#AskMark Volume 3

Thursday, 15 April 2021

Welcome to the third in the series of our new weekly #AskMark, in which our founder Mark McCourt responds to your questions. This week, Mark answers questions on differentiation, and how parents of young children can best prepare them for beginning school.

Don't forget you can submit your own questions too - simply tweet @LaSalleEd using the hashtag #AskMark .

Mahnaz Siddiqui (@MahnazSiddiqui) asked:

What is differentiation in mathematics? What does it look like?

Thank you for your question, Mahnaz. Differentiation is a huge topic, which could easily fill a book on its own. I’ll try to do it some justice in a short response, but would love to hear your thoughts too – feel free to add to what I have written here in the comments below.

If we are working with more than one pupil, then it is always the case that 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 that exist within a class. Understanding that the class is made up of individual human beings with vastly different lives means that teachers can appreciate the burden upon them to ensure that all pupils have a successful experience of learning whatever new idea the teacher is planning for them to grip.

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 that 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 that 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 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 you are aiming for. This is just a way of checking that the meaning is being received. We should not be fooled into thinking that 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 that pupils begin to learn.

It is important that we do not stop at the point of them knowing – at the point they were able to give the correct articulation. Imagine a pupil learning to play the piano, for example. The teacher could tell them the keys that need to be pressed and in what order, with what pressure and at what pace in order 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 can 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 mathematical skill. We want pupils to be so competent in doing the new mathematics that they achieve a fluency in doing so. That is to say, that they can perform without the need to give attention.

The next step in differentiation is clearly the amount of doing that we ask of individual pupils – they will all achieve fluency at different rates. 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 might argue that 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.

To ensure that learning is maintained, we now ask the pupils to engage in practise.

Effective practise occurs in phases too. Firstly, teachers should create opportunities both in the classroom and beyond, for pupils to engage in purposeful practise – this type of practise is goal driven. Considering the mathematical skill that the pupil has been working on and now has automaticity with, teacher and pupil examine carefully the common errors that the pupil makes.

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 practise 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 practise can be carried out independently at home because the pupil has a success metric to give them continual feedback and spur them on.

Purposeful practise 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 practise – 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 practise. But it does have its limitations, since purposeful practise 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 practise by deciding what type of deliberate practise to provide to the individual pupil. Deliberate practise 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 practise 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 practise is designed to help further assimilate the new learning with the pupil’s developing schema of knowledge. Now, practise problems are randomly mixed with problems of earlier learnt ideas – this removes recency and cue from the pupil’s practise 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 practise 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.

David Burns (@mrburnsmaths) asked:

My daughter started Primary 1 in August 2020. What advice would you give a new parent, like me, to help support and develop mathematical ideas/concepts?

Thank you for your question, David. I am sure it will come as no surprise to you if I offer this very short, initial response: Get a huge bag of Cuisenaire rods and let your daughter play with them.

And always be on the lookout to extend her natural play into opportunities to behave mathematically. For example:

  • How many orange rods does it take to surround your favourite toy (Perimeter)
  • Can you make a huge yellow and green snake? What other snakes can we make? (Sequences)
  • Or, a favourite; making sandwiches using two long rods and 'filling' them with rods that add up to the length of the 'bread'

And beyond Cuisenaire? Well, all children want to learn about the environment around them, whether it is learning to count objects around them or learning about the value of numbers. Take all the opportunities you can to convey your joy of all things mathematical – make mathematics an integral and natural part of everyday life and conversation. This could be as simple as allowing your daughter to learn about measures by cooking in the kitchen or talking about money and time.

Most importantly let her enquire and discover the mathematics all around her.

Got your own question for Mark? Simply tweet @LaSalleEd using the hashtag #AskMark and he’ll answer as many of your questions as he can.

Next week: Catering for mixed ability classes, including the challenges faced by KS1 teachers; how to cater for pupils who finish independent work ahead of others; and the impact of White Rose in Primary schools.

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