She Already Knew What “Concrete” Meant — We Just Had to Prove It Works Online

She was ready to quit. Then the stars on the screen made sense.

This article expands on a private letter sent to our SingMath parents. Read the original letter here.

The most common worry I hear from parents considering online math tutoring is not about scheduling or cost. It is a deeper, more philosophical concern — one that tells me the parent has done real research. It goes something like this: “How can you teach the concrete math phase online? Doesn’t the CPA approach require hands-on materials?” It is a legitimate question. And the answer, I believe, changes how we think about what learning actually is.

The CPA approach — Concrete, Pictorial, Abstract — is the architectural spine of Singapore Math. Every concept is introduced first at the Concrete level, then represented Pictorially, and only then expressed in Abstract symbols. The sequence is not optional. Skip the first stage and the later stages have no foundation. Parents who understand this, who know that the C is the load-bearing wall of the structure, are right to wonder whether a screen can bear that load.

I want to argue, carefully and with evidence, that it can. But only if we first correct one widely held misunderstanding about what “Concrete” actually means in this framework — and what it was never meant to mean at all.

The Hidden Assumption Inside “Concrete Math Online”

When most people hear “Concrete-phase learning,” they picture a child gripping a set of colored counting cubes. The physical sensation — the weight in the hand, the click of block on block — feels like the point. It feels irreplaceable. And so a screen, with its frictionless glass surface and its two-dimensional images, seems like a fundamental downgrade.

But here is what Jerome Bruner, the developmental psychologist who formalized this progression in the 1960s, was actually describing. He called the first stage “enactive” — from the Latin for “to act upon.” His core claim was that children must act upon a representation of an idea before they can hold that idea abstractly. The action creates the internal structure. The physical object was simply the most practical vehicle for that action available to a mid-century classroom teacher. It was a means, not the destination.

Here is the revealing truth about physical manipulatives in undirected use: a child can handle a counting block, stack it, sort it by color, and return it to the bin — all without ever achieving genuine mental contact with the numerical concept that block was meant to represent. The hands were engaged. The mind, quite possibly, was elsewhere. Physical touching is a delivery mechanism for cognitive engagement. It is not the same thing as cognitive engagement.

What we are actually trying to produce, in the Concrete phase, is a vivid, navigable, tactile-feeling mental model — a structure the child can return to and reason from when the abstract symbol on the page stops making sense. That mental model can be built just as effectively, and in certain ways more precisely, through a rich visual object on a screen, when the teacher knows exactly what she is doing with it.

Number bonds made visual: decomposing 12 into 7 and 5, with stars as the anchor objects.

How We Build the Concrete Phase in Our Virtual Classroom

Our approach to the Concrete phase in a virtual small-group session rests on three deliberate layers. I want to name them clearly, because each one serves the same function a physical manipulative would — and I would argue they serve it with greater precision, because each element is designed by a teacher rather than handed to a child and left to chance.

Layer One: The Anchor Object. We do not use abstract shapes or generic icons. We use objects that children already have a rich, sensory, lived relationship with — apples, fish, stars, groups of children. When we place fifteen illustrated stars on a shared digital worksheet, we are not asking a child to imagine something neutral. We are invoking something their visual memory already holds in full color and detail. This is the first act of making mathematics feel real.

Layer Two: The Enactive Act. The child is then asked to do something to those objects. Circle a group. Draw a line between them. Drag a set across the screen. Move their cursor to point. This is where the cognitive engagement Bruner described actually occurs. The child’s hand — mediated by a mouse or trackpad — is performing an intentional, directed operation on a mathematical relationship. That operation creates the internal structure we are after. This is the moment I watch for in our sessions. The slight pause before the circle is drawn. The quiet that tells you the mind has arrived at the problem.

Layer Three: The Language Bridge. Only after the child has acted do we name what they did. We introduce the mathematical term — “number bond,” “part and whole,” “groups of” — as a label for an experience that already exists inside them. In our context, a number bond is the relationship between a whole and its two parts: the idea that 12 is always made of 7 and 5 (among other combinations), and that knowing this lets you calculate flexibly rather than count on your fingers. When we name it, the name lands on lived experience. It does not land on empty air.

The Small-Group Advantage: Concrete Learning Is Not a Solo Act

There is one more piece to this that I rarely see discussed, and it may be the most distinctive advantage of our virtual small-group format. Our sessions are capped at six students. That limit is not a logistical convenience — it is a pedagogical decision made precisely for moments like these.

When a child in our session draws her circle around a group of stars and says something aloud — even something tentative, even something that turns out to be slightly wrong — the four or five other children in that small group are watching and listening. They are encountering that idea through a peer’s eyes, not just through a teacher’s instruction. This peer enaction — watching someone else act upon a mathematical idea and narrate it — has its own powerful effect on how the idea settles into the minds of the observers. The Concrete phase, in our classroom, happens six times simultaneously, at six slightly different angles.

Bar modeling on a shared screen: Concrete thinking moving into Pictorial in real time, with peers watching.

Why the Transition to Pictorial Is Smoother Online

One unexpected dividend of building the Concrete phase on rich screen-based visuals is that the transition into the Pictorial phase — where we move to drawn representations like bar models — is almost seamless. There is no perceptual rupture. In a physical-manipulatives classroom, a child must make a conceptual leap from a three-dimensional block to a two-dimensional bar drawn on paper. That leap requires cognitive bridging. It is manageable, but it is a jump.

In our virtual space, the illustrated stars were already on the same screen where the bar model will appear. The medium is continuous. The child who has been acting on visual objects on a shared digital surface moves to bar modeling on that same surface without switching worlds. The Concrete and Pictorial phases breathe into each other, and the bar model — which represents a quantity as a rectangular bar subdivided into its parts — arrives not as an abstraction but as a natural compression of the visual work already done.

The girl whose mother nearly pulled her out of class — she is still with us. In a recent session, working on decomposing numbers with number bonds, she arranged a set of stars into groups, drew her circles, and then said, spontaneously, to the five other children in her group: “It looks like a flower.” She was describing the visual structure of a number bond diagram — a central circle with two parts branching off it — and she was right. She had built a picture inside her mind. That picture will be there when the abstract symbols arrive. It will be there in second grade, and third, and when she gets to fractions, it will still be there.

That is concrete learning. It happened on a screen, in a small group, through a shared digital worksheet. And it was every bit as real as anything a child can hold in her hand.

With deep respect for every parent who asks the right questions,

Leeling

Director of Education & Academic Standards
SingMath Tutoring, home of Singapore Math Classes & Camps