Concrete-Representational-Abstract (CRA) is a continuum of representations for concepts. Bruner is famous for advocating this. Other terms used are concrete-pictorial-symbolic.
Typically math teachers rely on symbolic/abstract (P=2L + 2W below). This is often too abstract for many students and they do not develop conceptual understanding. Pictorial/representation (the rectangle below) is still too abstract for many. Concrete is a level students can intuitively access. In the example below I taught a lesson perimeter and area of a rectangle to 6th graders. They were first asked to create a pen for their animals and then count how many fence pieces were needed. This led to the rectangle and eventually the formula.
A teacher candidate asked about CRA for students in higher level courses. Some even many students in these courses do need all three. When the concepts are more challenging and abstract the concrete representation is harder to develop often times but it can be done. For example, area under a curve and integration can be modeled by using a loaf of sliced bread.
Typically math teachers rely on symbolic/abstract (P=2L + 2W below). This is often too abstract for many students and they do not develop conceptual understanding. Pictorial/representation (the rectangle below) is still too abstract for many. Concrete is a level students can intuitively access. In the example below I taught a lesson perimeter and area of a rectangle to 6th graders. They were first asked to create a pen for their animals and then count how many fence pieces were needed. This led to the rectangle and eventually the formula.
A teacher candidate asked about CRA for students in higher level courses. Some even many students in these courses do need all three. When the concepts are more challenging and abstract the concrete representation is harder to develop often times but it can be done. For example, area under a curve and integration can be modeled by using a loaf of sliced bread.
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