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The first vignette will illustrate the use of concrete manipulatives for teaching the concept of equivalent fractions while the second vignette will exemplify the use of concrete experiences, in the form of children’s literature, to empower students to discover the concept of odd and even numbers. This will be expounded through two classroom vignettes. Many complex problems are easier to solve with algebra. The Concrete Pictorial Abstract (CPA) approach is a system of learning that uses physical and visual aids to build a child’s understanding of abstract topics.

We will discuss the enactive, iconic and symbolic modes of representations (Bruner, 1966) and multiple embodiment principle (Dienes, 1971) as our theoretical underpinnings for the C-P-A approach. The model drawing approach takes students from the concrete to the abstract stage via an intermediary pictorial stage. In this approach, teachers provide students with opportunity to interact with concrete manipulatives or concrete experiences to construct meanings and connect this learning experience with the pictorial and abstract representations of the mathematics concepts. A Concrete Pictorial Abstract (CPA) approach attempts to help improve the understanding of abstract topics.

To facilitate students to build their understanding of abstract mathematical concepts from meaningful contexts and relevant everyday experiences, the Concrete-Pictorial-Abstract (C-P-A) approach is widely used in Singapore, especially at the primary level. Many folks are familiar with the Concrete-Pictorial-Abstract model of representation (seen below), or at least the idea behind it.

Multiple representations of knowledge and their connections are fundamental pedagogical considerations to enable students to develop conceptual understanding of mathematics.