What da Vinci Knew
That School Forgot

There is a moment that many homeschooling parents describe, usually a little sheepishly, as if admitting to something they shouldn’t feel proud of. It goes something like this: their child is stuck on something — really stuck, not just distracted — and instead of reaching for a worksheet or opening an app, the parent sits down beside them. They pick up the pencil together. They talk it through. And somewhere in that conversation, something shifts. The child gets it. Not because they were told the answer. Because they found their way there.

Most parents brush this off as a lucky afternoon. But it isn’t luck. It’s one of the oldest forms of learning there is.

Around 1466, a teenager named Leonardo arrived at a master craftsman’s workshop in Florence with no formal schooling and an insatiable curiosity about how things worked. Over the next decade, that teenager would become Leonardo da Vinci — not through lectures or textbooks, but through exactly that kind of moment, repeated daily: sitting beside a master, picking up the tools, talking through what he observed, wondering, questioning everything he saw, and finding his way to understanding.

Da Vinci didn’t learn mathematics in isolation. He learned it by watching how water moved, how a human arm bore weight, how light softened at the edge of shadow. His mathematical understanding grew from genuine curiosity about the world — and from the conversations those observations sparked with people who cared what he thought.

I was nine years old when I drew Galileo Galilei sketching the moon. I showed my father, half proud, half uncertain. He could have said “that’s lovely” and moved on. Instead, he saw something in what I’d drawn — not just the picture, but the curiosity behind it — and he opened a door.

He explained who Galileo was. He connected the telescope to what we could actually see that very night. Later that week, we went outside together and looked up. We talked about the universe for what felt like hours — about the people throughout history who had looked at the same sky and asked the same questions I was asking.

My father didn’t teach me astronomy that night. He made me feel that astronomy was mine to explore. That my questions were worth following. That the connection between a drawing, a telescope, and the night sky wasn’t a coincidence — it was an invitation.

Mathematics, in our house, wasn’t a subject. It was woven into conversations about history, art, and the natural world. I didn’t know at the time that this was unusual. I only understood it much later, after fifteen years in classrooms and lecture halls and curriculum meetings, watching children be handed the symbols before they’d ever felt what those symbols meant.

My father gave me what da Vinci’s master gave him: time to wonder, someone to wonder alongside, and the confidence that the wondering itself was the point.

Across the developed world, families are making a quiet but significant choice. They are deciding that the education their children deserve looks more like that kitchen table conversation than like a classroom of thirty.

To understand what’s driving this, I recently sat down with three people who have spent decades inside the homeschooling world: Dr. Brian Ray, president of the National Home Education Research Institute in the United States; Jamie Heston, board member of the Homeschool Association of California; and Tina Smith, treasurer of Australia’s Home Education Association.

What they described wasn’t a rejection of education. It was a return to something older and, the research suggests, considerably more effective.

To understand what homeschooling families have quietly preserved, you first have to understand what institutional schooling gave up — not through malice, but through the genuine demands of scale.

It traded time for coverage. Curriculum documents list dozens of topics per year level. Teachers must move forward whether children are ready or not. The bell rings. The next topic begins.

Tina Smith, who homeschooled her own children and now supports hundreds of families through the HEA, describes this with characteristic directness: “With homeschooling, you wait until the child learns. Some children need to do an activity a hundred times. Some only need to do it once. You don’t move on until the foundation is there — because without the foundation, everything you build on top of it will have weaknesses.”

It traded conversation for efficiency. Walk into most mathematics classrooms and you’ll find silent seatwork — children completing worksheets individually while a teacher circulates. Mathematical dialogue, the kind that actually builds understanding, has been quietly replaced by procedures to execute alone.

Jamie Heston has seen this from both sides — as a homeschooling parent and as someone who consults with families making the transition. “Parents don’t look at their children and imagine them as empty vessels to pour knowledge into,” she says. “They’re not sitting at the front of the kitchen table, teaching. They’re working alongside them. Facilitating. Bringing resources and exploring them together.”

It traded concrete experience for abstract symbols. We rush children to the notation — the numbers, the operations, the algorithms — before they’ve built the felt sense of what those symbols represent. Then we wonder why the procedures dissolve so quickly, and why so many children conclude that they are simply “not maths people.”

And — perhaps most critically — it never developed a grammar for mathematics.

Here’s something worth sitting with for a moment: mathematics has a grammar, just like language does.

When your child learned to speak, they didn’t memorise a rulebook. They absorbed the structure of language through conversation, stories, songs, and time. They felt how sentences worked — how meaning built and connected — before they could explain a single grammatical rule. The grammar was always there, underneath everything. They just came to understand it naturally, through use.

Mathematics works the same way. There are deep patterns that connect everything — from counting to place value to fractions to algebra. Patterns that a child can feel and explore long before they can name them. A child who genuinely understands why ten ones become one ten, and can feel that relationship in their hands, is already doing algebraic thinking. They just don’t know it yet.

For reading, we have phonics — a grammar that makes the relationship between sounds and symbols explicit, so that parents and teachers can guide children through it systematically. It changed literacy outcomes dramatically once we had it.

For mathematics, we had nothing equivalent. Until now.

Dr. Andrew Waywood spent twenty years identifying the systematic structures that connect mathematical concepts from the earliest counting all the way through to advanced algebra. Not new teaching strategies. Not a better curriculum. The actual grammar of how numbers relate to each other — made visible, teachable, and explorable for the first time.

When I first understood what Andrew had discovered, something clicked for me that I’d been circling around for fifteen years of teaching. The reason so many children struggle with mathematics isn’t that they aren’t capable. It’s that we’ve been asking them to speak a language whose grammar we never made visible.

Here is the fascinating thing about the homeschooling families I’ve spoken with and worked alongside: they haven’t invented new pedagogies. They’ve preserved the old ones. The ones that have always worked.

Dr. Ray describes watching homeschooling families at work, and the image he reaches for is unexpectedly beautiful: “Mom is moving around like a bee in springtime — going from child to child, talking. Sometimes she’ll have two or three sitting together, reading aloud and discussing. Conversation, conversation, conversation. That’s what I wrote down before we spoke. That’s what successful home educators do.”

He describes how he taught his own children science — not through formal lessons, but through daily life. “I just ate and breathed science with them. We talked about it constantly — in the car, on road trips, working in the garden. Fermentation, decomposition, bacteria, viruses. I didn’t do formal teaching. I just talked, and they talked back.”

This is how da Vinci learned. This is how I learned astronomy from my father. This is, if you examine it honestly, how you learned most of what you truly know — not from a worksheet, but from a conversation that mattered, with someone who cared what you thought.

Tina’s children learned place value and multiplication through Lego — building cities, trading blocks of different colours, negotiating exchanges. Mathematics wasn’t a subject in her home. It was a tool for navigating a world they were genuinely exploring.

Jamie’s children learned number through card games, board games, and the slow accumulation of real-life situations where quantities mattered. “You’re using a variety of resources — software, videos, documentaries, museums, libraries, co-ops, group classes,” she says. “Nobody’s sitting at a kitchen table teaching everything. You’re facilitating. You’re bringing the world to them and exploring it together.”

These aren’t radical innovations. They’re timeless human principles. And homeschooling families prove daily that they still work.

Here is where the story takes an unexpected turn — and where it gets genuinely interesting.

Research consistently shows that home-educated children outperform their institutionally schooled peers across nearly every academic domain. But there is one significant exception: mathematics.

When I raised this with Dr. Ray, he was candid. With Tina, it sparked an observation I’ve heard echoed by families across Australia: “Some parents hated maths themselves — and they don’t realise how much maths they use in their adult life. They’re aware of their weakness and they seek help. Or they leave it to a co-op because they don’t feel confident with the subject.”

Jamie sees it in California too — gifted, resourceful, deeply engaged parents who will read history aloud with their children for an hour, then hand them a worksheet for mathematics.

Why? Because for every other subject, their instincts are enough. They know how to have a conversation about a novel. They know how to explore a science question. But mathematics — without its grammar made visible — doesn’t yield to conversation in the same obvious way. It feels like it requires a different kind of expertise that most parents don’t have and can’t quickly acquire.

This is the paradox at the heart of homeschooling’s relationship with mathematics. And it points directly to what both homeschooling families and schools are missing: not better intentions, not more resources, not more time — but the grammatical framework that makes mathematical conversation possible in the first place.

The families I’ve spoken with — and the fifteen years I spent inside classrooms, lecture halls, and curriculum meetings — all point to the same conclusion. The principles that make homeschooling successful everywhere else can work for mathematics too. They just need one thing to anchor them: a grammar that makes the invisible structure of numbers visible, explorable, and conversational.

That’s what we set out to build.