Exploring Singapore's Concrete-Pictorial-Abstract (C-P-A) Approach

At the beginning of each school year, I give my students a mathematics attitudes and perceptions survey in order to measure their interest, confidence, persistence, and growth mindset towards mathematics. One of the underlying themes I have observed is that my students often claim mathematics to be the most challenging subject and have unnatural apprehension towards it. They feel that it often only contains numbers, formulas, rules with abstract theorems and no space for creativity and "fun." 

To mitigate this, I started embedding the Concrete-Pictorial-Abstract (C-P-A) approach into many units to help my students create and build on their previous knowledge. I have found that students enjoy this approach because it starts with something tangible/concrete that they can observe with their five senses, while using various resources to model their thinking with their peers. Once students have some background, it encourages them to create visual representations and derive rules, relationships, and justifications from their lived experiences. By the end-of-the-year, the majority of my students tend to have more personal interest and a positive attitude towards mathematics. 

Given that I was not formally trained in the C-P-A approach, I decided to apply for the Fulbright Distinguished Awards in Teaching Research Program in order to learn firsthand from Singapore, a country that has advocated for the C-P-A approach in their national curriculum since the early 1980s. In addition, Singapore is well-known for its highly successful, effective, and challenging ways of teaching mathematics for deep conceptual understanding. The International Mathematics and Science Study (TIMSS) 2019 and the Programme for International Student Assessment (PISA) 2018 have consistently ranked Singapore's students first or second in the world in mathematics for the last quarter century. 

The Concrete-Pictorial-Abstract Approach

The Concrete-Pictorial-Abstract pedagogical approach was introduced in the early 1980s and has been widely used for mathematics teaching in Singapore (MOE & NIE, 2012). The C-P-A approach encourages students to process, analyze information, and develop a sequence of logical steps to problem solve. When teaching mathematics, it is important that teachers do not quickly converge rules and formulas to get to the "Abstract" and bypass other modes of representation because students' sense-making will be diminished. Classic theories of cognitive development have demonstrated that students' thinking is initially concrete and grounded in their experience of the physical world and a context that is already familiar to them (Piaget, 1970). Students will have the most effective learning when they move back and forth between each of the stages to reinforce their conceptual understanding. 

Students should have concrete learning experiences through different forms of activities, situations in real-life contexts, or through the use of manipulatives. Manipulatives are dynamic objects designed to represent explicitly and concretely mathematical ideas that are abstract. Common manipulatives include base ten blocks, fraction tiles, pattern blocks, Cuisenaire rods, and money. They have both visual and tactile appeal and can be manipulated by learners through hands-on experiences (Moyer, 2001). Children whose mathematical learning includes physical resources/manipulative experiences will be more likely to bridge the gap between the world in which they live and the abstract world of mathematics (Dienes, 1960). 

"Pictorial" refers to a student's ability to draw pictorial representations or sketches, such as a bar model (model method), that link a concrete resource and an abstract notation. Visuals should be varied when reiterating the teaching of concepts so students can "perceive the concept irrespective of its concrete embodiment" (Lee et al., 2020). 

"Abstract" refers to a student's ability to use abstract symbols, rules, and algorithms to model problems and numerals (i.e. +, –, x, / to indicate addition, multiplication, or division).

The C-P-A approach does not have to be linear. Students are encouraged to concurrently use all three representations during the problem-solving process. For example, students can manipulate fraction discs on their tables, while indicating/shading the pictorial circular models in their notebooks. In addition, students can connect their representations with mathematical statements using abstract symbols (Chang et al. 2017). There should be opportunities for students to make sense of the interactions with the representations and self-reflect on their connections in order for knowledge and information to be internalized. Teachers should stay consistent in their practice of carefully selecting and sequencing representations and learning tasks for maximum effectiveness.

I-Ling Hsiung is a participant of the Fulbright Distinguished Awards in Teaching Research Program (Fulbright DA), a program sponsored by the U.S. Department of State’s Bureau of Educational and Cultural Affairs (ECA) with funding provided by the U.S. Government and administered by IREX. The views and information presented are the participant's own and do not represent the U.S. Department of State, the Fulbright Program, or IREX.