This strategy consists of three phases:
· Concrete Phase
· Representational Phase
· Abstract Phase

In the concrete phase, students with learning disabilities in math are provided with manipulatives and other material or physical learning tools that will provide them the opportunity to explore a mathematical concept or process by actually doing it with the tools. This is the stage of “getting their hands dirty” with the intent that having an actual experience will enable the construction of the knowledge being targeted.
In the representational phase, students with learning disabilities in math begin to develop mental images of the manipulatives by drawing on other means for understanding the target knowledge. Another way to think about this phase is to say that students with learning disabilities in math are encouraged at this time to step back from the manipulatives and other concrete tools and focus on the mathematical concept or process involved in performing actions with the tools.
In the abstract phase, students with learning disabilities in math could manipulate concepts or processes in the absence of the tools that were important in the early phase of learning.
Examples:
1. Algebra: In teaching multiplication of two binomials, the Concrete-Representational-Abstration works in the following manner:
Concrete: Use algebra tiles to explore the product of two binomials, say,
(x + 2)(x + 3). If you do not have algebra tiles available at this time, then you may still do it in virtual space. Access the following website to work out some problems:
http://www.coe.tamu.edu/~strader/Mathematics/Algebra/AlgebraTiles/AlgebraTiles2html.

Representational: students with learning disabilities in math should then be allowed to use paper and pencils to multiple two binomials based on what they learned from the first phase.

At this stage, you should assist students with learning disabilities in math make a connection between using the tiles
(finding areas) and the FOIL method, another representational mode:

(x + 3)(x + 1) = x2 + x + 3x + 1 = x2 + 4x + 1

First

Outer

Inner

Last

Abstraction: At this stage, students with learning disabilities in math should be able to multiply two binomials without having to rely on the tiles. Even if they continue to rely on the FOIL method to obtain products, they should see the process of finding products of binomials as a result of the distributive property.

2. Geometry: There are several ways of teaching congruence between two triangles. One route is by way of transformations. Let us focus on reflections.
Concrete: Prepare a handout which shows two “congruent” triangles with an imaginary line of reflectional symmetry. Ask students with learning disabilities in math to either use a patty paper to trace both figures. Then ask if they are congruent (same shape, same size). You may also want to use the MIRA to explore reflections first. If you want an online resource, access the following website for activities and more information about how to use the MIRA: http://homepage.mac.com/efithian/Geometry/Activity-05.html.

Representational: students with learning disabilities in math can then be given pairs of triangles to determine whether they are congruent or not. At this stage, they may use paperfolding, where the creased line acts as the line of symmetry between two triangles. If the two triangles fit exactly, then students with learning disabilities in math can make conclusions regarding their congruence.

Abstraction: students with learning disabilities in math at this time can then be taught the significance of the reflectional line of symmetry as the perpendicular bisector of the segments that joint a point on one triangle to its corresponding image on the other triangle. They should see in visual terms the importance of the line as being perpendicular and bisecting the segments.