Five phases for fluency: modelling and explaining a mathematical process

Fluency of mathematical procedures is crucial in developing all round mathematical understanding and as such, our modelling and explanations need to be expert.  Here are 5 phases of teacher modelling to set children up to succeed.

1. Talk through a completed example

The first stage is to have a fully completed example ready.  Here, we’re showing children the process in its entirety.  It must be stripped back of any unnecessary information so as to avoid overloading children’s working memory.  At this stage there is no need for contexts to set the question within, there is no need for superfluous text on presentation slides and there is no need for images to jazz up presentation slides either.  These are distractions; we must focus children’s attention on the aspects of the process that are important because they can only attend to so much at a time.  We do however need to prime children to notice certain parts of the explanation and at this stage it needs to be language – the words that we are using to label key parts of the process.  Before the explanation, ask children to listen out for two or three mathematical words (written up on the working wall too) and make a show of exaggerating them when they come up in the explanation, pointing to the relevant part of the model.
That model must have a concrete and or pictorial aspect to it.  Modelling with concrete manipulatives so that children can see clearly how you’re manipulating them can be tricky.  A visualiser or simply a video of you having completed an example beforehand can solve this problem.  Explaining and displaying images at the same time is highly desirable when it comes to getting children to remember things so a pictorial representation of the process is a must.  This could be prepared beforehand or created as you go.  The language that we use over the images must be concise and precise with the explanation planned and rehearsed. 

When the explanation is complete, the process needs to be named so that children can talk about it.  It’s at this point that talking about a learning objective probably makes most sense to a child - they’ve seen it in action rather than having it stated to them at the beginning of a lesson.  This explanation will not have been simply downloaded into children’s memories.  They’ll need multiple interactions to be able to competently do so themselves and so here is a good time to ask children what they noticed.  We need to manage what children are thinking about and so a question for talking partners to summarise what they have seen and heard is wise.  They need to pay particular attention to those mathematical words that we primed them to notice because they’ll need to use that language themselves.  A good way to end this first phase of the explanation is to return to the original list of words on the working wall and add images that can help children remember their meaning.

2. Finish an example together

If the first phase is teacher led, the second is collaborative.  It is important to leave up the completed example that was done previously and work on a partially completed example that is physically next to the completed one.  If this second example is minimally different from the first, with only one or two differences, we can draw similarities better with what we have already showed children.  Here, we need to model tracking between the completed example and the one that we are working on now because that is a good habit for children to do when they are working later and it’s the whole point of having a working wall – for children to use to help them with their current work.

As we work through the example together, this is the time to generate success criteria, instructions to follow that can be applied to other similar problems.  Questions such as: ‘What did I do here (pointing to a part of the completed example)?’ help to focus children’s thinking on the necessary steps in the process.  We, of course, will have planned the exact wording of these criteria and can guide the refinement of what children say.  What’s important about success criteria is that children can remember them.  Recording them on the working wall as images and then elaborating on them with speech ensures that they then become memory prompts.  As in the first stage, it would be beneficial at the end of this phase to pause and get children to think through the success criteria again, making notes or talking to a partner.

3. Complete an entire example together

At this point, we’re ready for a shared example in its entirety, modelling using the success criteria and questioning children to ensure they’re thinking in the right way.  Have the success criteria side by side with the example and model tracking between the two, just as the children should be doing when they work independently later on.  If the variation is slight from the previous example, we can ensure initial success which is important for motivation and confidence.  If children get stuck, there are still worked examples to refer to that were completed previously and would now be on the working wall.  If the first phase was teacher led, and the second a collaboration, then this phase involves handing over the reins a little more but being ready to catch them if them if they fall.

4. Children have a go

By now, children should be raring to go, having experienced success with familiar examples.  There’ll be worked examples and success criteria now on the working wall for children to refer to.  Consider paired work where each partner is assigned a role.  One completes the question, following the success criteria, and the other acts as the metacognitive voice.  They could say things to their partner such as: ‘Ok you have done the first step, next you have to…’ or ‘Wait! You missed the part where you have to…’.  Of course, this kind of behaviour needs to be modelled (perhaps by a teacher / TA partnership) and practised.
Also consider working individually in silence to remove distractions.  There need only be a small number of examples to complete here because the reason for this work is to check that children have understood what you want them to do – this is not their independent work yet.  The number of examples might be few but the complexity of the questions needs to increase gradually.  It is tempting to jump straight in with a group of children and to explain the whole process again for them but at this stage it’s best simply to watch and listen.  We need to find out who’s coping well and who isn’t so that we can challenge them further or address misconceptions.
For some children who might be struggling with the mathematical process, consider giving them questions to complete that you’ve already worked on as a class.  Alternatively, they could be provided with examples that need to be finished, gradually increasing, question by question, how much they do themselves.  For children that have grasped the concept quickly, this time could be used to show them something at a deeper level and set them off working independently on something more challenging.

5. One more shared example

You’ve set them up for success with some clear modelling, concise explanations and precise success criteria.  Inevitably, misconceptions will arise and you’ll have spotted them during the previous phase.  In this last phase, it’s time to address those misconceptions explicitly.  ‘Doing it wrong’ is a useful strategy here, either with some prepared misconceptions (because great teachers have the depth of subject knowledge to predict where children typically go wrong) or creating a live example.  It gives the opportunity to recap the success criteria one more time and because of you’ll have been watching carefully what children were doing in the previous phase, you’ll know exactly which children to direct questions to.
After these 5 phases, children will be ready for the independent work that you’ve set for them and the working wall will be heaving with useful vocabulary, models, images and success criteria.

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