Concrete, Pictorial, Abstract...What Is That?

You and I more than likely learned math through procedures with concepts we didn't understand. If I memorized the formula, I would get the answer correct; that is why I like math. However when I was asked to solve the concept within a real-life situation and explain my answer, I failed miserably. Why?? The reason  is I truly didn't understand the concept to begin with. 

In our Strategic Intervention Solutions, "Making the Shift in 21st Century Mathematics Initiative" training series, we find in many classrooms there is an illusion that students understand the concepts being taught. We encourage all of the teachers in our Mathematics Initiative to teach all mathematical concepts through Concrete, Pictorial, and Abstract means. 

Jerome Bruner proposed that we should use three modes of representations when teaching mathematical concepts: 

1.) Action Based (Concrete)

2.) Image-Based (Pictorial)

3.) Language or Symbol-Based (Abstract)

This CPA instructional model leads to better mastery and better retention. My famous saying is that we "Have to Go Slow to Go Fast," and that is especially true within this thought process.

Example of CPA: 

Problem 24+17=