If we are not careful, our worked examples could contradict this advice by providing too much structure and guidance, and removing the ‘problem’ from the ‘problem-solving’.
Blackpool Research School have recently worked with ten schools across the north of England in developing strategies and approaches which support Year 7 pupils in developing their problem-solving strategies and skill. This has included the use of a ‘Think aloud’ and a ‘debrief’, alongside careful consideration of the ways in which teachers exemplify problem-solving as the expert mathematician in their classroom.
Teacher modelling
At the start of our project, fewer than half of the pupils involved reported that they have the chance to look at other pupils’ work in mathematics lessons, sharing methods they use and discussing the decisions they make.
Effective teacher modelling of worked examples is one way to meaningfully integrate this into maths teaching.
For example, in a lesson recently observed in Cumbria, children listened to their teacher ‘thinking aloud’ as they modelled a worked example to a problem, used a heuristic ‘checklist’ to pick up on the traits of an expert problem-solver. Additionally, their teacher questions about their decision-making process during a debrief.
After trying some problems themselves, the teacher invited pupils to share their methods and decisions using a visualiser, with the process coming full circle as the rest of the class used their heuristics checklist to consider their peers’ approach to the problem. Over time, this will develop into a less structured approach in which pupils routinely compare strategies as part of a well-established problem-solving toolkit.
Comparing multiple worked examples
Evidence from What Works Clearinghousetells us that through providing regular opportunities for pupils to examine multiple different strategies for solving problems, they can become more confident, efficient, and flexible in selecting appropriate approaches. This shifts the focus away from ‘the answer’, and towards a deeper understanding of the different approaches available to us when tackling an unfamiliar problem.
As teachers, we can produce worked examples of different ways of solving the same problem, and encourage pupils to notice and discuss the differences. We can focus on the decisions that have been made, the efficiency of each method, and the success of the approach being used. Carefully constructed, these examples can really illustrate to pupils the range of approaches available to them, and that there is often no one way – or best way – to tackle mathematical problems.
This process in turn supports the ‘debrief’, as they begin to notice the different approaches taken by their peers and become better equipped to have mathematical discussions about these.
Spotting the errors
In our project, fewer than 40% of the pupils involved reported that, having completed a problem, they reflect on how they got to their answer and whether or not it makes mathematical sense. One way in which this can be encouraged is by using incorrect worked examples and by supporting learners to explain why the solutions are wrong and at what point in the problem-solving process a mistake was made.
Again, this shifts the focus away from the answer and towards the problem-solving process. It can help pupils to develop deeper understanding than if they solely consider correct worked examples. It also illustrates that not everything prepared by the teacher is always correct, and that making mistakes in mathematics is to be expected (an important message to convey).
The project is at a pilot phase, and there remains much for us to learn about how we can explicitly support our pupils to become more resilient and effective problem solvers. But early indications are that the project is beginning to change teacher classroom habits in a positive way. Crucially, pupils are also developing skills which make it more likely that they will ‘have a go’ at an unfamiliar problem than they were previously.