ConcreteRepresentationalAbstract Strategy

1
Concrete-Representational-Abstract Strategy

• F. D. Rivera, Ph.D.
• Department of Mathematics
• San Jose State University
• Module 15, Session 3

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Concrete-Representational-Abstract Strategy

• Witzell, Smith, and Brownell (2001)
• This strategy consists of three phases
•         Concrete Phase
•         Representational Phase
•         Abstract Phase

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Concrete 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.

4
Representational Phase

• 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.

5
Abstract Phase

• 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.

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Example Algebra

• In teaching multiplication of two binomials, the
  Concrete-Representational-Abstract 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/Alge
  bra/AlgebraTiles/AlgebraTiles2html.

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Example Algebra

• 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.

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Example Algebra

• 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.

9
Example 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).  

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Example Geometry

• 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/Geomet
  ry/Activity-05.html.

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Example Geometry

• 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.

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Example Geometry

• 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.