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What is "Concrete Pictorial Abstract" in Primary Maths?
A Concrete Pictorial Abstract (CPA) approach attempts to help improve the understanding of abstract topics. In particular, it explains concepts by:
(1) using concrete representations such as counters,
(2) using pictorial representations such as drawings, and
(3) using abstract representations such as numbers.
Concrete Representations
The children can be introduced to an idea or a skill by acting it out with real objects.
These real objects can be seen, physically handled and touched, enabling students to investigate different concepts.
These real objects are sometimes referred to as maths manipulatives.
For example, this might be done by combining a number of red apples with a number of green apples and counting the total number of apples to represent a sum.
This is a ‘hands on’ approach using real objects and it is the basis for conceptual understanding.
It goes without saying that skilled teachers vary the apparatus the children use in class. For example, one day they might use counters; another day they might use straws.
Examples of Maths Manipulatives include:
- ordinary household items such as straws or dice,
- specific mathematical resources such as dienes or numicon.
Pictorial Representations
Pictorial representations can be used when a child has sufficiently understood the concrete experience and can relate to the concept via pictorial representations. In the case of a addition, this could be drawing objects and adding them up.
This stage is supposed to provide scaffolding between concrete representation and abstract representation.
It is sometimes referred to as the “seeing” stage.
Abstract Representations
Abstract representations are sometimes called the “symbolic” stage.
It’s where learners use numerals and recognisable mathematic notation.
So no drawing apples or counters, just 4 + 7 = ?
Bringing Concrete - Pictorial - Abstract Together
Although CPA has three distinct stages, a skilled teacher will go back and forth between them to reinforce & investigate concepts.
Learners should be encouraged to represent maths problems in a variety of ways. Can learners working at greater depth represent problems in three different representations.
Varying the manipulatives used to solve problems helps learners to feel they have a solid foundation and understanding before progressing to the next area of maths to be studied.
CPA is NOT Hierarchical
While CPA has been explained here in terms of addition, it can be used to cover huge amounts of maths (and science).
Indeed, my maths/physics degree was dominated by lecturers turning up with pendulums (including the start of non-linear dynamics – which led to some chaos theory content). Us students would model the pendulum by drawing little pictures to clarify our thinking and then hopefully producing the relevant equations. We would then investigate strange aspects of the maths and check what was really going on with the pendulum. (A pendulum pointing perfectly upwards over the pivot point should in theory stay there.)
As explained here, teachers and learners will constantly move between the stages reinforcing & checking concepts and investigating new ones.
Concrete Pictorial Abstract & LANGUAGE
Mark McCourt one of the leading voices on Maths Mastery (please note that CPA is NOT maths mastery) and includes “Language” as a fourth interconnected aspect. We’re a big fan of embedding mathematical terminology in lessons and we’ve even built an app that tests students knowledge of the language of maths.
Not Just for Weaker Students - For Everyone
I have heard the opinion that CPA is just for weaker or younger students. I’m not a believer in this. I’m not particularly a believer in weaker students, just students who haven’t grasped a concept yet (and surely these students under a true Maths Mastery approach shouldn’t particularly exist!).
I really like the video below as it demonstrates a Concrete-Pictorial-Abstract (CPA) approach to solving a quadratic equation. (Although she calls it a Concrete – Representational-Abstract Approach.) As such, it clearly demonstrates that even students working at greater depth can use a CPA approach in their work.
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