Concrete – Pictorial – Abstract
Concrete – opportunity given to children to use concrete objects and manipulatives to develop an understanding of what they are doing
Pictorial – children should then build on this concrete approach by using pictorial representations. These representations can then be used to reason and solve problems.
Abstract – with the foundations firmly laid, children should be able to move to an abstract approach using numbers and key concepts with confidence
To ensure children are able to be fluent with a good ‘sense of number’, pupils need arrange of experiences where they can talk though strategies, discuss possible answers and justify their outcomes. Children need to know ‘how’ and ‘why’ numbers work the way they do and be able to explain this.
Reasoning is a daily expectation within Maths lessons to allow children regular opportunities to explore numbers, make conjectures, investigate patterns, explain possibilities and represent findings.
We use a 5 step progression to reasoning which shows the stage within reasoning that children are working at, based on the NRICH ‘hierarchy’ of reasoning skills:
Step 1: Describing-Simply saying what they did, say what they see
Step 2: Explaining-Offering a reason for what they did.
Step 3: Convincing-Convincing themselves or a friend that they have a solution or case.
Step 4: Justifying- say why they are convinced
Step 5: Proving: provide an argument that is mathematically sound, often based on generalisations. Making a watertight case, and proving it using resources if necessary. At a higher level, this leads to developing an algebraic proof.
The hierarchy of these steps informs our questioning and expectation of the children.
Bar modelling has become a vital part of teaching problem solving at Chilwell Croft. Drawing a bar model helps children to find the unknown elements in word problems (in the context of part/part/ whole relationships) which supports the development of algebraic thinking.
It allows for deeper analysis and understanding of the problem by teaching children to:
- Identify all information given and its relationship to other pieces of information
- Identify missing information that needs to be found
- Understand what mathematical operation to use to find the answer