Representation and Structure

Key Messages:

  • The representation needs to clearly show the concept being taught, and in particular the key difficulty point. It exposes the structure.

  • In the end, the students need to be able to do the maths without the representation

  • stem sentence describes the representation and helps the students move to working in the abstract (“ten tenths is equivalent to one whole”) and could be seen as a representation in itself

  • There will be some key representations which the students will meet time and again

  • Pattern and structure are related but different: Students may have seen a pattern without understanding the structure which causes that pattern

Representations used in learning expose the mathematical structure that is being taught. This use aids understanding and, while the physical representation is used as pupils progress through coherent steps in learning, the aim is that pupils will be able to do the maths without the physical representation once learning and understanding is secured. For example the use of place value counters and place value charts in year 6 is a key step when learning about numbers up to ten million. Use of this representation connects with prior place value knowledge and helps to secure further understanding of the structure of the mathematics. The use of representations is valuable at all ages to secure understanding in new learning steps and to connect with prior learning.

Bank of support

Effective use of representation to support the understanding of mathematics is at the core of mathematics learning.

Recently we are focussing more on the mathematical structures as well as the representations we use in our classrooms.

Both the NCETM PD Materials and the Ready To Progress materials have a strong focus on developing representation and structure in tandem in order to secure a deep understanding that builds effectively on prior learning.

Stem sentences

  • help pupils to develop and use precise and accurate language in mathematics
  • provide suggested sentence structures for pupils to use to capture, connect and apply important mathematical ideas.

Once pupils have learnt to use a core sentence structure, they should be able to adapt and reason with it to apply their understanding in new contexts.

Enigma Maths Hub has collated sentence structures from the NCETM PD Materials for each of the spines. Download these documents below. 

Number, Addition and Subtraction
Multiplication and division
Fractions – Linked to NCETM PD Materials

Thinking Boards

  • Capture Think board 1
  • Capture Think board 3
  • Capture Think board 4
  • Capture Think board 5
  • Capture Think board 6

Think boards can help pupils to understand the structures linked to concepts. 

See how the think boards help pupils to connect representations, symbols, pictories and stories. 

Could this be useful for your pupils? 

Interesting articles and further reading