## 1. They give a tactile understanding of numbers

Students often struggle with math because they struggle with abstraction. Enter the CPA method.

The CPA method is based on Jerome Bruner’s foundational enactive/iconic/symbolic concept (more commonly referred to as CPA — concrete/pictorial/abstract). A mainstay of maths teaching in Singapore, the CPA approach helps students master new ideas by introducing abstract concepts in a tactile, physical way. Through three steps, students master new concepts via physical objects, pictorial representations, and conventional abstract methods.

Here’s what it is, and how to use it with number bonds.

### Concrete

The concrete stage brings abstract concepts to life via physical objects. This is often referred to as the “doing” stage. Instead of being shown how to solve a problem, children are introduced to concepts by handling physical objects.

For example, provide a student with five apples. In how many different ways can they separate the apples onto two plates? Do the two plates combined still add up to five? This exercise shows students the different ways that numbers can be combined or split up.

Here’s another example.

### Pictorial

Pictorial exercises act as a bridge between tangible, concrete problems and abstract ones. This is the “seeing” stage of CPA.

The progress from concrete to pictorial is gradual. For example, you might first replace real apples with physical models (like cubes) to represent the apples. From there, use pictures to represent the cubes.

Familiar images, models, or diagrams encourage students to mentally connect familiar physical objects (like apples) with more abstract objects (like cubes and pictures of cubes). It’s just like the concrete stage, except the tangible objects are being represented with models, and not physically handled.

### Abstract

With concrete and pictorial exercises mastered, students progress to the abstract step. This is the “symbolic” stage where teachers introduce symbolic numbers, notation, and mathematical symbols.

Having moved from concrete examples to pictorial ones, students make efficient connections between abstract examples and the concrete foundations they started with.

When it comes to mastering abstraction, variety is key. Each time a new maths idea is introduced, start with a concrete model and gradually move to pictorial representations before arriving at abstract symbols. Reinforcing the journey through the stages encourages a stronger mental connection between each phase.

Here’s another exercise. Afterwards, encourage pupils to write out the same story using numeric symbols and number bonds .