Maths Mastery - The C-P-A approach - Using the CPA approach to support your child in maths

Sep 2, 2026

Maths Mastery explains how the concrete–pictorial–abstract (CPA/C-P-A) approach builds deep mathematical understanding through mastery learning. It outlines why learners may struggle with abstract ideas and shows how to move from physical manipulatives to visual representations and then to symbols, numbers, and equations. The document also highlights that effective teaching is not strictly linear: teachers revisit stages to reinforce meaning, support all abilities and ages, and encourage flexible use of manipulatives in every lesson to help children make sense of new concepts and challenging questions.

LanguageEnglish

CategoryExam

FormatPDF

Pages67

Reading Time2 h 49 min

AccessFree to download

Maths Mastery

Today's Aims

Why CPA?

Let's break it down!

Making CPA Work

Nothing new here!

# Maths Mastery Developing Mathematicalunderstanding: The C-P-A approach ## Today's Aims: ·Today we will discover how the concrete, pictorial,abstract(CPA)approach helps pupilsto develop a deep understanding of maths aspart of mastery learning ·Explore ideas and reflect on how you can usethe CPA approach to support your child inmaths ## Why CPA? Children (and adults!)can find maths difficultbecause it is abstract. The CPA approach builds on children's existing knowledgeby introducing abstract concepts in a concrete and tangibleway.It involves moving from concrete materials,to pictorialrepresentations,to abstract symbols and problems. The CPA framework is so established in Singapore mathsteaching that the Ministry of Education will not approve anyteaching materials that do not use the approach. CPA is at the heart of MATHS MASTERY. ## Let's break it down! Concrete Pictorial Abstract (CPA)is a three step instructionalapproach that has been found to be highly effective in teachingmath concepts. Concrete =The 'doing'stage-physically moving objects to explore aconcept.This helps bring the maths to life.Every abstract concept is firstintroduced using physical,concrete objects. For example,if a problem involves adding pieces of fruit,children can firsthandle actual fruit.From there,they can progress to handling abstractcounters or cubes which represent the fruit. Pictorial=The 'seeing'stage-images used to represent theobjects. This stage encourages children to make a mental connection between thephysical object they just handled and the abstract pictures,diagrams ormodels that represent the objects from the problem. Building or drawing a model makes it easier for children to grasp difficultabstract concepts(for example,fractions).Simply put,it helps childrenvisualise abstract problems and make them more accessible. Abstract=The 'abstract'stage-symbols and numbers areused to model the problem or calculation.The teacher usesoperation symbols (+,一，x,/)to indicate addition,multiplication,or division. 4+5=9 ## MAKING CPA WORK Although we've presented CPA as three distinct stages,askilled teacher will go back and forth between each stage toreinforce concepts. ## Nothing new here! Giving children objects and drawings to help themto understand key concepts isn't anything new. So,what is it that makes this approach so valuableto the study of maths and particularly to theteaching for mastery? Firstly,CPA is not about getting theanswer quickly. ·Concrete manipulatives are used to helpchildren work through new concepts and challenging questions and provide a transitionto pictorial and abstract. After all,maths lessons aren't about teachingtricks;they are about giving children the tools tounderstand the problem in front of them. ·Interestingly,in a mastery classroom,theredoesn't have to be a linear progression fromconcrete to pictorial to abstract. Instead,teachers apply a cyclical approach. For example,even when a pupil has worked outthe answer using an abstract method,it is worthasking them to use concrete manipulatives toconvince others that they are correct. Secondly,CPA is for everyone! ·Useful for all abilities and ages.Concrete manipulativesare a common feature of KS1 classrooms across thecountry.By KS2,they used to barely exist and wereonly occasionally brought out for children who werestruggling. ·Mastery teaching encourages the use of concretemanipulatives in any lesson as there is value in KS2children having a variety of equipment to aid theirthinking.For these children,concrete objects can oftenkick-start learning about a new concept and aregradually abandoned as they progress through thelesson.

FAQs

What does the CPA (C-P-A) approach stand for in maths teaching?

CPA stands for Concrete, Pictorial, and Abstract. Learners move from using physical objects to representations and then to symbols and calculations.

Why is CPA considered effective for developing mathematical understanding?

CPA makes abstract concepts more accessible by connecting new ideas to concrete experiences and visual models, then gradually using symbols to represent the same concepts.

Does CPA require a strict linear progression from concrete to pictorial to abstract?

No. In a mastery classroom, teachers often use a cyclical approach, returning to earlier stages even after a pupil solves an abstract problem.

Sep 2, 2026

Uploaded by Ava Thompson