Embedding the CPA maths approach: concrete, pictorial, abstract

Author: Alexi Makris
Published: 05/08/2020

We are surrounded by objects, pictures, words, numbers and symbols in our everyday lives. The CPA approach, a key principle of the Mathematics Mastery programme, incorporates all of these to help students explore and demonstrate mathematical ideas, enrich their learning experience and deepen understanding.

This idea of teaching and learning through multiple representations – concrete (e.g. a manipulative), pictorial (e.g. a bar model) and abstract (e.g. a numeral or algebraic expression), helps students to cement their knowledge and truly understand what they’ve learnt.

The accompanying classroom resource demonstrates the CPA approach. The magic square encourages students to practise algebraic manipulation concretely – with the multilink cubes representing unknowns – as well as to solve equations concretely.

The task requires students to manipulate and experiment with unknowns, forming expressions and ultimately checking their solutions by the physical substitution of unknowns. It’s often most powerful for the concrete to be represented side-by-side with formal written algebra, to support students in developing their conceptual understanding.

Download Alexi Makris's resource Embedding the CPA maths approach: the magic square.

Alexi Makris

Alexi Makris is Design and development lead at Mathematics Mastery, having taught maths in an outstanding inner London comprehensive.