1. Concrete Pictorial Abstract

2. The origins of CPA
Psychologist Jerome Bruner: enactive, iconic and symbolic modes of representation (1966)
Further adapted and used in maths education to become concrete, representational, pictorial and then to become concrete, pictorial, abstract (CPA)
Particularly prominent in Singapore and Shanghai (top scoring on international maths tests)

3. From Singapore Ministry of Education:
This [activity-based] approach is about learning by doing. It is particularly effective for teaching mathematical concepts and skills at primary and lower secondary levels, but is also effective at higher levels… From concrete manipulatives and experiences, students are guided to uncover abstract mathematical concepts or results.

4. Why use concrete and pictorial when the ultimate aim is symbolic?
Bruner (1966) : “For when the learner has a well-developed symbolic system, it may be possible to by-pass the first two stages. But one does so with the risk that the learner may not possess the imagery to fall back on when his symbolic transformations fail to achieve a goal in problem solving”

5. The concrete
Counters
Dienes
Place value counters
Numicon
Lots more…

7. The bar model

8. Girls = 18
Boys 6

9. Part part whole model 3 9

10. “A mathematical concept or skill has been mastered when … a person can represent it in multiple ways, has the mathematical language to be able to communicate related ideas, and can think mathematically with the concept so that they can independently apply it to a totally new problem in an unfamiliar situation” Helen Drury (2015)