Concrete, Representational, Abstract (CRA)
The Concrete, Representational (Pictorial), Abstract (CRA) model is based on Jerome Brunner’s theory of cognitive development: enactive (action-based), iconic (image-based) and symbolic (language-based). Typically, a child will start by experiencing a new concept in a concrete, action-based form. They move to making a representation of the idea, in pictures or more sophisticated diagrams, and then to using abstract, symbolic notation. Students do not always pass through these stages in a linear fashion, nor are they specific to particular age groups. Students are able to build visual models of complex ideas that help to deepen their learning.
Although concrete manipulatives are common in lower primary years they can (and should) be used throughout schooling. There are many virtual manipulatives (digital representations of concrete manipulatives, not games) that can be used when the physical manipulatives are not available.
When using manipulatives, it is important that the connections to the mathematics being learnt are made clear, as well as providing clear instructions on how to use the manipulative.
Watch this video, which describes ways to use a range of strategies including different representations to help students make meaning in their maths lessons.