This calculation policy is designed to ensure consistency and progression in the teaching of addition, subtraction, multiplication and division across the school. It is aligned with the 2014 National Curriculum.
Children will use mental calculation approaches as their first port of call when it is efficient and appropriate to do so. When necessary, an efficient written method needs to be used accurately, confidently and with clear understanding.
Within each section there are examples of concrete (the practical items that pupils can hold and manipulate to help them explore abstract mathematical concepts and the relationships between them), pictorial (models and representations) and abstract (the symbolic stage).
Concrete is the “doing” stage, using concrete objects to model problems. Instead of the traditional method of maths teaching, where a teacher demonstrates how to solve a problem, the CPA approach brings concepts to life by allowing children to experience and handle physical objects themselves. Every new abstract concept is learned first with a “concrete” or physical experience.
For example, if a problem is about adding up four baskets of fruit , the children might first handle actual fruit before progressing to handling counters or cubes which are used to represent the fruit.
Pictorial is the “seeing” stage, using representations of the objects to model problems. This stage encourages children to make a mental connection between the physical object and abstract levels of understanding by drawing or looking at pictures, circles, diagrams or models which represent the objects in the problem.
Building or drawing a model makes it easier for children to grasp concepts they traditionally find more difficult, such as fractions, as it helps them visualise the problem and make it more accessible.
Abstract is the “symbolic” stage, where children are able to use abstract symbols to model problems (Hauser).
Only once a child has demonstrated that they have a solid understanding of the “concrete” and “pictorial” representations of the problem, can the teacher introduce the more “abstract” concept, such as mathematical symbols. Children are introduced to the concept at a symbolic level, using only numbers, notation, and mathematical symbols, for example +, –, x, ÷ to indicate addition, multiplication, or division.
Although presented as three distinct stages, it would be expected for teaching to go back and forth between each representation to reinforce concepts.
We hope these will help you understand the methods. If you are still unsure, please do see your child's teacher who will be happy to help.