Concrete representation - modelling a concept with concrete apparatus, at this stage children should be doing using manipulatives to support.
Pictorial representation - learning is moved on and key concepts are now represented in other ways such as drawing pictures/ diagrams. At this stage the children are 'seeing' using representations of objects to model problems.
Abstract representation - maths is now modelled in a symbolic way using numbers, symbols.
https://makingeducationfun.wordpress.com/2012/04/29/concrete-representational-abstract-cra/
http://www.aveson.org/asl/singapore-math-faq
The concrete-pictorial-abstract (CPA) approach is at the heart of the Singapore Maths program in which the bar method is used to provide a powerful pictorial representation. Please watch the video for good introduction.
A good interactive resource to support your teaching of the bar method are the 'Thinking Block' tools as shown below in which there is a great modelling tool.
In a the most recent edition of "Teach Primary" Beth Budden, a teaching and assessment lead at John Ball Primary School in Lewisham, write an interesting article about how she uses Blooms Taxonomy to design the learning journey in stages and build a mastery approach. In summary:
Stage 1: Remembering (fluency) - Retrieve an recall the required information. At this stage Beth talks about being aware of what the children already understand and where the gaps lie. It is about the children gaining confidence and growing efficiency with numbers.
Stage 2: Understanding - The children need to understand 'how to' and may need lots of practice.
Stage 3: Applying (problem solving) - The child needs to apply what they have learnt in context e.g real life maths.
Stage 4: Analysing (reasoning)- at this stage the children need to be able to deconstruct what they have learnt through reasoning. The children should be able to write or verbalise their learning.
Stage 5: Evaluating - this stage is all about the children being able to make judgements about a process, understanding how they or a peer made an error and how to correct it.
Stage 6: Creating - creating their own questions/ problems, writing their own success criteria. The children should also be able to coach another children through the steps to complete the problem.
