Our Story

Two children have stayed with me across fifteen years of teaching.

The first was behind. Not by a little — significantly, visibly, in the way that everyone in the classroom knew including her. She had gaps that had compounded quietly over years, each one making the next concept harder to reach, until mathematics felt like a foreign country she had no map for and no hope of entering. The work she was given had been simplified to the point of condescension. She could see what was happening. Children always can. She had stopped trying not because she lacked ability, but because no one had ever shown her the structure underneath the confusion — the part that would have made everything make sense.

The second was ahead. Significantly, visibly, in the way that also made the classroom an uncomfortable place to be. He finished the work early. He was given more of the same work. He learned, quickly, that being capable in mathematics meant being punished with repetition — more worksheets, more of what he already understood, never anything that asked him to go deeper or further or to genuinely wonder. He didn't stop trying either. But he stopped expecting mathematics to surprise him. He stopped expecting it to be worth his full attention.

One of these children might be yours. This is what I built Linear Abacus for.

Both of these children were being failed. By the same system, for the same reason: a curriculum built for the middle, moving at one pace, in one direction, with no language for what genuine mathematical understanding actually looks like — or how to build it in a child who is nowhere near the middle.

I built Linear Abacus for both of them.

I grew up in a family where mathematics was not a subject. It was a conversation — one that never really ended. My sister is a mathematician. Around our table, the question was never can you do this but do you see why this works, do you see where it comes from, do you see how beautiful this is. We talked about the history of mathematics the way other families talk about history or literature — with genuine curiosity about how ideas evolved, how they were discovered, how imperfect and human and fascinating the story was.

Mathematics, to us, was never a set of rules. It was a continuous story — one that is still being written, still unfolding, still full of questions worth asking. Place value alone — the Hindu-Arabic numeral system, base ten, the centuries of human endeavour it took to arrive at what we now hand to a six-year-old as obvious — is one of the great chapters in that story. When you know where something comes from, it looks completely different. Numbers are not arbitrary. They are answers to questions that human beings spent a very long time asking.

I wanted every child I taught to feel that. Not just the ones who arrived already curious. All of them.

For fifteen years I looked for a framework that could make the experience of genuine mathematical understanding teachable — that could give a parent or a teacher the language to open that conversation with any child, regardless of where they were starting from. Something that could meet the child who was behind and the child who was ahead, because both of them needed the same thing: not more practice at the surface, but access to the structure underneath.

I found it in a university staffroom.

Andrew Waywood was not trying to build a product or advance a career. He was doing what he has always done: thinking deeply, for the love of it, about how mathematical understanding actually develops. He had identified something no one else had named — the genuine grammatical structure of the number system. Not a new teaching strategy. Not a curriculum framework. The actual architecture of how numbers relate to each other, made visible and teachable for the first time.

We talked about mathematics the way neither of us had talked about it with anyone in years. Not for work. For the pleasure of it. Two people who believed that understanding was the point — not performance, not procedure, not the result on a page — finally in the same room.

Andrew once told me he really enjoys thinking deeply. He means it in a way most people in education have forgotten is possible. He is not chasing recognition or position. He wants to give something real to the world, and then get back to the thinking. In fifteen years working in education at every level — classroom teacher, university lecturer, curriculum writer, consultant — I had met very few people who meant that without reservation.

I meant it too. I still do. I have faced scrutiny for thinking differently, for refusing to simplify what deserves to be taken seriously, for insisting that children are capable of far more than most curricula ask of them. I would do it again.

What we built together — the Linear Abacus, the books, the lessons, the methodology we call TALK DO WRITE — is our answer to both of those children. The one who was behind and had been told, quietly and persistently, that mathematics simply wasn't for her. And the one who was ahead and had been told, just as quietly, that there was nowhere further to go.

There is always somewhere further to go. That is one of the most beautiful things about mathematics.

If you are homeschooling a child who has fallen behind — who carries some version of the belief that they are simply not a maths person — this is for you. And if you are homeschooling a child who finished the curriculum in half the time and is hungry for something with more depth, more structure, more genuine challenge — this is equally for you.

The Linear Abacus does not have a ceiling. It has a foundation. And from a real foundation, every child can build further than anyone expected — including them.

Genovieve Grouios MEd
Founder, Linear Abacus