True understanding takes place when a child grasps not only how but also why a process works. Research by Jerome Bruner (Bruner, 2000) states that instructional strategies build understanding for students when they move from the concrete (manipulatives or math tools) to the pictorial (visual models or drawings) to the abstract (symbols). CPA is an instructional technique used across the district in our mathematics classrooms.
At the concrete level, tangible objects, such as manipulatives, are used to approach and solve problems. Examples of concrete tools include the following: unifix cubes, base-10 blocks, fraction tiles, counters, or measuring tools. Almost anything students can touch and manipulate to help approach and solve problems can be considered a concrete tool.
Once students have explored concepts using concrete objects, students are moved to the pictorial level. At this stage, representations are used to approach and solve problems. A student has sufficiently understood the hands-on experiences performed and can now relate them to representations, such as drawings, pictures, illustrations, diagrams, charts, and graphs. Such images are visual representations of the concrete manipulatives.
At the abstract level, symbolic representations are used to approach and solve problems. These representations may include numbers or letters, as these symbols provide a shorter and more efficient way to represent numerical operations. At this stage, a student is now capable of representing the problems by using mathematical notation (e.g., 12= 8 + 4).
Building from concrete to abstract, teachers enable students to gain rich understanding of a concept, apply their new knowledge in different situations, and acquire ownership of what they have learned.