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Representation and Structure

Teachers provide access to the mathematics through carefully selected representations.


Representations expose mathematical structure.

Procedural Variation

Calculations are

connected and pupils

use the relationships to

make connections.


Conceptual Variation 

Representing the same

concepts in different ways

to draw out essential features.  


Making connections between the different representations is important.

Mathematical Thinking

Children and teachers need to think mathematically. This means to:

  • look for pattern,

  • make connections,

  • and reason about relationships.


Having instant recall of number bonds to ten and within ten and being able to apply these to multiples


40 + 20 = 60  


and bridging ten  8 + 5 = 8 + 2 + 3 = 13  


is crucial for mastery of the curriculum.


Learn the facts so that the mind is freed up to think about concepts.


The NCETM have produced PD material with exemplified small steps mapped out from Year 1 to Year 6.    

They have split the curriculum up into a small number of areas called ‘spines’ – 
Spine 1: Number, Addition and Subtraction, 

Spine 2: Multiplication and Division and
Spine 3: Fractions.

Each spine is composed of a number of segments, and a recommended teaching sequence for segments across the three spines.

An explanation of the structure of these materials, with guidance on how teachers can use them, is contained in a Getting Started video.

Download the full documents here

Spine 1

Spine 2

Spine 3

Zoe Hall at NPAT who is one of our Enigma mastery specialists has collated these documents to make connections between representation and structure as well as variation.

Download the full document

Download the full document

Download the full document

Click here for full document

Primary Mastery Blog
Representation and Structure
Fluency and MT