Theoretical Background of the C-P-A Approach

Jerome Bruner

The C-P-A instructional sequence is based on American psychologist Jerome Bruner's proposition of enactive-iconic-symbolic representations of cognitive growth (Bruner, 1965). Bruner believed the mathematical learning process should be a progression from the enactive to the icocnic and then the symbolic stage, with continual exposure. However, Bruner's modes are not distinct and separated chronologically; in his conception, elements of the symbolic mode are developed alongside the "primarily enactive and iconic stages of instruction, leading towards a proficiency of operation within the symbolic system" (Yew Hoong et al., 2015). Bruner also stated that it is possible for a student to bypass the first two stages if the tools in the first two modes are not powerful problem solving resources. Overall, it is a constructivist approach where students construct and reconstruct their knowledge based on real-world interactions and experiences. This leads to a much deeper conceptual understanding of mathematical concepts and skills since they are learning by doing. 

Bruner's Three Modes of Representation: 

• Enactive involves doing and motor responses.

• Iconic involves seeing and using mental images/visual mediums to think about ideas that are not physically present .

• Symbolic is abstract and uses symbols to encode knowledge.

Kho Tek Hong

Dr. Kho oversaw Singapore's mathematics syllabus formulation since the 1970s and highlighted the importance of the C-P-A approach in 1981 (Hoong et al, 2015). He believed students must be given ample opportunity to think deeply about mathematics and understand concepts at a relational level. 

Dr. Kho led the Primary Mathematics Project (PMP), which produced instructional materials, tools, and strategies for the professional development and teaching approaches of teachers. Bruner's theories were chosen to drive much of their approach to teaching mathematics in Singapore. Specifically, Dr. Kho and his team developed and introduced the model method, a two-dimensional pictorial diagram that helped students represent known and unknown quantities and their relationships in a problem, in the 1980s. Drawing a pictorial model as a representation of a problem played an important role in the comprehension of a problem as students got the opportunity to "draw, think, examine, reflect and discuss the models" and "engage in active construction of meaning, mathematical reasoning, monitoring their own thinking process as well as self-regulating their own learning" (Kho et al., 2014). It became a distinguishing feature of Singapore's primary mathematics textbooks and was explicitly integrated as part of the Learning Experiences in Singapore's mathematics syllabuses. 

In all of Singapore's instructional materials and textbooks, chapters have a specific order that include 1) a real-life setting and context for a situation, 2) a visual representation of the situation and 3) an abstraction from visual forms to a symbolic form. Teachers are encouraged to provide necessary learning experiences (concrete experiences) to facilitate student learning.