How We Learn Maths

About twenty thousand years.

That's how long it took for us humans to get from counting on our fingers to inventing a positional decimal system—a compact and elegant way of representing any number in the universe using just ten digits. It's astonishing. And then we expect second graders to leap right into this system as if it's obvious.

This is one of the most complex, ingenious tools our species has ever developed. It took whole civilizations, religious debates, astronomical observations, and philosophical revolutions. Take infinitesimals, for instance. There was a time when suggesting that a quantity could be smaller than anything we can see but still not quite zero was considered not just illogical—but heretical. Mathematicians and philosophers debated whether God would allow such quantities to exist. Leibniz and Newton risked ridicule and damnation to give us calculus. In a way, mathematics has always sat at the intersection of mystery, order, and power.

And yet, we reduce it to mechanical drills.

We rob it of awe. Of imagination. Of meaning. And then we wonder why children fear it.

We do something that makes no sense at all: we present math not as a way of seeing, measuring, modeling, and understanding the world—but as a cold, brittle sequence of symbolic manipulations. We ask kids to stare at those symbols and divine their meaning before they've had a chance to experience the ideas they stand for.

Of course they feel lost. Of course many develop math phobia. We are asking them to run before they've even learned what walking is for.

Math is beautiful. But more than that, it is a way of attending to the world. It allows us to zoom in, zoom out, and find structure in apparent chaos. We can track how a seed grows into a tree, how resources are shared, how long a journey will take, how something changes over time. It's the closest we've come to a universal language of thought.

But it only becomes meaningful when it is rooted in experience. That's where we begin.

Before anything else, children need to feel math. We mean that quite literally.

We start not with numbers, but with things. Pebbles, lids, seed pods. Countable, graspable, movable things that allow the mind and the hand to connect. The first step to understanding "three" is not seeing the numeral 3. It's seeing three tomatoes, three pebbles, three turns on the swing. And noticing something common across all of them.

That's what we mean when we say "three-ness." It's an idea, not a symbol. A universal abstraction that emerges from the particulars of experience.

Last week, we sat with Anika as she arranged a collection of pebbles in the garden. She placed them in groups of three, counted them, rearranged them into groups of four, and counted again. Her brow furrowed when she noticed something interesting. That the numbers were the same even if the groups were different.

What Anika discovered through play—the concept of conservation—took mathematicians centuries to formalize. She was exploring an idea that ancient civilizations puzzled over, that Piaget studied intensively, that forms the foundation of algebraic thinking. And she found it herself, in the garden, with a handful of stones.

That's also why we don't rush into formal notation. We hold back. We wait for the concept to become vivid before the symbol enters. Because if you begin with the symbol, you're asking them to learn the map without seeing the territory.

It's hard to overstate how much damage meaningless math problems have done.

"Eva picked 23 flowers. She gave 8 to her friend. How many does she have left?"

Who is Eva? Why is she picking flowers? Who gives away 8 flowers anyway?

Instead of using math to make sense of the world, we've turned it into a strange, disconnected game. But math is already all around them—if we show them how to see it.

Take the elevator puzzle we explored in Math Lab. A girl walks up a few floors and then takes the lift the rest of the way. Why? The kids come up with dozens of ideas—playful, surprising, logical, emotional. And somewhere along the way, they're comparing distances, estimating floors, calculating possibilities. They're doing math. Deep math.

But they're also doing something else. They're using math to make sense of a human story. To figure out a hidden motivation. They are discovering that numbers are not just about quantities—they are clues.

Math becomes not a requirement, but a revelation.

When we set the table for lunch, we're working with one-to-one correspondence—one plate for each person, one glass for each plate.

When we divide fruit for snack time, we're exploring fractions—how do we share three apples equally among six children?

When we track the growth of plants in our garden, we're gathering data, measuring change, representing findings visually.

Last month, our older children became fascinated with building a marble run. What began as play evolved into deep mathematical exploration. They tested angles to find the perfect slope for different speeds. They measured lengths to ensure connections would work. They predicted where marbles would fall and adjusted their design accordingly.

No one told them they were "doing math." They were solving problems that mattered to them, using mathematical thinking as a natural tool. The concepts of angle, velocity, prediction, and spatial reasoning weren't isolated lessons but interconnected parts of a meaningful project.

This is where it gets exciting.

Once the basic concepts are grounded, we invite kids to stretch them. We play board games where resource tracking and probability sneak in as part of strategy. We make our own games. We create puzzles with elevators, with shadows, with strange machines and made-up rules. They play Settlers of Catan and design villages. They learn what it means to optimize without even hearing the word. They tinker with ratios, angles, speed, and fairness.

The goal is not speed. It's not even correctness. It's playfulness. The willingness to explore a question just because it's interesting. Because it opens a door.

In a conventional classroom, the answer is everything. At Comini, the answer is the least interesting part.

Dev sprawled on the floor with a collection of colored tiles, creating an intricate pattern. "Look," he said, pointing to his design. "It's like the Fibonacci sequence, but with colors."

Dev had discovered the famous number pattern weeks earlier while exploring spirals in nature. Now he was representing it visually, creating his own mathematical expression. This wasn't assigned—it emerged from his fascination with patterns.

Mathematics begins with play. With arrangement and rearrangement. With noticing what happens when things change. This playful exploration builds the foundations for later, more formal mathematical understanding.

Our spaces are filled with materials that invite mathematical play: pattern blocks, tangrams, counting materials, measuring tools, games that involve strategy, probability, and logical thinking. These aren't labeled as "math manipulatives." They're simply part of our environment, available for exploration alongside art materials, books, and natural objects.

One of the most corrosive habits in math instruction is teaching procedures before purpose.

"Borrow from the tens place."

"Carry the one."

"Flip the inequality."

Why?

Math is full of "why." Why do we need to add things? Why do we use base 10? Why do two negatives make a positive? These are not trivial questions. They are essential.

Children ask them naturally. And when we take the time to explore them—really explore them—we're not delaying their learning. We're deepening it.

We see this over and over again: kids struggle with the idea of place value. Or that numbers can form a pattern when laid out in grids. And yet it becomes second nature once they have spent time seeing how we can bunch up things in bundles of tens for convenience. And we give them the space to absorb this at their own pace.

This transformation from abstract to concrete isn't just a teaching technique. It reflects how mathematical understanding actually develops. Neuroscience shows that mathematical concepts are grounded in physical experience—our brains map numbers to space, connect operations to movement, understand relationships through embodied experience.

There's a strange obsession with efficiency in math education.

But what if the goal isn't speed? What if the point is to stay with the question? To wander, to circle, to wonder? To see what else it might be?

We've seen it in Math Lab. A group stuck on a puzzle for days. Ideas fly. Guesses crash. Someone leaves and returns with a theory. A breakthrough happens not in the moment of instruction but on the walk to the bathroom or while fiddling with LEGO bricks. And when it clicks—their faces light up.

That's learning. Not the filling of blanks, but the moment when a new idea becomes your own.

Eventually, when introduced to standard units like grams, Leela had a rich foundation of experience to connect these abstractions to. The symbols had meaning because they represented relationships she had physically explored.

Math isn't confined to textbooks. It's in the floor tiles, in the beats of a song, in the arc of a thrown ball.

We measure time and distance and price and ingredients. We estimate, we compare, we notice patterns. A child clapping out a rhythm is counting without knowing it. A child stacking blocks is intuiting geometry.

We don't need to create artificial math problems. We just need to pay attention to the real ones.

"Where do we see math in the real world?" I asked our group during morning circle.

"In shops when we pay for things," offered Rohan.

"When we measure ingredients for cooking," added Meera.

"In music, when we count beats," said Arjun, air-drumming to demonstrate.

And they're right. Math isn't a separate subject confined to textbooks. It's everywhere—in the rhythm of our days, in the shapes of our world, in the transactions and transformations that happen constantly around us.

We do use curriculum maps. They help us track what a child has explored and what lies ahead.

But we never confuse the map for the territory.

Too many children are taught to follow the map—lesson by lesson, worksheet by worksheet—without ever realizing what it is a map of. The result? They know how to perform operations, but they don't know what they mean. They know formulas, but not what they describe.

We've seen 9-year-olds do long division who cannot tell you whether their answer is too big or too small. Because they haven’t actually developed an interior model for numbers. They're following instructions.

The world doesn't need instruction-followers. It needs sense-makers. Pattern-finders. Problem-framers.

So yes, we track progress. But we track it in context. With actual understanding. And with the deep belief that not all journeys follow the same route or pace.

This respect for natural progression doesn't mean we never challenge children or introduce new concepts. It means we do so in response to their readiness, building on their current understanding rather than imposing an arbitrary sequence from external standards.

The result? Children who see mathematics as a tool they can use, not a performance they must give. Children who approach new problems with confidence rather than anxiety. Children who understand that mathematical thinking isn't about being "smart" or "dumb" but about developing strategies that work.

"Does alternative schooling mean alternative careers?" One parent had asked us that question, and I think it captures so many concerns and anxieties beautifully. Implicit here is a concern: Yes, these mindless drills are boredom-inducing and rob joy, but how else would one master subjects needed for exams? And math presents this challenge the best.

It's a real question, with layers of worry beneath it.

If we let go of drill-based math practice, are we giving up on our children's ability to perform in exams, pursue engineering, become "successful"?

Here's our answer: mastery doesn't come from drills. It comes from depth, from time spent in meaningful exploration. It comes from curiosity and confidence and the ability to return to a challenge again and again. It comes from real understanding—not from test-cramming.

And yes, the kids who spend years learning math with joy, with wonder, with persistence—they will do just fine in exams. More than fine. Because they won't panic. They won't freeze. They'll know what the symbols mean, and what lies beneath them. They will have the resilience to try, fail, and try again.

Math is not a gatekeeper. It's a gateway.

So much of what we do in our Math Lab is hard to describe.

There's laughter, music, arguments, experiments. Someone is rolling dice. Someone else is drawing a spiral. Another child is trying to build a perfectly symmetrical city in a sandbox.

And yet, somehow, it's all math.

It is pattern, precision, possibility. It is logic with wonder. It is a way of holding the world in your hands and asking—what else might be true?

Perhaps most importantly, children develop identities as mathematical thinkers. They see mathematics not as something that happens to them but as something they do—a perspective they create, a language they speak, a tool they use to understand their world.

Learning is a whole-body, whole-being experience. It's physical. It's emotional. It's social. It's messy. And when we ignore that, we don't just make learning harder. We make it harmful.

We start with meaning and work toward abstraction, not the other way around.

That's what we want them to feel.

Not fear.

Not failure.

Just the simple, world-altering joy of discovering how numbers—ancient, stubborn, miraculous—can help us make sense of everything.