Teaching Numeracy Skills with the Concrete-Pictorial-Abstract (CPA) Approach

The transition from counting on fingers to understanding complex addition can feel a bit challenging for a young child. At Hess Preschool, we ensure numeracy is an adventure of discovery that nurtures your child’s potential rather than a test of memory.

To achieve this, we utilise the Concrete-Pictorial-Abstract (CPA) approach. This systematic three-stage method moves children from tangible, hands-on exploration to visual representations and, finally, abstract symbols.

By grounding mathematical concepts in the real world, this model ensures that preschool students develop an understanding of how numbers work.

What is the CPA Method in Singapore Math?

The Concrete-Pictorial-Abstract method is a three-stage teaching method used across schools in Singapore to guide children toward a conceptual understanding of numeracy. A central methodology within the MOE’s Nurturing Early Learners (NEL) Framework, the CPA approach provides young students with a simple introduction to math.

Instead of asking a preschooler to memorise symbols on a page, the CPA approach grounds every mathematical idea in real-world play. By moving from physical objects to visual aids, children gain the confidence needed to eventually master abstract symbols in math.

The Three Stages of the CPA Approach in Detail

The CPA approach follows a systematic progression that helps prevent children from rushing into complex ideas before they are ready. By moving through these three distinct phases, children can connect the tangible with the symbolic concepts in math.

1. The Concrete Stage: Hands-On Exploration

Considered as the “doing” phase, the concrete stage allows children to interact with physical and tangible objects, which are often referred to as manipulatives. These are real objects that children can touch, move, and count one at a time.

At Hess Preschool, this stage aligns with our thematic playgroups and active learning approach. Whether children are counting seashells during an “Under the Sea” theme or sorting blocks as part of a classroom project, they are learning through purposeful engagement.

For example, a preschool student might move items from one tray to another while counting to help them keep track of counted and uncounted objects. Also, the primary goal of this stage is to establish a strong physical and sensory connection to mathematical ideas.

2. The Pictorial Stage: Visual Representation

Once a preschool student has developed a reliable understanding of counting at the concrete phase, they transition to the pictorial stage. In this “seeing” phase, children use visual representations to represent the objects they previously handled.

This is also the stage where Nursery and K1 students begin to visualise math in creative ways. For instance, teachers can guide them as they draw their own “number stories” or sketch the same blocks they were just playing with.

Additionally, visual organisation tools, such as tens frames, can be introduced to help children see how numbers are grouped. While they can no longer physically move the objects, being able to touch a drawing or a tally mark helps them form a clear mental model of the concept.

Ultimately, this bridge can allow the child to “carry” the mathematical idea in their mind without needing the physical toys in front of them.

3. The Abstract Stage: Introducing Symbols

The final phase of the CPA model is the abstract stage, where symbols and mathematical notation are introduced. Children learn to use numerals or number words to represent the quantities they previously mastered through touch and sight.

In our K1 and K2 classes, children begin to transition toward writing simple “number sentences.” For example, after a student has physically grouped items and drawn them in their journals, they will learn that the written symbol “5” or the word “five” represents that specific quantity.

This symbolic stage is also vital in an early preschool setting because it enables children to solve problems efficiently using the language of mathematics.

Concrete-Pictorial-Abstract Examples

At Hess Preschool, we integrate the Concrete-Pictorial-Abstract approach into our core numeracy curriculum. Some examples of how we help our students develop a practical understanding of mathematical concepts include:

Example 1: Teaching Counting and Number Sense

We use multi-sensory instruction and let our preschool students handle counters or playdough to create a specific quantity. This hands-on exploration also ensures one-to-one correspondence, such as counting five physical bears by touching and moving them.

Moving to the pictorial stage, children draw five circles to represent those bears, forming a visual mental model. Finally, the abstract stage introduces writing the numeral ‘5’ to represent the quantity.

Example 2: Introducing Early Addition

Early addition is mastered by first using large, soft blocks in the concrete stage, where students physically add two red blocks to two blue blocks and count the total.

When they move to the pictorial phase, the children draw two red squares and two blue squares in a diagram, circling the entire group to represent the sum of four.

In the final abstract stage, they learn to translate this experience into a formal number sentence by writing the equation: 2 + 2 = 4.

Example 3: Exploring Geometry and Measurement (STEAM Education)

Within our STEAM education framework, children explore measurement through project-based learning by using rulers or tape measures to measure objects during a tower-building project. For the pictorial stage, preschool students draw these objects and annotate the measurements on a visual chart to compare different heights.

The abstract stage then concludes the lesson as children record lengths using standard metric units. This systematic process can also help them link geometry and physical construction to a deeper sense of number and scale.

CPA Approach, Bilingualism, and Thematic Learning at Hess Preschool

The educational experience at Hess Preschool is characterised by a thematic curriculum with clear learning objectives, where we carefully balance teacher-guided activities with child-led exploration. We also believe that numeracy should be meaningful, which is why every mathematical concept is introduced through hands-on learning and purposeful engagement.

CPA in a Bilingual Learning Environment

The CPA model works well in a bilingual preschool environment by providing a stable, visual bridge for mathematical concepts. Whether a lesson is conducted in English or Chinese, the concrete and pictorial representations remain consistent.

This shared visual understanding helps ground a child’s learning, ensuring that language differences do not become a barrier to mathematical mastery. Instead, the physical and visual aids provide a foundation for understanding quantities and logic.

CPA and Discovery Learning

The CPA approach encourages discovery through active, hands-on exploration. When preschool students physically combine five cubes with two cubes to make seven, they are actively constructing their own learning rather than passively receiving information.

By physically seeing the results of their “math experiments,” children develop a genuine sense of curiosity and a “can-do” attitude toward problem-solving.

Low Teacher-to-Child Ratios

Successfully implementing the Concrete-Pictorial-Abstract approach in a preschool setting requires careful teacher observation to ensure each child masters a concept before moving to the next stage. By having a low teacher-child ratio, we can provide this crucial, personalised assessment and scaffolding.

This attentive environment also prevents gaps in fundamental understanding, as teachers can tailor their support to each student’s specific pace.

Making Math Meaningful for Preschool Students