If we are working with more than one pupil, then it is always the case there will be variation in the experiences the pupils have had to date, in their understanding of ideas, in the maturity of their knowledge schemata, in how quickly they can make sense of a new idea, and in how keen they are to do so.
Differentiation is simply a teacher’s response to all of the variations existing within a class. Understanding the class is made up of individual human beings with unique and different lives means teachers can appreciate the burden upon them to ensure all pupils have a successful experience of learning whatever new idea the teacher is planning for them to understand.
So, differentiation is just a way of saying how the teacher reacts to the pupils in front of them. This is a continual process and changes from class to class, idea to idea, even day to day. Perhaps it is helpful to consider the phases a teacher and class progress through as they work together on a new idea.
To begin with, the teacher will seek to establish the pupils’ ‘readiness’ for learning a new idea – this could be through some sort of diagnostic activity or through discussion or through detailed prior knowledge of the pupils. Clearly, pupils will have differing levels of readiness – some will have forgotten things, some will have missed key moments, some will have independently prepared more than others, etc. The first stage in differentiation, then, is the actions the teacher takes based on an individual pupil’s readiness. For some pupils, the teacher may react by providing corrective instruction, working carefully to undo and overcome a misconception, for example. For other pupils, perhaps some pre-teaching will help them to connect partially forgotten ideas. Other pupils will be perfectly well equipped to proceed with new learning having demonstrated their readiness by mastery of the diagnostic activity – the teacher might react here by extending the pupil’s domain expertise in a prerequisite idea by asking them to work on an unfamiliar problem or they might simply allow the pupil to progress to the new learning. This will depend on the teacher’s plan for classroom management and whether or not they wish all pupils to receive the introduction to the new idea together.
When pupils are ready to learn a new idea, the next step is instruction. We know understanding new ideas relies on understanding earlier, pre-requisite ideas. This is how we construct new knowledge – by linking it to what is already understood and using this understanding to ‘bridge’ to new meaning. The teacher can do this by using story-telling and metaphor. To enable metaphors to come to life and have mathematical meaning, the teacher uses models. The models are explored in examples and these examples form the way of narrating the instruction.
The second step in differentiation is, therefore, when teachers react to how readily (or not) pupils are making sense of the instruction. They do this by changing the examples, the models and the metaphors they are using to animate their instruction. The order in which these changes are made is really important. I have previously written about how to react during the instruction phase in this blog, Models, Metaphors, Examples and Instruction.
All pupils (all people, in fact), grip new ideas at different speeds. The purpose of instructing pupils is to bridge from a mathematical idea in my head and understood by me to one that the pupil is able to make meaning of. Working out whether or not the individual pupils in front of us are making appropriate meaning is best achieved through dialogue – as we narrate an example, we then ask them to work on a similar problem and narrate back at us their thinking. In other words, we are using the to-and-fro of examples and problems as a conversation between teacher and pupil – the pupil is forced to articulate their meaning.
The next step in differentiation is, therefore, to react to the pupils in front of us by varying the number of examples they are asked to respond to until each individual is communicating the meaning the teacher is aiming for. This is just a way of checking that the meaning is being received. We should not be fooled into thinking their ability to articulate the correct meaning is an indication that any learning has taken place. At this stage, it hasn’t. But we do now know that we are able to ask the pupils to work independently on problems. We can now ask them to do some mathematics.
Doing mathematics is an absolutely vital step in learning mathematics – it is through doing mathematics pupils begin to truly learn mathematics.
It is important we do not stop at the point of them simply knowing – the point they were able to give the correct articulation. Imagine a pupil learning to play piano, for example. The teacher could tell them the keys they need to press and the order in which they must be pressed, with what pressure and at what pace so as to produce a certain tune. And the pupil could articulate back at the teacher the precise instruction – they know how to play the tune. But that doesn’t mean they can play the tune.
A teacher could explain, through the use of several examples and problems, how to multiply over a bracket, say, and a pupil might articulate back at the teacher the precise instructions – they know how to do it. But that does not mean they can multiply over a bracket. This is why we now give the pupils ample opportunity to actually do the mathematics. We want pupils to be so competent in doing the new mathematics they achieve a fluency in doing so. That is to say, they can perform precisely without the need to give attention.
The next step in differentiation is clearly the amount of doing we ask of individual pupils – they will all achieve fluency at different rates. Initially, this doing might be thought of as naive practice. Once the new skill is something pupils are comfortable with, it is time to start learning.
This might sound a trifle odd and some people will argue, surely, if the pupils are fluent, they have learnt what they need to. But this is just the first step. Learning only occurs at the boundary of our current ability. All pupils have pretty much unlimited potential, but they only continue towards expertise if they continue to operate at their limits. Automaticity is a poor aim for any lesson – it represents a pupil who is no longer learning.
Effective practice occurs in phases too. Firstly, teachers should create opportunities both in the classroom and beyond, for pupils to engage in purposeful practice – this type of practice is goal driven. Considering the mathematical skill the pupil has been working on and now has automaticity with, teacher and pupil examine carefully the common errors the pupil is making.
For instance, the pupil who can fluently multiply over a bracket may well forget to multiply the second term in the bracket two times in every, say, ten questions. We now have a goal – it is highly specific to the pupil and, through dialogue with the teacher, the pupil can set about undertaking more practice with an awareness of that goal – they can be looking out for the common mistake they make and can try to reduce the number of times they falter to, say, just two times in every forty questions. Purposeful practice can be carried out independently at home because the pupil has a success metric to give them continual feedback and spur them on.
Purposeful practice keeps the pupil at the limit of their competence and, therefore, creates the cognitive conditions for learning to occur. So, the next step in differentiation is how the teacher reacts to the pupil’s need for purposeful practice – varying the amount of practice, the goals and the feedback to best realise the individual pupil’s limitless potential to learn. A pupil can significantly improve their mathematical skill through purposeful practice. But it does have its limitations, since purposeful practice leaves the pupil to determine how best to overcome their common mistakes.
The next stage in differentiation is, therefore, how the teacher responds to the pupil’s progress with their personal purposeful practice by deciding what type of deliberate practice to provide to the individual pupil. Deliberate practice is also goal driven, but draws upon what is already known in a domain to improve performance. With the pupil above, who has been forgetting to multiply the second term, the teacher can coach them in overcoming the problem by telling them about tried and tested ways for doing so. In other words, in the deliberate practice phase, the teacher trains the pupil in the approaches that experts in the domain have developed and used to overcome the very specific problem they are facing.
The final stage of practice is designed to help further assimilate the new learning with the pupil’s developing schema of knowledge. Now, practice problems are randomly mixed with problems of earlier learnt ideas – this removes recency and cue from the pupil’s practice exercise and forces them to retrieve previously learnt skills and to identify when to select certain mathematical tools.
The final stage in differentiation is, therefore, the teacher’s reaction to a pupil’s agility in selecting appropriate methods in mixed problems – all pupils will improve their method selection at different rates, so the teacher carefully judges the amount of practice required and supports the individual pupil as required.
This view of differentiation can be thought of as the oft quoted idea of learning being like building an enormous edifice. Constructing a mighty building requires very careful placement and gradual levels of scaffolding. Here, the teacher is the scaffold, providing all the necessary support and rigour needed for the pupil to fulfil their potential.
And just like the construction of an edifice, it is key that the scaffolding is removed at the right moment to let the building shine.