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How Singapore math uses the CPA (Concrete-Pictorial-Abstract) approach
The Concrete-Pictorial-Abstract (CPA) approach is a three-stage method for teaching math. Students first manipulate physical objects (concrete), then move to drawings and visual representations (pictorial), and finally work with numbers and symbols alone (abstract). Adapted from American psychologist Jerome Bruner's theory of representation, Singapore added CPA to its math curriculum in 1981, and it remains one of the curriculum's foundational pedagogical approaches.
Watch "Singapore math methodology, concrete-pictorial-abstract (CPA) in action"
You already know CPA, you just didn't know it had a name
Every child goes through a version of CPA learning language, whether or not they ever encounter Singapore math. A baby points at a cookie and reaches for it (concrete). A toddler can point to a picture of a cookie and say the word (pictorial). Eventually, a child can say "I want a cookie" with no cookie or picture in sight at all (abstract). Nobody teaches a baby to skip straight to the word, the stages happen in order, built on each other.
Math works the same way. Handing a child an abstract equation before they've built the concrete and pictorial understanding underneath it asks them to memorize a symbol with no meaning attached, which is exactly the kind of rote learning Singapore's curriculum is designed to avoid.
CPA in action, two examples
Understanding a number. Concrete, count 5 physical objects. Pictorial, count 5 pictures of the same object. Abstract, recognize the symbol "5" represents that same quantity.
Addition. Concrete, physically join 5 blocks and 3 blocks into a pile of 8. Pictorial, draw a bar split into a section of 5 and a section of 3. Abstract, write the equation 5 + 3 = 8.
The pictorial stage, specifically bar modeling, is where Singapore's approach goes further than most curricula. Instead of moving through pictorial representation quickly on the way to abstract equations, Singapore deliberately prolongs it for years, using bar models as a visual bridge all the way through upper elementary. (Full breakdown, What is bar modeling?)
The most common misuse of CPA, and how Singapore math actually does it
Bruner's original theory described three systems of representation, enactive, iconic, and symbolic. What most parent-facing explanations of CPA leave out is that he never framed these as a checklist to complete in order. Researchers who later studied how that transition actually works gave it a name, concreteness fading, a gradual and explicit shift from concrete materials to abstract symbols, not a hard handoff between three separate stages.
That's exactly how Singapore math uses CPA, and it's also where most CPA explanations get it wrong, whether they're describing a Singapore classroom, a UK mastery maths lesson, or a workbook explainer online. The common misuse is running a child through concrete work until it's "done," switching entirely to pictures, then switching entirely to numbers, as if each stage graduates them out of the last one. CPA is not a strict, one-directional sequence, concrete then pictorial then abstract, done. Kids don't have to fully graduate from concrete before touching pictorial or abstract representations. Singapore's approach uses all three together, reinforcing each other, especially while a concept is still forming. A child working through an addition problem might touch blocks, draw a quick bar, and write the equation within the same few minutes, moving back and forth between representations instead of marching through them once.
Manipulatives follow the same logic. They're not a phase kids age out of. If a manipulative helps a student see the math, there's no grade level where that stops being useful. Plenty of upper-elementary and even algebra-level concepts benefit from a concrete or pictorial pass before going fully abstract, precisely because the concrete-to-abstract link has to be rebuilt, explicitly, every time a concept is genuinely new.
The mistake that stalls kids at the manipulatives stage
Here's a very common pattern. A child can solve a problem with base-ten blocks, but the moment the blocks disappear, they're lost. Parents often read this as a manipulatives problem or an understanding problem. It's usually neither, it's a mapping problem.
Take subtraction with regrouping, 23 − 8. A student can break a ten into ten ones, combine it with the three ones, and subtract 8 using blocks. But if the connection between what the student just did with the blocks and what the standard algorithm is doing on paper is never made explicit, the two representations stay disconnected in the child's head. This is exactly what concreteness fading is designed to prevent. Every action taken with concrete materials has to be deliberately mapped to the matching step in the pictorial and abstract representations. That link doesn't form automatically just because a child used manipulatives at some point, and it doesn't form if the stages are taught as three separate units instead of one continuously fading thread.
Frequently asked questions
What does CPA stand for in math?
Concrete-Pictorial-Abstract, a three-stage teaching approach where students move from physical objects, to drawings, to abstract numbers and symbols.
Who created the CPA approach?
It is adapted from American psychologist Jerome Bruner's theory of representation (enactive, iconic, symbolic). Singapore was among the first education systems to integrate it into a national math curriculum at scale, adding it in 1981.
What is concreteness fading, and how is it different from CPA?
Concreteness fading is the research term for how CPA is supposed to work in practice. It describes a gradual, explicit transition from concrete to abstract representations, not three separate stages taught in sequence. CPA names the three types of representation. Concreteness fading describes the process of moving between them correctly.
Is CPA unique to Singapore math?
No, the underlying theory is used broadly, including in UK mastery maths classrooms. What is distinctly Singapore's is pairing CPA with a deliberately prolonged pictorial stage through bar modeling, and applying all three representations as one continuous, overlapping process rather than three steps completed in order.
Is CPA only for young children?
No. The concrete and pictorial stages remain useful at any grade level when a concept is genuinely new or difficult. CPA isn't a phase kids age out of, it's a tool that applies whenever understanding needs to be built from the ground up.
Why can my child solve problems with manipulatives but not without them?
Usually the connection between the concrete action and the abstract procedure was never made explicit. Every step done with physical objects needs to be deliberately linked to the matching step in the written method.
Written by Wenxi Lee. Wenxi Lee is the founder of Singapore Math Circle. She grew up in Singapore and went through 12 years of the curriculum that helped Singapore rank #1 in the world. She is a trained mathematician from Washington University in St. Louis and holds a PhD in math education from the University of Illinois at Chicago.