Structure from the start: making maths meaningful

Every pupil deserves to leave school able to think mathematically — to reason, make connections, and apply their understanding to a wide range of contexts and problems. Mathematics isn’t just a collection of skills and procedures. It is a coherent system of interrelated structures that explain how ideas connect and build over time. When pupils see those connections clearly, they don’t just remember more, they understand more. 

Why structure matters now

This matters now more than ever. The recent Curriculum and Assessment Review (CAR) and the Maths Horizons reasoning and problem-solving work both place mathematical structure at the heart of learning - not as an abstract ideal, but as a practical route to equity and excellence.  

When pupils understand how ideas connect, they gain the confidence to reason, the curiosity to explore, and the independence to persist. When we make structure explicit, we do more than deepen understanding - we level the playing field. Pupils who might otherwise rely on memorisation gain access to the same reasoning tools as their more fluent peers. In that sense, attention to structure is a matter of equity as well as pedagogy.  

Seeing what stays the same

Additive and multiplicative structures run throughout the primary curriculum. Numbers and contexts may change, but the structural relationships stay constant, operating in the same way through all stages of maths learning. Some pupils intuitively perceive these structures, allowing them to reason and connect ideas with fluency. Others may not yet see these connections and therefore experience mathematics as disjointed or difficult. 

A curriculum that makes these underlying structures explicit helps all learners see the logic that holds mathematics together. Paying attention to underlying mathematical structure helps pupils make connections between problems, solution strategies, and representations that may, on the surface, appear different, but are actually mathematically equivalent (EEF, 2017). 

Great teaching draws pupils’ attention to what changes and what stays the same. By making deep structures explicit, we help all learners - not just the most confident - succeed and thrive.   

Making structure visible

Take the additive structure.  

There are 5 frogs, some on a log and some in a pool. 3 are on the log. How many are in the pool? 

This simple story illustrates the part–part–whole structure where the parts (frogs on the log and frogs in the pool) combine to make the whole (total frogs). 

The part–part–whole structure connects addition and subtraction: 

• Addition combines parts to make a whole. 

• Subtraction separates parts from a whole (and can be used to find a missing part). 

• The parts can be exchanged while the sum remains the same - the structure of commutativity.

• Understanding these links helps learners move from concrete counting to flexible, abstract reasoning in arithmetic. 

The multiplicative structure works differently but teaches the same lesson: mathematics makes sense when we look for relationships, not just answers. 

Understanding multiplication as units of - with one factor representing the unit size and the other the number of units - unlocks reasoning about scale, ratio, and proportion. Pupils who can see this structure can flexibly interpret a problem like 9 × 754 as nine units of 754 or 754 units of nine. When pupils notice those relationships, they begin to reason, not just calculate. 

Seeing structure in problem solving

Recognising mathematical structure is at the heart of problem solving. It’s what turns complexity into clarity. Consider this example from a KS2 SATs problem:

Two pupils tackled the same question:

• Pupil A went straight to the arithmetic and struggled to find the answer.

• Pupil B looked for structure, saw “units of 754.”, and reasoned their way to the answer.

Pupil A:

Pupil B:

That act of seeing the underlying relationship is what reasoning is all about. Understanding mathematical structure places pupils in a powerful position. It allows them to choose more efficient strategies and, ultimately, make the mathematics easier. A reasoning-rich curriculum helps every learner build that skill. Through well-sequenced examples, talk tasks, and explicit connections, pupils learn to identify what changes and what stays the same — and that’s how understanding deepens. 

Greater depth does not always mean harder mathematics — it often means seeing the mathematics more clearly. We can give all pupils access to this “easier” yet deeper mathematics through explicit teaching of mathematical structures.

The opportunity ahead

A focus on structure within the curriculum will help pupils to see mathematics as an interconnected system of ideas, think logically and reason with confidence in an ever-changing world. This closes the disadvantage gap, not by lowering challenge, but by revealing coherence. When we make structure visible, we make understanding possible - for every pupil, in every classroom. 

References :

Education Endowment Foundation (2017) Improving mathematics in Key Stages 2 and 3. London: Education Endowment Foundation.

Dr Debbie Morgan is an expert on the Maths Horizons specialist team.  

She is the Director for Primary Mathematics at the National Centre for Excellence in the Teaching of Mathematics (NCETM) and has a background as a primary teacher, headteacher, and university lecturer. She holds a CBE for services to education and has been the mathematical consultant for the BBC's 'Numberblocks' program. Her work focuses on developing mathematics education policy and implementing teaching for mastery approaches in primary schools.