Research School Network: From Foundations to Independence Problem Solving in Secondary Mathematics

In the first two blogs in this series, we explored how the foundations of mathematical problem solving are established in the Early Years and strengthened through explicit strategy instruction in Key Stage 2. In our cathedral, the foundations are secure and the structure is taking shape.

By secondary school, the focus begins to shift. As pupils encounter increasingly complex mathematics, they continue to build fluency through explicit instruction, teacher modelling and practice. They develop their reasoning skills, learning to justify and explain their thinking. Alongside this, they tackle increasingly challenging problem-solving opportunities, moving beyond familiar, topic-based questions towards situations where the most appropriate strategy is less obvious.

The challenge is no longer simply teaching strategies, but helping pupils select, monitor and adapt them when faced with unfamiliar problems.

When pupils get stuck 

We’ve probably all seen it.

The class has been taught the relevant content. The teacher has modelled examples. Pupils have practised similar questions successfully. Then comes the problem-solving task.

Some pupils get going straight away. Others pause, stare at the page and aren’t quite sure where to begin.

Often, the issue is not that pupils do not know the mathematics. It is that they do not yet know what to do when the route to the answer is not immediately obvious. That is what makes problem solving in Key Stages 3 and 4 so important.

At North East Learning Trust, we spend a lot of time thinking about this in mathematics. We know that successful problem solvers do not emerge by accident. They are developed through careful teaching and repeated opportunities to think deeply about mathematical ideas.

This is particularly important for disadvantaged pupils. If we want all pupils to succeed, we cannot assume that the habits of expert thinking will develop by chance; they need to be taught deliberately.

Looking beyond the surface 

One of the biggest challenges pupils face is recognising the mathematical structure beneath the surface of a problem.

If a question looks familiar, they may know what to do. If it is presented differently, they can quickly lose confidence, even when the mathematics itself is well within their grasp.

That is why we need to help pupils look beyond surface features. This reflects a broader progression in mathematics learning. Pupils first become fluent in key methods and procedures through modelling, practice and feedback. They then learn to reason mathematically, explaining, justifying and connecting ideas. Problem solving builds on both of these, requiring pupils to draw on their knowledge and reasoning when the route to a solution is less obvious.

We want pupils to notice relationships, not just contexts. We want them to ask themselves:

What kind of problem is this? What is it really asking me to think about?

The role of metacognition 

This is where metacognition becomes so important.

The Education Endowment Foundation’s Metacognition and Self-Regulated Learning Guidance Report highlights the importance of planning, monitoring and evaluating learning. In mathematics, these processes often come to the fore during problem solving, when pupils need to decide how to approach a problem, check whether their strategy is working and adapt when they reach a dead end.

Successful problem solvers tend to ask themselves a series of useful questions. Over time, these questions become habits of mind.

In mathematics, that might sound like: 

• What is the problem asking me to find? 
• What information is important here? 
• Have I seen something similar before? 
• Which strategy might help? 
• Is my current approach working? 
• Does my answer make sense? 

These questions seem simple, but they are powerful. They help pupils slow down, make sense of a problem and evaluate their progress rather than rushing straight to a calculation.

From teacher prompt to self-regulation 

The challenge, of course, is that these questions do not always come naturally to pupils. This is particularly important for disadvantaged pupils, who may be less likely to have developed these habits independently. It is one reason metacognition and self-regulation are considered such powerful approaches for improving outcomes.

That is why these questions need to be taught explicitly, practised regularly and revisited often.

Initially, these questions may come from the teacher. They might appear on a classroom display, be embedded within worked examples or become part of everyday classroom routines. Over time, however, the goal is for pupils to begin asking them for themselves.

That gradual shift from teacher prompt to pupil self-regulation is where much of the real work lies.

Building mathematical independence 

We want pupils to leave mathematics lessons with more than a correct answer. We want them to recognise patterns, draw on prior knowledge, make sensible decisions and respond thoughtfully when they get stuck.

We want them to understand that successful problem solving is not about instant success; it is about knowing how to think when the method is not obvious.

Across this series, we can see a clear progression. In the Early Years, children begin by exploring, noticing and reasoning. In Key Stage 2, they develop a growing repertoire of strategies. By Key Stages 3 and 4, the emphasis shifts towards helping pupils use those strategies with increasing flexibility and control.

Like the cathedral itself, mathematical expertise is built over time. Strong foundations matter. Effective architecture matters. But ultimately, our goal is to develop pupils who can tackle unfamiliar challenges with confidence, monitor their thinking and adapt when necessary.

Success in mathematics is not simply about knowing what to do when the method is obvious; it is about knowing how to think when it isn’t.