Concrete, Pictorial, Abstract

The Concrete, Pictorial, Abstract approach to learning was devised by psychologist Jerome Bruner, primarily as a way to teach beginners math to grade school students.

When teaching addition, the CPA framework would start with a physical presentation of real objects. Children are given small blocks in several distinct piles and asked to count the piles. The piles are then combined and the students count them again. They learn that adding a pile of two blocks to another pile of two blocks leads to four blocks, and they start to derive an understanding of math of their own volition.

The children are then shown pictures of the same process, often using squares lined up as bars. So a bar of two squares is shown near another bar of three squares. The plus and equal symbol are used to represent the act of adding the bars, and then a bar or five squares is shown. Later on the the students are shown pictures of two bars and asked how many squares the resulting block should be.

The final stage is to abstract the concept into numerical and operational symbols. The countable physical and pictorial representations are replaced with numbers, and the students are asked to perform the same operations as before. The approach builds number since and a more grounded understanding of abstract mathematical concepts.