“It’s better to solve one problem five ways, than it is to solve five
different problems”.
The origin of the CPA approach
First put forward in 1966 by psychologist Jerome Bruner, the CPA approach was presented as an educational method to scaffold learning. As a result, educators instantly noted the harmonious relationship it had with the teaching of mathematics. Closely related, Benjamin Bloom introduced the concept of ‘mastery’ in 1968. Bloom promoted the idea that learners must have entire access and complete mastery in preliminary knowledge before advancing further. Essentially, students must completely understand what they have learned so that gaps in knowledge don’t show themselves later on.
Returning to the CPA approach, Bruner recognised that maths had the potential to be a complete mystery for many children. With appropriate scaffolding, things would fall into place and a deeper understanding could occur. He discovered that the usage of the CPA approach in mathematics allowed students to break down complicated concepts into digestible steps, leading to a greater understanding and, in turn, stronger retainment and internalisation. There’s a common misconception that Singapore created the CPA approach, yet the Singapore maths curriculum has just been influenced externally. However, due to the Singapore maths curriculum acting as a catalyst, this approach has gained more popularity and success over the recent decades.
What is the CPA approach?
As touched upon before, the CPA approach is a method of learning that implements physical or visual aids to build upon abstract concepts. Within maths, learners are introduced to a mathematical concept with the use of physical resources, such as Dienes, blocks, cubes, etc. Once the student has grown confident solving problems with concrete aids, pictures are then provided. These are typically pictorial representations of the physical objects that were previously used.
From here, the learner is tasked with finding the solution to problems solely with the abstract (numbers or symbols). These are the stepping stones that a student takes throughout the CPA approach, strengthening the connection and understanding of numbers and their contextual reality. As a result, the concept should be easier to secure and understand.
Should the CPA approach be used in maths?
The short answer: yes. With the CPA approach, pupils have proven to gain a much more meaningful understanding of the concepts they are exploring. This is primarily due to the fact that they don’t have to rely on rote memorisation, as a deeper understanding of the context and process is fostered. Think of it this way: with language, we must have an understanding of the contextual significance of any word.
When we merge the letters t-r-e-e together, without understanding what a tree actually is, then the word is an abstract concept with absolutely no meaning to us. Maths should be thought of in the same vein. Many children may find mathematical concepts to be just as difficult to grasp without the physical resource or image to supplement it.
Concrete
So, let’s break down CPA through a deeper understanding of each stage. Firstly, concrete representations are introduced in the form of physical objects. Students can pick them up and arrange them, allowing further understanding of the concept in its physical form. This can come in the form of typical household objects, or dedicated mathematical resources.
Some may only reserve concrete objects for struggling learners. This is a mistake. For deeper understanding, they can be used by learners of all abilities. Collaboration and conversations can also occur at this stage, allowing learners to speak through their reasoning.
Pictorial
The next stage involves pictorial representations of the concrete objects they have grown comfortable with. This is the bridge that brings them away from the physical world, while still offering a visual aid. Transitioning from the concrete stage to the pictorial stage is important for teachers to understand that children are able to form that connection between a physical and abstract resource.
Abstract
The final stage leverages lessons learned from the previous two steps to tackle abstract problems. These are problems typically using symbols and numerals. To gain the most from their learning, learners are encouraged to move between every stage, ensuring that they have a complete understanding of the entire process.
How to implement the CPA approach in the classroom
Due to the success that has been recognised in many schools across Singapore, it’s hard to avoid the effectiveness of implementing it in the classroom. But what are some examples of ways it can be incorporated? For instance, teachers should be encouraged to introduce a topic in a physical nature, using concrete objects during a lesson that will hook learners in. Collaboration should also be promoted, as this will allow children to bounce ideas off of each other, make observations, and verbalise reasoning behind their decisions.
The entire process should be closely assessed, ensuring that students have a full understanding of each stage. Tactful teachers will recognise when to allow a child to progress, or if they need to revisit concrete and pictorial representations. Consistent, key vocabulary should also be reinforced, so that learners are able to make links and strengthen reasoning skills. Questions should also be encouraged, for both student and teacher, to ensure that all stages are being met in the CPA approach.
How to move learners from concrete to abstract
As we have briefly touched upon, traditional teaching may reserve concrete objects for early education, or for struggling children. However, it has become more evident that, regardless of a child’s ability level, mathematical concepts are secured at different rates. The important thing here is to facilitate the transition between each stage. For instance, the entire process should be extremely flexible. Children shouldn’t be pushed forward if they’ve failed to grasp concrete or pictorial representations.
Scaffolding should also be there to assist learners, supporting their progress with cues. Once a student is confident with a concept, this scaffolding should slowly be taken away. On the other side of the coin, challenges should be encouraged for a greater depth of learning. As they progress through the CPA approach, the problem should also increase in difficulty, creating a solid bedrock of understanding. All of this can be gauged and implemented by an effective teacher who has a close connection with their students’ learning journey.
