“The important thing about a problem is not its solution, but the strength we gain in finding the solution.” – Seneca
Sweets come in packets of 12, 15 or 20. How many packets did you buy in total if you got exactly 86 sweets?
For many pupils, the first hurdle is knowing where to begin. When faced with uncertainty, pupils may freeze or wait for adult support. Others make quick guesses.
Some pupils will recognise that 15 and 20 can only create multiples of 5 sweets. Then, they will explore how this knowledge can assist them. PISA 2022 data suggests that only 12% of pupils in England demonstrated strong mathematical reasoning and problem-solving behaviours like this.
Why isn’t the development of problem-solving behaviours more widely embedded in education? Part of the answer lies in how we structure the experience of solving problems.
There is no recipe book
Problem-solving isn’t a mystical ability that develops through exposure to problems, nor is it a generic skill bolted onto the curriculum. It can’t be taught by following step-by-step “recipes” that reduce rich problems to mechanical procedures. It emerges when pupils use their deep knowledge of a topic and independently draw on a rich web of ideas.
An experienced chess player doesn’t calculate every possible move; they recognise familiar patterns on the board and draw on strategies they’ve practised. The same is true for effective problem-solvers who draw on a network of deep knowledge and familiar strategies.
This means two things:
- Problem-solving rests on knowledge and is teachable.
- The curriculum should teach pupils how to use their knowledge.
Same same but different
Maths circles allow pupils to focus on fewer problems in greater depth, where the emphasis is on deep reasoning, collaboration and discussion. Classrooms, understandably, face different constraints: time pressures, curriculum coverage and assessment demands. We are also doing a different job in classrooms – we’re building the knowledge base that pupils need to be able to go and solve problems.
But we can and should still find time to explore great problems in the classroom. Whether in a maths circle or lesson, teachers can encourage pupils to be:
- Adventurous — try unfamiliar or challenging problems and have a go even when unsure.
- Articulate — explain ideas and approaches clearly, using correct mathematical vocabulary.
- Accepting — be open to different approaches and ideas; listen and learn from others’ successes and mistakes.
These habits can be nurtured through consistent practice.
Practical shifts = big differences
Pedagogical tweaks can change how pupils experience problem solving and build the organised knowledge it rests upon. This includes encouraging pupils to share multiple approaches and giving the class time to wrestle with problems rather than immediately demonstrating the solution.
Take the problem at the start. Teachers might ask:
- Could you make a total of 86 sweets just using bags of 15 and 20?
- If you only bought one bag of 12, how many sweets would you need from bags of 15 and 20?
In practice, this means slowing down rather than racing through numerous problems. It’s about making the invisible work of problem-solving visible by exploring chosen strategies and asking “why?”
Instead of simply providing answers, thoughtful prompts help draw out pupils’ thinking. This approach gives children space to grapple with challenges and become comfortable with uncertainty.
A culture of mathematical thinking
We’re not suggesting that every lesson becomes a problem-solving workshop. Pupils need to learn the underpinning knowledge and techniques that make problems accessible to them. But it’s important that maths doesn’t become just about learning techniques that pupils never get to use. Once mathematical knowledge is acquired, we then get the joy of using it!
Pupils should start to see problems as invitations to think hard and try things out. Over time, these experiences help pupils build the rich web of knowledge that makes effective problem solving possible. They become comfortable with getting stuck, and come to relish that as the moment they know a problem is going to be worthy of their time. With that, they begin to become mathematicians.
“I enjoyed the problems that were given and getting stuck in finding the solution”, shared a Year 8 pupil reflecting on a recent maths circle.
Our role isn’t to hand pupils a set of universal strategies, but to make the thinking process visible. By encouraging adventurous, articulate and accepting behaviours, we demystify problem-solving. By prizing reasoning and persistence, we can develop a wonderful generation of problem-solvers who are ready to tackle whatever complex challenges come their way.