A CPA approach to Maths

A CPA approach to Maths

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As explained in previous posts, Operation Maths is built on a concrete, pictorial, abstract approach, or CPA approach. Developed by American psychologist, Jerome Bruner, it is based on his conception of the enactive, iconic and symbolic modes of representation. Research has consistently shown this methodology to be the most effective instructional approach to enable students to acquire a thorough understanding of the concepts required. This CPA approach is also the mainstay of maths teaching in Singapore.

What exactly is CPA?

Concrete Pictorial Abstract (CPA) is a three step instructional approach that has been found to be highly effective in teaching math concepts.

  • Concrete stage: Also known as the “doing” stage, this involves physically manipulating objects to demonstrate and explore a concept.
  • Pictorial stage: also referred to as the representational stage in some literature, it can be explained as  the “seeing” stage and involves using images to represent the objects previously used in the concrete stage.
  • Abstract stage: also known as the “symbolic” stage and involves using only numbers and symbols to represent and solve a computation.

What does CPA look like?

Below are some examples of a how a CPA approach might look, at each of the main  class levels:

Concrete Pictorial Abstract
Infs Use logic bears, toys, etc to show a set of five and explore the various ways to partition and then re-combine Use counters and ten frames, cubes , blocks, cuisinere rods and/or draw images to represent the concrete Use digits and/or symbols to represent the relationships established during  the previous two stages eg 2 + 3 = 5
1st/2nd Adding and subtracting, without or with renaming using base ten materials eg straws, cubes, base-ten blocks Use or draw images to represent the concrete manipulative Use a written algorithms for addition and subtraction
3rd/4th Explore multiplication using rows of base ten blocks (area model of multiplication) Draw images to represent the concrete manipulatives Use a written algorithm for multiplication
5th/6th Explore operations with fractions using concrete manipulatives eg paper plates in halves, quarters, eighths Draw images to represent the concrete manipulatives Use a written algorithm and/or branching


What does CPA look like in Operation Maths?

The best way to fully appreciate the CPA approach in Operation Maths is to look at some examples.

Addition without renaming (Operation Maths 1)

The children should use real base ten blocks to model the calculations, before progressing to using the pictorial representations and then, finally, to the column method of the written algorithm


Multiplication involving two-digit numbers (Operation Maths 4)

Using base ten blocks to demonstrate multiplication as an area array

Moving on from the actual blocks; drawing a pictorial representation

Moving on from the area models, using grids

Using the partial products method

Ultimately arriving at the traditional algorithm; the abstract stage


Adding fractions and mixed numbers (Operation Maths 5)

Suggestions for concrete activities in the Teacher’s Resource Book (TRB)

Examples shown of how to use the concrete materials, as well as showing how branching and number lines could be used

Fraction pie pieces in the Discovery Book; there are also blank number lines as an extra pictorial resources given on the inside cover of the Pupil’s Book

Other examples of a CPA approach in Operation Maths

These are only a few small selection of examples of the CPA approach across the Operation Maths programme. Other examples are:

  • The inclusion of free five, ten and twenty frames with the infants to 2nd class books which enable teachers to include frames as one of the concrete activities.
  • The inclusion of free place value manipulatives with the 3rd to 6th class books which enable teachers to include explore and use these resources to demonstrate place value, addition, subtraction, multiplication and division.
  • The free mini-white boards (MWBs) facilitate the drawing of quick jottings to represent concepts and calculations.
  • The TRBs suggest ways in which the teacher can organise concrete activities and use real objects to explore concepts, including suggestions for stations and Aistear themes in the junior end TRBs.
  • The inclusion of base-ten money as photocopiables in the senior end TRBs i.e. the images of 1c, 10c and €1 coins, €10 and  €100 notes. These can be used to add variety to the resource examples and also provide the means to explore decimal numbers in a concrete way.
  • Within the RUCSAC approach to problem-solving, the stage “create to show what you know” specifically prompts the children to use concrete materials and/or pictorial representations to represent the problem.
  • The use of visual strategies for problem-solving,  such as bar models, number lines (for whole numbers, decimal numbers and fractions), number bonds and branching, also provide a pictorial way to bridge the gap between the concrete and the abstract.
  • Many of the digital eManipulatives, accessible on edcolearning.ie, are themselves pictorial representations of real objects; the sorting and shop eManipulative, the fraction eManipulative, the bar-modelling eManipulative and  the counting stick eManipulative can be all used to demonstrate concepts in a graphic way.
  • The Maths Around us videos also use real life objects to show ways to represent mathematical concepts

Some final thoughts…

My own experience of primary maths was typified by the abstract stage; in maths texts of the time, the exercises and even the explanatory sections were almost entirely digits and symbols based, with little or no visual imagery. In recent times, teachers are more aware of the importance of incorporating concrete activities into maths instruction and do so regularly. However, I do think that this is more evident in the junior classes and that teachers of the senior classes sometimes struggle to find ways to demonstrate concretely the more complex concepts required by the curriculum in those classes.

I also believe that the pictorial stage is often neglected and that instructional activities often jump from the concrete stage straight to the abstract stage. If we think of the three stages as stepping stones on a child’s journey to mathematical understanding, many of the stronger, more mathematically-able children are able to make the leap from the concrete  to the abstract. However, for the less able, this can be too big a leap and they don’t successfully manage the jump. For these children especially, it is vital that we ensure the pictorial stage becomes a regular intermediary part  of the instructional sequence.

Thankfully, teachers no longer have to struggle to come up with ways to represent complex concepts or search for ideas for concrete  and pictorial experiences for their classes; instead, Operation Maths is ticking all those boxes and then some!

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