fractals and baked prolifically. She priced plane tickets and poured over fundraisers. She took part in the invention of several student currencies and at least one Minecraft bribery scandal. She mastered slime ratios. She played with physics on ice skates and rock climbing walls. She built a set of stairs for her elderly dog, planning, plotting, and cutting the wood, joining it at careful angles. She got as good at the kind of quick mental math that makes things go smoothly at the D&D table as she was at finding the logic puzzles that make a session delightful. She worked steadily at mastering the advanced time-math of logisticing the NYC public transit system – how can we go and come back on a limited number of subway swipes, how many blocks is it, how many trains is it, how far to the bus, how much buffer time must we leave ourselves to make it on time to meet our friends at the Museum of Natural History, or the Queens Hall of Science, or the Museum of Math, where we went one rainy Wednesday and rode impossible tricycles with square wheels around a ring of floor with curved petals like a flower.

This litany doesn’t cover a fraction of all the learning I witnessed her doing, and little of what she taught me. I can only wonder how it measures up to what she thinks she learned, and what her future self will need to know. Only time will tell. There’s growing to be done, still.

“How we are at the small scale is how we are at the large scale,” teaches adrienne maree brown in her Emergent Strategy1 explanation of fractals. “The patterns of the universe repeat at scale. There is a structural echo that suggests two things: one, that there are shapes and patterns fundamental to our universe, and two, that what we practice at a small scale can reverberate to the largest scale.”

How do children learn math? The same way we learn anything, dear one: in relationship.

We can infuse those relationships with rigid and dehumanizing hierarchy or we can do it a different way – by practicing curiosity, in relationship with curious others, and witness how those relationships create a fractal pattern that mirrors every fractal in nature, growing along the trellis of scheduled and unscheduled time, branching out by the most irrational number, changing the world as we, ourselves, change.

[1] brown, adrienne maree. Emergent Strategy. AK Press, 2017.

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