The Concrete-Pictorial-Abstract Approach

Training was provided at West Twyford Primary School in Ealing on Thursday 1st February 2018.  The school uses a CPA approach in the teaching of maths mastery.

What is Maths Mastery?

When children are taught to master maths, they develop their mathematical fluency without resorting to rote learning and they are able to solve non-routine maths problems without having to memorise procedures.

What is CPA?

Concrete, pictorial, abstract is a key approach to the teaching, which develops a deeper understanding in the pupils.  It was developed by Jerome Bruner, an important thinker who will be familiar to you from teacher training.  The approach is within the Singapore method of teaching maths for mastery.

Concrete – children use concrete objects to model problems.  They come to understand mathematical concepts through the experience of handling physical objects.  A range of resources may be used, eg lollipop sticks, Numicon, strips of paper, 10 frames, counters, cups.

Pictorial – visual representations are used to model problems.  This encourages children to make a link between the physical object they handled and the abstract model they will be moving to.  Pictures of the concrete models may be used, for example.

Abstract – children use abstract symbols to model problems.  Children only move to this stage once they have demonstrated secure understanding at the concrete and abstract stages.  This stage involves the use of only numbers, notation and mathematical symbols.

Lesson Structure

Features of the CPA approach were observed in a lesson observation.  A typical lesson is structured as follows:

Anchor Task – Children explore through discussion and use of concrete resources and pictorial representations.

Journal Task – Children apply the concept

Reading and Reflecting – Children evaluate the efficiency and suitability of methods

Guided Practice – Further exploration with guidance and support.

Independent Practice – Completion of tasks in a workbook.


Our main reflections were the following:

  • We noted how all children used concrete objects at the initial stage of learning a concept, moving to pictorial and abstract when suitable.
  • Children worked in mixed attainment groups so they could collaborate and support each other in their learning. This is particularly beneficial as it ensures that lower attaining children have regular opportunities to participate in ‘maths talk’ with children who are higher attaining.
  • Using a problem as the basis for the lesson – the anchor task – ensured that children were engaged in mathematical reasoning from the outset of the lesson.
  • Questioning was key in deepening understanding and developing children’s reasoning skills.

Next Steps

The following actions can be applied in Montpelier:

  • An audit of maths resources
  • Use of the CPA model in planning, with discussion of its application in planning meetings
  • Sharing best practice in using the CPA approach across each phase
  • All children to have access to concrete objects when appropriate
  • A renewed focus on questioning as a means to deepening understanding


Kelly and Andrew