The GLA Vision:
A GLA Mathematician...
- can recall prior knowledge and use suitable technical vocabulary to articulate their explanations
- is able to talk effectively about their Maths and the strategies they have used in their mathematical processes
- is able to use different resources to support their learning and show resilience in their approaches
- loves Maths and is confident with reasoning, problem solving and thinking in different ways
- is fully equipped for the mathematical challenges of everyday life, preparing them for the world of work and have economic awareness.
Modelling - CPA
Concrete Concrete (Enactive) Stage:
This stage should always be used during the learning of new concepts or when building further onto learnt concepts for every child in the classroom. It involves the physical manipulation of objects to explore structure, find commonalities and rehearse the mathematics. When pupils are acting on the mathematics with the manipulatives they are also more likely to form the language to communicate concepts and ideas. This allows teachers to gain a greater understanding of where misconceptions lie and the depth of understanding a child exhibits. It also allows pupils to develop their ability to communicate mathematically and to reason.
Pictorial Pictorial Stage:
This stage involves the use of images to represent the concrete situation enacted in the first stage. It can be pupils’ drawings of the resources they are acting on or a representation such as the bar model, number line or a graph. This stage acts as a ‘bridge’ to support pupils to make links between the concrete and the abstract and develops their ability to communicate and to represent their mathematics.
Abstract Abstract (Symbolic) Stage:
This is the use of words and symbols to communicate mathematically. It is difficult for pupils to get to this stage without the other two stages working alongside. This is because words and symbols are abstractions. They do not necessarily represent a direct connection to the information. For example, a number is a symbol used to describe how many of something there are, but the symbol of a number, in itself, has little meaning. Why should a ‘5’ represent five any more than the digit ‘2’ stand for five? The other stages support pupils’ understanding of this stage.
"A misconception is an understanding that leads to a ‘systematic pattern of errors’. Often misconceptions are formed when knowledge has been applied outside of the context in which it is useful."
It is important that misconceptions are uncovered and addressed rather than side-stepped or ignored. Pupils will often defend their misconceptions, especially if they are based on sound, albeit limited, ideas. In this situation, teachers could think about how a misconception might have arisen and explore with pupils the ‘partial truth’ that it is built on and the circumstances where it no longer applies. Counterexamples can be effective in challenging pupils’ belief in a misconception. However, pupils may need time and teacher support to develop richer and more robust conceptions. (EEF Improving Mathematics)
The GLA Maths Sequence of Learning
An elicitation should be conducted with all children 2 weeks before a unit of work is taught. The purpose of the elicitation is to determine how well prior learning has been retained and identify any misconceptions. This should use a carousel approach in order to gain as much knowledge of children’s current understanding. The elicitation should include a short written task; a teacher/TA led discussion mat or activity; and a practical game or exploration activity
Teachers will use the elicitation outcomes to inform differentiated planning, teaching strategies, resources and activities. Teachers will be using Power Maths as the main planning/teaching tool. Power Maths is a whole-class mastery programme which builds on previous knowledge. Within each lesson, a child should review prior learning, learn new material in small steps and ask questions. The C-P-A model should be used to enable children to make links in their learning and transfer skills in to different areas of maths. They should have opportunity to practice new material in groups and independently in order to achieve mathematical fluency. Power Maths provides opportunities for all children to be stretched and challenged with problem solving and reasoning tasks throughout a learning sequence.
Delivery and Environment
Teachers should provide models to support children’s understanding of key concepts in maths. Working walls should be used to support children when working independently. The environment should be rich with mathematical vocabulary, modelled examples and concrete examples.