The document introduces the Concrete-Pictorial-Abstract (CPA) approach for teaching mathematical concepts. It explains that the CPA approach introduces abstract concepts in a more tangible way by starting with concrete objects, then representations, and finally abstract symbols. The concrete stage uses physical objects. The pictorial stage uses drawings and diagrams. Finally, the abstract stage uses only numbers and symbols. This progression helps children build understanding before moving to abstraction.

1. Introduction to Conceptual
Understanding and the
CPA Approach (Concrete,
Pictorial, Abstract)

2. Children and adults can find maths difficult because it is abstract. The CPA approach helps
children learn new ideas and build on their existing knowledge by introducing abstract
concepts in a more familiar and tangible way.
Concrete is the “doing” stage, using concrete objects to model problems. The CPA approach
brings concepts to life by allowing children to experience and handle physical objects
themselves. Every new abstract concept is learned first with a “concrete” or physical
experience using cubes, bears, numicon, beads, counters etc.
Pictorial is the “seeing” stage, using representations of the objects to model problems. This
stage encourages children to make a mental connection between the physical object and
abstract levels of understanding by drawing or looking at pictures, circles, diagrams or
models which represent the objects in the problem.
Abstract is the “symbolic” stage, where children are able to use abstract symbols to model
problems. Once a child has demonstrated that they have a solid understanding of the
“concrete” and “pictorial” representations of the problem, the teacher introduce the more
“abstract” concept, such as mathematical symbols. Children are introduced to the concept at
a symbolic level, using only numbers, notation, and mathematical symbols, for example +, –,
x, / to indicate addition, multiplication, or division.

3. To ensure that all pupils:
• become fluent in the fundamentals of mathematics, including
through varied and frequent practice with increasingly complex
problems over time, so that pupils have conceptual understanding
and are able to recall and apply their knowledge rapidly and accurately
to problems
• reason mathematically by following a line of enquiry, conjecturing
relationships and generalisations, and developing an argument,
justification or proof using mathematical language
• can solve problems by applying their mathematics to a variety of
routine and non-routine problems with increasing sophistication,
including breaking down problems into a series of simpler steps and
persevering in seeking solutions.
Mathematics Programme of Study: KS1-4
September 2014

4. Key components of conceptual understanding
•Mathematical vocabulary and language
•Connections between old and new knowledge
•Questioning to extract reasoning
•Manipulatives/Resources
•CPA Teaching Approach

5. • The following slides will show examples of
manipulatives which aid the development of
conceptual understanding.

6. www.youtube.com/watch?v=2Ss8xAXs_ns
CPA Teaching Approach

8. Numicon
Numicon

9. Counters
Base 10 Grid and Counters

10. Cuisenaire rods

11. Addition and Subtraction
• Part + part = whole

12. Bead Strings
What if the
bead string
is worth 1?
…0.1?

13. Addition – Numicon, Straws, Arrow Cards

14. Partitioning using Diennes
47 + 76 =
7 + 6 = 13
40 + 70 = 110
123
47
40 7
47 + 76 =
40 + 70 = 110 7 + 6 = 13
110 + 13 = 123

15. Column method
T U
6 7
+ 2 4

16. The movement between concrete, visual and
symbolic representations of a problem is often
quite complex, and the connections between the
different forms and the implications of using
different representations need to be explored.
NCETM, (2014)

17. “Research has shown that the optimal presentation sequence to teach
new mathematical content is through the (CPA) approach.”
Sousa, (2008)
“students who use concrete materials develop more precise and more
comprehensive mental representations, often show more motivation
and on-task behaviour, understand mathematical ideas, and better
apply these ideas to life situations”
Hauser(2009)
“I utilize the CPA approach ………………….. to foster a deeper
understanding of mathematics so that students are gaining greater
conceptual knowledge rather than procedural knowledge. Through this
approach, students are experiencing and discovering mathematics
rather than simply regurgitating.”
Gujarati,(2013)