Mathematics is more than just numbers, formulas, and procedures – it’s a way of thinking. At its core, maths is about problem-solving: breaking down complex situations, identifying patterns, and applying logical reasoning to find solutions. It is great to see problem solving as a core focus of our national curriculum and long may that continue.
That’s why explicitly teaching problem-solving strategies is so important. It equips students with the skills and confidence to tackle unfamiliar problems, nurturing a mindset that extends beyond the classroom.
Celebrating the work of teachers
First, I’d like to take a moment to appreciate the incredible work that teachers do every day. Teaching problem-solving isn’t easy. It requires patience, a deeper subject knowledge, and the ability to guide students through uncertainty while maintaining engagement. Teachers are already using rich tasks, real-world contexts, and open-ended questions to develop mathematical thinkers. Their dedication lays the foundation for students to build resilience and curiosity in the face of mathematical challenges. However, during teacher training, we don’t train teachers to teach problem-solving explicitly so it is no surprise that many rely on teaching in the way they were taught.
The case for explicit strategies
We often encourage students to “think logically” or “just try a different approach” when they get stuck. But what if they don’t know how? If we want students to become true problem-solvers, we need to give them specific strategies and model their use. Some key strategies we could be encouraging could include:
- Working backwards – Starting from the solution or end state and retracing their steps to understand the structure of a problem.
- Drawing a diagram – Visualising relationships and using powerful techniques such as bar modelling.
- Looking for patterns – Identifying sequences or repeated structures to make predictions.
- Making a simpler case – Often this method helps students see the structure of the problem in a more manageable, smaller number set before tackling the actual numbers.
- Using logical reasoning – Eliminating impossible cases and testing different possibilities systematically.
By explicitly teaching these strategies, we empower students to approach problems with confidence and flexibility.
To try and exemplify some of the methods I talk about here I wanted to take a question from last year’s KS2 SATs problems. It was towards the end of the first reasoning paper.
Working backwards method
Encourage them to start from the end (in this case 8 pm) and retrace steps to find what happened in the earlier.
- At 8 pm, there are 40 empty seats.
Since the hall has 1,250 seats in total.
Seats that are filled at 8 pm must be 1250 – 40 = 1210
2. At 7 pm, we know 880 seats were filled.
3. So, the number of seats filled between 7 pm and 8 pm must be the difference between these numbers.
1210 – 880 = 330
Drawing a diagram method
A different strategy is for children to create a simple visual (like a bar or boxes) to show changes from 7 pm to 8 pm.
- Draw a bar to represent all 1,250 seats.
- Mark the portion that is already filled at 7 pm (880 seats).
- Show that by 8 pm, only 40 seats are empty.
- The difference between these two “filled” sections (1,250 – 880 – 40 = 330) represents the newly occupied seats.
Making a simpler case method.
- Create a smaller-scale version of the same scenario. Suppose the hall had 100 seats instead of 1,250. At 7 pm, 70 seats are filled. By 8 pm, there are 5 empty seats.
- Work through the simple version. If there are 5 empty seats, then 95 seats are filled at 8 pm. The number of seats filled between 7 pm and 8 pm is 95 − 70 = 25
- Spot the pattern or logic in the simpler version. The difference between the number of seats filled at 7 pm and at 8 pm gives how many were filled during that hour.
- Try to scale back up to the actual problem: Use the same approach for the real hall with 1,250 seats. There are 40 empty seats at 8 pm, so 1,250 – 40 = 1,210, replacing the smaller, easier to handle, numbers with the larger ones. Compare with the 880 seats filled at 7 pm to find 1,210 − 880 = 330
A call to evolve our approach
Mathematical problem-solving isn’t just for high attaining students or future mathematicians, it’s a critical life skill. In an ever-changing world, where problem-solving is at the heart of innovation and decision-making, we owe it to our students to equip them with these tools.
Teachers are already doing incredible work, but as a profession, we should continue pushing the boundaries. Let’s move beyond simply teaching students how to solve specific types of problems and focus on developing them as problem-solvers—capable, adaptable, and resilient in the face of the unknown.
So, what needs to change to bring this to life in classrooms?
1. CPD for teachers about explicit problem-solving strategies.
2. Teacher guidance within textbooks and resources that highlight the most appropriate strategies for each problem.
3. Resources that are specifically designed to allow students to practice and experiment with the strategies they have been taught.
What are your go-to problem-solving strategies in the classroom? How do you help students develop their mathematical thinking? I’d love to hear your thoughts.
