It is important for teachers to explain how symbols can provide a shorter and efficient way to represent numerical operations. These representations can include numbers or letters. At the abstract level, symbolic representations are used to approach and solve problems. It is important for the teacher to explain this connection.Ībstract. These pictures are visual representations of the concrete manipulatives. These can include drawings (e.g., circles to represent coins, pictures of objects, tally marks, number lines), diagrams, charts, and graphs. At the pictorial level, representations are used to approach and solve problems.
Almost anything students can touch and manipulate to help approach and solve a problem is used at the concrete level. Examples of concrete tools include: unifix cubes, Cuisenaire rods, fraction circles and strips, base-10 blocks, double-sided foam counters, or measuring tools.
At the concrete level, tangible objects, such as manipulatives, are used to approach and solve problems. Through this approach, students are experiencing and discovering mathematics rather than simply regurgitating it.Ĭoncrete. I utilize the CPA approach with my preservice elementary teacher candidates in the mathematics methods courses I teach and utilized it when I taught elementary school to foster a deeper understanding of mathematics so that students are gaining greater conceptual knowledge rather than mere procedural knowledge. Regardless of the terminology used, the instructional approach is similar and is based on the work of Jerome Bruner (Bruner, 1960). This approach also goes by other names: the concrete-representational-abstract approach or the concrete-semiconcrete-abstract approach. Research has shown that the optimal presentation sequence to teach new mathematical content is through the concrete-pictorial-abstract (CPA) approach (Sousa, 2008).
Due to the importance of diverse approaches to differentiate instruction, this article explores mathematics teaching and learning through the concrete-pictorial-abstract (CPA) approach.Ĭoncrete-Pictorial-Abstract (CPA) Approach Students who have difficulties with mathematics can benefit from lessons that include multiple models that comprise a mathematics skill or concept at different cognitive levels (Sousa, 2008) since students with learning difficulties rarely learn from only seeing or hearing mathematics (Shih, Speer, & Babbitt, 2011). Students can benefit from mathematics approached in varied forms through hearing, seeing, saying, touching, manipulating, writing or drawing concepts within mathematics (Shih, Speer, & Babbitt, 2011). In order to meet the array of students’ needs in twenty-first century classrooms, it is important for teachers to incorporate multiple representations of mathematical ideas in their instruction because this increases the chance that they reach all students through their diverse learning styles (e.g., auditory, visual, kinesthetic).
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