Issues with the C-P-A Approach

Issue 1: Time v. C-P-A (How do we regularize the use of C-P-A in Singapore classrooms?)

There are anecdotal evidences that suggest that a common approach in the classroom is to skip directly to teaching of the "Abstract," which includes the rules, symbols and algorithms, because of time constraints to complete the mathematics syllabus in time for formal exams (Leong & Chick, 2011). However, this can be detrimental for students' sense-making and their ability to understand the underlying mathematical concept. 

Proposed Solution to Issue 1

A proposed solution is to choose certain mathematics units, with a duration of four to six lessons in each unit, to test whether the C-P-A sequence is effective on students' learning. Multiple lessons would allow all students to develop and transit to the next "predominant mode of representation" (Hoong et al., 2015). Teachers must be involved in the C-P-A design process because it allows them to develop a better understanding of C-P-A and they will be able to concretize its use in their actual classroom instruction. 

Issue 3: Is the C-P-A Approach for Everyone?

The C-P-A approach is a general principle with a broad sequence of stages that does not distinguish between specific students. Keep in mind that mastery will not always be a linear progression from concrete to pictorial to abstract. Some students will be able to use abstract representations automatically in one area, but the challenge comes when they are asked to use concrete manipulatives or experiences to convince their peers of their problem solving results. Familiarity with other modes encourages students to "fall back" if they cannot recall working in the abstract mode (Hoong et al., 2015). 

Proposed Solution to Issue 3

Each child learns differently and when you teach to problem solve, you should tailor the lesson to reflect this. Students can choose to use the level of representation that they are comfortable with or you can encourage students to use more than one level of representation based on their readiness. However, all students will be able to develop a much deeper understanding of mathematics when they do not use rote learning. 

In the "Low Attainers in Primary Mathematics" (LAPM) research project done by the National Institute of Education (NIE) and supported by the Ministry of Education (MOE), it aimed to determine how schools and teachers motivated low attainers in learning mathematics and how schools address the diverse learning needs of these learners. The MOE identifies low attainers with the use of the school-based School Readiness Test (SRT) and other formative and summative assessments. Many strategies were used to motivate low attainers in learning mathematics including modifying teaching units, small group instruction and activity-based learning. However, teachers highlighted the effectiveness and use of concrete manipulatives and experiences (Toh and Kaur, 2019). Many empirical studies have concluded that lower ability mathematics learners tend to be kinesthetic learners (Shahrill et al., 2013). Low attainers were encouraged to use manipulatives and mathematical language to demonstrate and prove their truth when developing their mathematical thinking of a concept. The concrete experience of talk-and-interaction between the teacher and the student was key in guiding students to make correct connections between the manipulatives and the concept knowledge they are meant to convey. Manipulative materials were not only engaging and fun (and helped enhance their disposition and attitude in class), but also key in enhancing students' mathematical connection abilities. 

For middle-ability and high-ability students, it is important to spend time developing the mental connections between the three steps by intensifying the complexity of a problem. The goal for teaching the C-P-A approach is to encourage students to to internalize the problem solving process and provide them the ability to imitate/extend the process to any real-life context (Yuliawaty, 2011).