How the Concrete Pictorial Abstract (CPA) model can support advanced numeracy

Many secondary mathematics teachers quietly assume that manipulatives, concrete resources, visual models and hands-on tasks belong only in primary classrooms, where children are still making sense of numbers. By the time students reach the secondary stage, the thinking is that they should be working directly with symbols, equations and formal notation.

While understandable, this assumption limits students – particularly those who have spent years memorising procedures they have never really understood. The Concrete Pictorial Abstract (CPA) approach is a pedagogical principle rooted in how humans build understanding – and it’s just as powerful at secondary level as in primary.

What is the Concrete Pictorial Abstract approach?

The Concrete Pictorial Abstract (CPA) model describes three interconnected stages through which learners develop mathematical understanding:

1. ‘Concrete’ – students handle physical objects: counters, algebra tiles, fraction bars. The point is not the object itself but what it does – it gives a concept weight and texture before any symbol appears on the page. Learners build meaning through touching and using the objects.
2. ‘Pictorial’ – the physical experience is expanded as an image: a bar model, number lines or diagrams. This bridges the gap between the tangible and the symbolic. Students carry a visual representation in their minds that they can return to when abstract notation becomes slippery.
3. ‘Abstract’ – formal symbols, equations and algebraic notation; the language of mathematics as it appears in textbooks and exams.

Where does the CPA model come from?

The CPA approach has its roots in the work of Jerome Bruner, an American psychologist who argued that understanding always begins with doing. In this framework, learners first make sense of new ideas through physical action, then through images and visual representation, and finally through formal symbols and language. He insisted that this sequence applies regardless of age, and there is no point at which students outgrow the necessity of starting the learning process with the concrete.

This framework was adopted and refined through the development of Singapore’s mathematics curriculum, which has consistently placed at or near the top of international rankings since the 1990s. Singapore Maths made the CPA model central to how all students – not just the younger ones – encounter new mathematical ideas. Its influence on the Mathematics Mastery movement in England means that many primary teachers are already working within this tradition. The challenge is to carry it forward into secondary.

Why abstract-only instruction often falls short

Ask your students to expand a pair of brackets and many will do so accurately. But if you ask them why this works, silence often follows. Abstract-only instruction creates this gap – students can perform a procedure through a memorised set of steps, but are unable to explain, adapt or apply it to new contexts.

Mastery learning, which underpins much of the current thinking in mathematics education, emphasises the importance of conceptual understanding as a foundation. When students learn to manipulate symbols without first building meaning, their knowledge is fragile. It can be recalled under familiar conditions but crumbles when the problems are presented differently.

The CPA approach addresses this directly by giving students multiple entry points to a concept – physical experience, visual representation and formal notation. This creates a deep, connected understanding required for genuine mathematical fluency.

These three stages are not rungs on a ladder to be climbed one after another, but should be considered modes of thinking that work together. A student might move between tiles, a sketched model and the symbolic equations within a single lesson – and that movement between modes deepens understanding. For students who have spent years being told they aren’t ‘maths people’, this approach to learning can provide the support and confidence they need.

Reframing manipulatives at different learning stages

The practical barrier that secondary teachers often raise is that students view manipulatives as childish. This concern is real, but one that can be managed with deliberate framing.

The most effective approach is to introduce CPA as standard classroom practice for everyone - not only for those who are struggling - so the stigma dissolves. When they’re on every desk in the classroom, algebra tiles are tools for thinking, rather than markers of weakness.

CPA also normalises physical and visual modelling beyond the maths classroom. Engineers prototype in physical models, architects build scale representations, and scientists sketch and diagram constantly. Mathematics manipulatives are part of this same process of using the concrete to reach the abstract. It reframes them as part of a professional’s toolkit and not just what younger children need.

Colleague scepticism can be addressed similarly. Sharing the research information of Jerome Bruner, Singapore Maths or the broader mastery literature gives CPA legitimacy as a solid pedagogical choice.

Practical starting points for secondary teachers

Incorporating CPA does not require a complete overhaul of your schemes of work. A single manipulative, introduced thoughtfully in a single topic, is a meaningful starting point.

Algebra tiles for expanding and factorising expressions

Algebra tiles make the area model of multiplication visible and tangible. When students can physically arrange tiles to represent (x+2)(x+3), they can see why the expanded form contains the terms it does. Factorising becomes a process of working backwards from an arrangement, which is far more meaningful than pattern-matching a rule.

Fraction bars for ratio and proportion

Fraction bars allow students to compare, add and scale fractions in a way that symbols alone rarely convey. In ratio and proportion work, they support the development of multiplicative reasoning by making the relationships between quantities visible.

Geometric models for trigonometry

Many students encounter trigonometric ratios as three formulae to memorise. A unit circle or a carefully constructed right-angled triangle situates these ratios within a geometric reality. Students who understand why the sine of an angle represents the ratio it does are far better placed to apply that knowledge flexibly.

Number lines and bar models for negative numbers, percentages and proportion

The humble number line remains one of the most versatile tools available to secondary teachers. For negative numbers, it makes operations intuitive rather than rule-dependent. For percentages and proportions, the bar model supports multiplicative thinking and helps students structure their reasoning before moving to formal methods.

A principle, not a phase

The CPA approach is not something secondary teachers borrow from primary colleagues. It is a description of how mathematical understanding is built at any age. Students who have spent years performing procedures without comprehension deserve the chance to encounter ideas differently. Starting with the concrete, moving through the pictorial and arriving at the abstract with genuine understanding is a clear, easy-to-follow route to mathematical fluency for all learners.

Further reading

Learn more in our other blogs about teaching mathematics: Everything you need for maths mastery teaching and Supporting a successful transition from MYP to DP mathematics: taught curriculum.