Concrete-Pictorial-Abstract, Explained
The Singapore math sequence: objects first, then pictures, then numbers.
Concrete-Pictorial-Abstract, usually shortened to CPA, is the teaching sequence most people mean when they say Singapore math. It’s a way of introducing every new idea in three stages rather than starting with the symbols.
Five math programs on this site use it.
The three stages
Concrete. The child works with physical objects. To understand that four groups of three make twelve, they build four groups of three counters and count them.
Pictorial. The objects are replaced with drawings that stand for them. The child now works with pictures of groups rather than the groups themselves, holding the same idea one step further from the table.
Abstract. Only now do the symbols arrive. The child writes 4 x 3 = 12, and the notation means something because they’ve already built and drawn it.
Every new concept runs through the same three stages, which is why CPA programs move more slowly through early material and cover fewer topics per year than a traditional spiral program.
Bar models
The most recognizable piece of Singapore math is the bar model, a rectangle drawn to represent a quantity. Two bars side by side show a comparison. A bar split into parts shows a whole and its pieces.
Bar models are the pictorial stage doing its job on word problems. A problem that would otherwise need algebra becomes visible: draw what you know, see what’s missing. Parents who didn’t learn math this way often find bar models the hardest part to teach, because it represents a problem in a different way from the one they grew up with.
Who it tends to suit
Children who need to see why something works before they’ll accept it. Owners consistently report that CPA builds unusually durable number sense, and that children who come through it handle mental math and word problems well.
It suits less well a family that wants to move fast, or a child who is already comfortable with symbols and finds the manipulative stage tedious. It also asks something of the parent. Four of the five programs here score 4 or 5 on parent involvement, and the pictorial stage in particular is hard to hand to a child alone.
The other thing owners report: the pacing feels slow in the early grades and then catches up. Programs that spend first grade on place value tend to arrive at fractions better prepared.
Where it shows up on this site
Dimensions Math (Singapore Math) is the direct descendant of the Singapore approach and the most rigorous of the group, spanning PK-8.
RightStart Mathematics leans hardest on the concrete stage, built around an abacus and card games rather than worksheets. It is one of only three math programs on this site scoring 5 on parent involvement.
Math with Confidence is the gentlest entry point, scripted for the parent and paced for a typical child rather than an accelerated one.
Math Mammoth uses the sequence in a workbook a child can largely work through alone, which makes it the outlier in the group at 2 on parent involvement.
Apologia Math applies the sequence inside a faith-integrated elementary program.
What it is not
CPA isn’t the same as being manipulative-driven. Manipulatives are the first stage, and the method is defined by moving deliberately through all three. A program that stays on blocks without pushing toward pictures and symbols isn’t doing CPA, and neither is one that hands a child a worksheet on day one.
It’s also not the same as mastery, though the two usually travel together. CPA describes how one concept is introduced. Mastery describes how the year is organized. Dimensions Math and Math Mammoth are both, while RightStart Mathematics, Math with Confidence, and Apologia Math use CPA within a blended structure.
And Singapore math isn’t one product. The name refers to an approach used in Singapore’s national curriculum, and several publishers sell versions of it. Dimensions Math is the one most directly connected to the original.
Read this alongside the rest
If you’re choosing a math program from scratch, how to choose a homeschool math curriculum puts this question in order with the others. See how our scoring works for what each attribute means, and browse Math to compare these five directly.