Counters, blocks, fraction tiles, bar models: making maths physical is one of the most popular ideas in primary classrooms — and one of the most misused. The research says representations genuinely help children understand abstract ideas, with an important catch: they only work when they're connected to the symbols and then deliberately let go of. Here's how to get the real benefit and avoid the traps.
A child has been told, for the third time, that to add a half and a quarter you find a common denominator. She can chant the rule and still gets it wrong, because to her ½ + ¼ is just symbols being pushed around. Then a parent tears a strip of paper, folds it in half, folds one half again into quarters, and lays the pieces side by side. Suddenly it's obvious: the half is two quarters, so a half and a quarter make three quarters. Nothing was re-explained. The maths simply became something she could see — and once seen, it couldn't be unseen.
That moment is what manipulatives and visual models are for. It's also why they're so beloved, so widely used, and — as the research quietly shows — so often used in ways that don't actually help. Making maths physical is powerful, but it isn't automatic magic. Whether a bag of colourful counters builds understanding or just fills a happy, mathematically empty half-hour comes down to a few principles most people are never told.
Every parent and teacher has seen it: a concept that wouldn't land as symbols suddenly lands when it's made concrete or drawn. Sharing biscuits to understand division. Stacking blocks to see place value. Drawing a "bar model" to untangle a word problem. The appeal is obvious and real. But the same classrooms also produce the opposite: children who happily play with the blocks and then can't do the sum without them, or who treat the manipulative as a fun toy entirely disconnected from the maths it was meant to illuminate. Same materials, completely different outcomes — and the difference is the whole story.
To an adult, "¾" or "3 × 4" carries a lifetime of meaning. To a child meeting it, a mathematical symbol is just a mark — a squiggle with rules attached. If we teach only at the level of symbols, we're asking children to manipulate things that, to them, mean nothing yet. Some cope by memorising the rules (which, as our piece on rote versus understanding explains, collapses later). Many simply flounder.
Imagine being taught to read sheet music purely as rules — "this dot means press this key" — without ever hearing the notes. You could follow instructions, but you'd have no feel for the music, and the moment a passage didn't match a memorised pattern you'd be lost. Teaching maths only in symbols is like that. Manipulatives and pictures are the "hearing the notes" stage: they let a child experience what the symbols actually mean before being asked to push them around on a page.
This is the gap representations fill: they give meaning a place to live before the abstract notation arrives. The famous question isn't whether to bridge from concrete to abstract — it's how, and how to make sure the child crosses the bridge rather than camping out on the near bank.
The intellectual root is Jerome Bruner (1966), who proposed that we grasp ideas in three ways: enactive (through action and objects), iconic (through images), and symbolic (through abstract symbols and language). Maths teaching built on this — Concrete–Pictorial–Abstract (CPA) — moves a child through physical objects, then pictures, then symbols, integrating each stage into the next so meaning carries forward. It's the engine of Singapore-style mastery and its celebrated "bar model".
Carbonneau, Marley and Selig (2013) pulled together 55 studies comparing maths taught with concrete manipulatives against maths taught with symbols alone. Manipulatives won — a moderate overall benefit. But the authors were careful: the effect depended heavily on how they were used. Benefits were larger with sustained, guided instruction and explicit links to the symbols; they were smaller — or absent — when manipulatives were used briefly, without guidance, or left disconnected from the formal maths. Manipulatives help; they are not a guarantee.
Because manipulatives look so obviously good, the caveats rarely get airtime — but they decide whether they work.
Pretty can be the enemy of useful. Research on "perceptually rich" materials finds that highly realistic, colourful, toy-like manipulatives can distract from the maths: the child plays with the cute apples and misses the structure. Plainer materials sometimes teach better precisely because there's less to pull attention away from the idea.
The bridge to symbols has to be built. A manipulative only helps if the child connects it to the formal maths. If the counters and the calculation live in separate worlds, the child learns to use counters, not to do maths. Every concrete step should be tied, out loud, to what it means in symbols.
You have to fade. The single biggest mistake is staying concrete too long. The goal is fluent, abstract maths; manipulatives are the route, not the destination. "Concreteness fading" — moving deliberately from objects to pictures to symbols — is what turns a nice hands-on moment into durable understanding. A child still reaching for blocks to do sums they should know is a sign the fade never happened.
Singapore made CPA famous and built the "bar model" into its curriculum — a pictorial tool that externalises a problem's structure so children reason about a diagram rather than juggling everything in their heads. It's representation used with discipline: always linked to symbols, always fading toward the abstract.
The Netherlands' "Realistic Mathematics Education" starts from meaningful, real-world situations and "models" (like the number line or bar) that gradually become more abstract tools for thinking. It's a different tradition from CPA but shares the core idea: meaning first, formal symbols as the destination.
England's "maths mastery" reforms imported CPA wholesale — concrete resources, bar models and careful progression to the abstract are now common in primary classrooms. The risk, as ever, is using manipulatives as decoration rather than with the guidance and fading the evidence requires.
Japanese lessons make heavy use of carefully chosen diagrams and a small number of well-understood representations, returned to repeatedly, rather than a cupboard of novelty resources. Depth with a few models beats breadth across many.
US classrooms use manipulatives widely, but with great variation in how well they're connected to the maths. It's the clearest illustration of the research's central caveat: the manipulative isn't the active ingredient — the guided link to the symbols, and the fade, is.
Use representations at every age, not just early years. Bar models, area diagrams and number lines illuminate fractions, ratio, algebra and beyond. Treat them as tools for thinking, not babyish props — and choose a small set of powerful models to use consistently rather than a drawer of novelties.
Plan the fade. Build concreteness fading into the sequence: concrete, then pictorial, then abstract, with explicit links at each step. Decide in advance when and how the support comes off, so students reach fluent symbolic work rather than getting stranded in the concrete.
Prefer plain over flashy. Given the evidence that perceptually rich materials can distract, favour simple, structured manipulatives (and clean diagrams) whose features map cleanly onto the mathematics. Cute isn't the goal; clear is.
The practice portal uses diagrams and visual models alongside the formal notation, so students see the meaning and the symbols together — the connection the research says matters most.
Open the practice portal →"Manipulatives automatically make maths easier and better."
They help on average, but only when used with guidance, linked to symbols and faded. Used badly, the benefit shrinks or disappears.
"The more colourful and fun the resource, the better."
Perceptually rich, toy-like materials can distract from the maths. Plain, structured resources often teach the idea more cleanly.
"Drawing and objects are babyish — older students should just use symbols."
Representations carry meaning at every level. Bar models and diagrams help with fractions, algebra and beyond — fade them, don't ban them.
A meta-analysis of 55 studies found teaching with concrete manipulatives beat symbols alone, with a moderate benefit — but the effect depends heavily on how they're used. Sustained, guided instruction with explicit links to the symbols helps; brief, unguided, disconnected use does little.
Rooted in Bruner's modes of representation, CPA teaches a concept first with physical objects, then pictures, then symbols, each stage scaffolding the next so the abstract notation carries meaning. It's central to Singapore-style mastery and its bar model.
No. Bar models, area diagrams and number lines illuminate fractions, ratio, algebra and beyond. The key is to fade them as understanding secures, rather than abandon them as "babyish".
Yes, if used poorly. Very colourful, toy-like materials can distract, and a child can grow reliant on the props without ever connecting them to the symbols. The remedy is concreteness fading — moving deliberately from objects to pictures to symbols.
Use everyday objects and quick sketches to make an abstract idea physical, always saying how it links to the symbols, then move toward doing it with symbols alone. Fold paper for fractions, draw a bar model for a word problem, then write the calculation.
A good tutor knows exactly when to make maths physical and when to let the props go. We find where your child is on that bridge in the free assessment.
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