The Concrete-Pictorial-Abstract approach in mathematics

What is the CPA approach and where did it come from?

The concrete, pictorial, abstract approach to mathematics has been a key feature of teaching and learning in Singapore since the 1980s. The approach is inspired by the work of Jerome Bruner, an American psychologist, who developed the process in the late 1960s. 

Singapore or mastery-style maths and its CPA approach was inspired by the research and work of Jerome Bruner

It has become popular in other English speaking territories in recent years due to a push by the British and North American governments for their school-aged children to perform better in mathematics according to the global PISA education performance rankings, which has led to the development and promotion of Singapore-style mathematics teaching in primary and secondary schools. 

Why is the CPA approach used in mathematics? 

The concrete, pictorial, abstract approach (or CPA method) is a process of using “concrete” equipment to represent numbers (including fractions) and operations, such as addition, subtraction, division and multiplication, followed by a pictorial representation to represent the equipment or derived structures (like bar and part-whole models), before moving on to the “abstract” digits and various other symbols used in mathematics.

Usually, those children who find maths hard to understand are feeling that way because of how abstract the subject is and how removed many of its symbols are from day-to-day life. Using a CPA approach allows children to get to grips with new concepts by making use of their existing knowledge and experiences by providing them with a more familiar and real-world entry point to new learning. 

APC, PAC or CPA? Does it really matter in which order you teach the C-P-A approach? 

Of course, any one-size-fits-all approach in education isn’t going to work 100% of the time! There might be occasions when leading with a pictorial representation for an earlier, similar concept might work better than kicking things off with free-form play-based learning with concrete equipment relevant to the learning at hand.  

That said, for the most part, it should go: concrete, pictorial, then abstract. 

Example CPA progression for one-digit addition without exchange

Bruner’s theory that underpins what was developed in Singapore and elsewhere in the 1980s onwards states that there are three means of representing tasks:

  • “enactive representation” (that requires objects and actions);
  • “iconic representation” (which requires sketching, interpreting and building on images); and,
  • “symbolic representation” (which is symbolic or language-based).

Seemingly, people have since thought that concrete, pictorial and abstract were catchier terms than enactive, iconic and symbolic representation… 

Another popular education term is “scaffolding” which Bruner coined while developing the three-stage process we’ve come to know as the CPA approach. The principle behind “scaffolding” is that when designing a teaching and learning sequence the teacher provides very carefully planned assistance to learners, removing support as children progress through a lesson or series of lessons. 

Bruner’s research aligns with the daily experience of teachers, children struggle most with the final element of the three-staged approach, in his terminology “symbolic representation” or in today’s teaching vernacular, the “abstract” phase. So, it follows that in the vast majority of occasions, a well-sequenced learning episode that concrete exploration ought to come first, followed by pictorial representation, then a phased introduction of the concept in its abstract form. 

More than that, Bruner poses that each phase should be integrated into the last, so rather than moving abruptly from concrete equipment to a pictorial representation, then from a pictorial representation to a totally abstract format, there is some scaffolding from the previous phases incorporated into the current phase before removing it and focusing solely on, for example, the abstract element of the relevant concept. 

The “C” in the CPA mathematics approach: Concrete

Concrete learning is the most physically active part of a lesson or series of lessons and involves children playing and working with mathematical equipment to explore a new concept or solve problems. 

It allows children to use equipment they are familiar with and that generally give a sense of quantity, shape or area tied closer to real-life than pictorial or abstract representations. 

Within Singapore-style or mastery mathematics new concepts are first introduced with “concrete” examples, such as when teaching equivalent fractions you would do so using equipment such as paper strip models.

A concrete introduction to equivalent fractions in Stage 4 – Spring Block 3 – Fractions – Lesson 2 – To be able to use diagrams to identify equivalent fractions

Another factor to consider when designing learning tasks with concrete equipment is to provide children with the broadest range of apparatus practicable to allow them to approach the new concept in many and various ways. 

By providing children with such a range of concrete objects, they will then have a broader and deeper foundation of the new concept to rely upon when moving on to the pictorial and abstract phases of the relevant learning objective. 

The “P” in the CPA mathematics approach: Pictorial

The pictorial phase uses what Bruner termed “iconic representations”. 

Here, we encourage children to move from manipulating concrete mathematical equipment to sketching representations and then on to familiar drawn models, such as bar models and part-whole models (sometimes first with equipment or sketches of equipment, followed by using abstract symbols like digits, numbers or other abstract mathematical symbols). 

By moving through various forms of pictorial representation, often blended with concrete equipment or abstract representations, children are able to draw and reinforce the conceptual links between physical objects, sketches, jottings and abstract mathematics. 

Example of the pictorial phase of the CPA approach when representing or partitioning three-digit numbers

In fact, by building in various different pictorial representations within lessons children are given a broader range of mental images to apply when visualising numbers, operations and other concepts when doing abstract mathematical work. 

Often when children go on to doing abstract work, especially when they get stuck, they will refer back to pictorial representations and use sketches to problem solve. 

For example, a child who has been shown various pictorial representations of fractions is likely to find it easier to add fractions with different denominators than a child who has missed that step in their learning. Having that mental model is key to conceptualising and completing such operations.

The “A” in the CPA mathematics approach: Abstract

“Symbolic representation” or the abstract phase of the CPA approach in mathematics is when children move from pictorial representation to using digits, numbers, fractions, operations and the like to form column operations or number sentences, such as 9 + 18 = 27, 32 × 3 = 96 or 1,234 < 3,978.

For a child to master a concept, it is key that their teachers have allowed them to manipulate concrete equipment and have drawn various relevant pictorial representations, such that they can reason and explain the abstract mathematics they are performing at the end of a lesson or unit of learning.

That said, mastery requires speed (in fact, automaticity in many instances) which abstract mathematics allows for in a way that pulling out a tray of multilink cubes or sketching place value counters don’t lend themselves to! 

For example, if children were to work through the following word problem: 

Jamal has 23 apples. Ruth has 7 apples. How many more apples does Jamal have than Ruth?

A child working through the problem would work through this far quicker by writing 23 − 7 = 16 than a child who gathered 23 as two ten straw bundles and three straws and started taking them away or even a child working using pictorial representations sketching 23 circles, a bar model or an empty number line (though an empty number line might very well be the mental model used to complete the abstract calculation!).

How should I move my class through the CPA phases?

For the most part, teaching with the CPA process will involve moving through the three phases with some distinction between them, those more familiar with the approach will segue from one to the next or even bounce back from abstract work to an open-ended, high-floor-low-ceiling concrete task.

One way to implement this “concrete sandwich” or to organise a CPAC lesson would be to introduce the concept with one sort of equipment, say Base 10 for place value or similar, then work through pictorial and abstract work, finishing with a free form, self-differentiating concrete task using place value counters instead.

That being said, the gold standard for the CPA approach is representing concepts and problems in various ways, such as making arrays with counters or cubes, sketching such arrays and area diagrams and similar for multiplication facts. By varying the equipment and structures children are given to introduce key facts and procedures, they are allowed to build strong mental models and connections between each of the concrete, pictorial and abstract phases as well as between earlier and later concepts in a given area of mathematics.   

At what ages is the CPA process appropriate for mathematics teaching and learning?

Children from the Early Years Foundation Stage and upwards use concrete and pictorial models extensively. There are many key types of equipment and structures introduced in EYFS and KS1 that are carried forward into KS2: ten and hundred frames, Cuisenaire rods, multilink cubes, Base 10 equipment, bar models, part-whole models… the list goes on!

We should begin to see concrete phases of learning shortened and pictorial and abstract phases become more dominant in lesson time as children progress through KS1 and into KS2. Nonetheless, concrete and pictorial representations are still key all the way through KS2 and even into secondary schooling where equipment like algebra tiles are becoming increasingly popular. 

2022 Key stage 2 – Test administration guidance

By Year 6, children should become more reliant on pictorial representations and abstract procedures, especially in time for the KS2 SATs where concrete equipment is not allowed in the hall!

Ofsted’s View on the CPA Approach

Shortly after publishing this exploration of the CPA approach, a teacher tweeted the following:

The tweet gained many retweets and comments from confused and confounded teachers.

Ofsted, picking up on the tweet’s traction, offered the following response:

Their response only served to elicit greater anger. In a large part driven by some new terminology coined by the person behind the Twitter account, who referred to “semi-concrete representations”. There isn’t such a thing nor a popularly held concept of a “semi-concrete representation”. So, it is understandable why some might be riled by this comment.

That said, Ofsted is correct in proposing that children should be encouraged to use them initially but they shouldn’t be overused to the point of dependency. It is the aim of a mastery curriculum for children to move from concrete and pictorial conceptualisation to abstract fluency.

Fluency is an operative word here as fluency dictates a deeper understanding than automaticity which conjures a mental image of regurgitation of facts and procedures without any deeper conceptualisation or the ability to use these facts and procedures across multiple contexts.

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