The third in the series of seven focuses on modelling.

What are your existing beliefs and assumptions about modelling in mathematics?

- Should pupils learn just one strategy for approaching a problem, or study multiple different strategies?
- While worked examples are almost always used in maths lessons, are they used effectively?
- Are pupils naturally good at classroom discussion, or will support will be needed to make discussion effective?

If all teachers of mathematics in your school were asked these questions, would there be broad agreement in their answers?

It is almost impossible to teach mathematics entirely without modelling. Teachers demonstrate new skills and ideas in almost every lesson they teach, and the use of a worked example – often copied into pupils’ books to be revised at a later date – is common practice.

Given that modelling of many types is commonplace in our classrooms, how might we nudge our practice to ensure we are maximising its impact? As always, it is helpful to consider the evidence base in order to guide our thinking. Our guidance report, based on an extensive review of the evidence on teaching KS2 and 3 mathematics, found that:

- Pupils should be taught to use and compare different strategies for approaching a mathematical problem;
- Teachers should use worked examples to enable pupils to analyse the use of different strategies;
- Initially, teachers may have to model metacognition by describing their own thinking; and
- We should provide regular opportunities for pupils to develop metacognition by encouraging them to explain their thinking to themselves and others. Pupils may need to be taught how to engage in effective discussion and teachers should model this.

So what might this look like in the classroom?

#### Worked examples

Every teacher of mathematics uses worked examples – they are part of the ‘bread and butter’ of mathematics classrooms the world over. But how we use them is critical to maximising their impact, and evidence suggests some relatively small changes in practice might make their use more beneficial to our pupils.

As well as the traditional ‘teacher-led’ examples, we should consider:

- providing fully worked examples of solved problems for pupils to interrogate and compare, particularly in the early stages of learning;
- encouraging pupils to explain the steps in a worked example: first to their peers and ultimately to themselves (‘self-explanation’);
- providing a series of examples which gradually remove steps until the pupil is working independently (‘backwards fading’); and
- using examples with incorrect steps and encouraging pupils to find the mistakes.

Modelling mathematical discussion

Many of the above examples use discussion: of steps in a problem, of mistakes in working out, or when comparing solutions. While we know that opportunities for explaining mathematical thinking are important, pupils may need to be taught how to engage in effective discussion in order to maximise its impact. Modelling classroom discussion is key, and the following case study from Tarleton Academy near Preston explains how one school have gone about achieving this.

### Modelling metacognition

As well as modelling the mathematics, it can be beneficial to model our metacognitive thinking when solving a problem with a class. Visualisers – or for the more technologically-minded a tablet and stylus – offer an ideal way of supporting this. While completing the mathematical problem, the teacher can ‘think aloud’ – sharing the thoughts of an expert to support pupils in their own development of these skills. The teacher can ask themselves questions such as:

- Have I seen questions which are similar to this before?
- What mathematical skills might support me in the solving of this problem?
- Which strategy shall I try first?
- Is my strategy working? Should I keep doing this, or change my approach?
- Does my answer make mathematical sense? How do I know?

Five ‘big questions’ for discussion

- Are worked examples being used as effectively as they could be in your school, with partially completed or deliberately incorrect examples used alongside fully completed examples?
- Are pupils given examples of multiple strategies to solve a mathematical problem, and encouraged to use and compare these?
- Do teachers regularly explain their own thinking out loud when working through an example, giving pupils an opportunity to experience how an expert approaches a problem?
- Are effective discussion strategies explicitly modelled in your classrooms?
- Are teachers in your school supported in maximising the impact of worked examples, in modelling their thinking to develop pupils’ metacognition, and in modelling effective discussion through appropriate professional development?