Virtual Manipulatives in CPA

While nothing can adequately replace the REAL thing, virtual manipulatives play an essential role in a student’s journey towards developing a profound understanding of mathematics. That’s a bold claim! Let’s go a little deeper…

You’ve heard of CPA? Concrete – Pictorial – Abstract. Sometimes it is referred to as CRA. (Concrete – Representational – Abstract)

These are the three ways a math concept can be represented.

  • Concrete – using actual physical objects represent the math concept
  • Pictorial – using pictures/drawings to represent the math concept
  • Abstract – using numbers, strategies, and/or algorithms to represent the math concept

So in which representation do virtual manipulatives belong? It is not entirely clear, but virtual manipulatives seem to occupy the space in between the representations. They serve as a transition from the concrete world to the pictorial…not quite concrete, but far more flexible than pictorial. The nature of the virtual manipulatives also allows the algorithm and the pictures to be placed side-by-side, making it easier for the students to shift towards abstract understanding.

Here is a small collection of virtual manipulatives to get you started.



  1. One other angle on these to think about is that the “abstract” includes verbal language as well as mathematical language. In my experience and tucked into the research is pretty strong evidence that real understanding depends on including language into the process. This means more than “explain what you did” at the end which can turn into a forced copying exercise. (I’ve also found that when I ask students about stuff… I sometimes find surprising ways of understanding or misunderstanding things.)

    When students can explain what’s happening in their own, natural language and then make connections between that and more mathematical language (words and numbers), deeper understanding is built.