Levels of Abstraction

Click on each topic to see each stage.

Concrete Stage

Students use physical materials (manipulatives) to understand the mathematical concept that is being taught. The teacher will say and use number words, but the focus is to represent the mathematics concepts/skills with a physical, visual model. Teachers should select the physical model that best represents the mathematics students will do in the abstract stage. Students can precisely represent their thinking by drawing pictures of the physical materials.

Representational Stage (Also known as the Bridging Stage)

In this stage students use physical models (manipulatives) and pictures/drawings of models to represent the mathematics being taught. Teachers purposefully connect the physical model to the numerals and symbols represented by the physical models. This is an extremely important stage since students need to make the connection of the physical models to the mathematical symbols. Students can precisely communicate the mathematics in a solution by drawing pictures and "labeling" the pictures of the models with the correct mathematical symbols and numerals.

Abstract Stage

This final stage occurs when students are able to solve the problem or complete the mathematical procedure just using numerals and symbols. This stage is another way to represent the solution using precision to accurately communicate the mathematics accurately.