#AskMark Volume 4

Friday, 23 April 2021

Welcome to the fourth edition of #AskMark, a weekly series in which our founder Mark McCourt responds to your questions. This week, Mark answers questions on the challenges facing Primary teachers seeking to implement a mastery approach, and what to do with the student who always finishes early...

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

Claire Rodger (@ClaireRodger6) asked:

What would you advise for primary schools who wish to implement a mastery approach? Classes are always mixed ability and attainment gaps are often very wide.

Thank you for your question, Claire. It is certainly the case that, in the UK and other Western jurisdictions, primary classes often contain pupils spanning a large attainment range. In year 4, say, it is common to find pupils who are operating at a mathematical level beyond what would typically be expected of a year 6 whilst also finding pupils who are operating at a mathematical level below what would typically be expected of a pre-school child. This is an enormous challenge for teachers to overcome.

There are approaches we can take to make such classrooms more effective, including in-class groupings, which change from subject to subject, day to day, for instance. Alternatively, when it is time for maths, we could use a common hook from which to spin off many different activities – so pupils are, in the face of it, working on the same problem, but at varying levels of complexity.

You ask about implementing a mastery approach in such a classroom. These two goals (having classrooms with very wide attainment gaps and running a mastery approach) are not compatible. A mastery cycle approach to teaching and learning relies on the group being closely enough aligned in terms of attainment to allow for the effective use of the elements of the model – that is, prerequisite quizzing and pre-teaching until all pupils are ready to progress with learning the new idea, whole class instruction, working on tasks specifically about the new learning goal (not spanning a large range of access points), ongoing formative assessment with immediate corrective teaching, extension tasks related specifically to the new idea, and testing at the specific level of difficulty of the new learning goal.

A mastery approach does not work with a large attainment gap.

This is why the main formulators of a mastery approach would often use non-grade settings – in other words, mixed aged classrooms – to enable groups to be a homogenous as possible in terms of their current attainment.

But this is not a very useful response to your question, is it? There’s little point in me just saying it can’t be done.

Primary schools can use a mastery approach. In fact, if a mastery approach is to work anywhere, then it is critical that primary schools use it. But it needs to be implemented from the very beginning.

Which perhaps brings us on to the next question...

Christopher Such (@Suchmo83) asked:

Mastery learning in mathematics relies on teachers addressing children's gaps in prerequisite knowledge before an idea is taught. It seems to be accepted, understandably, that when the gaps get too great, mastery approaches are a non-starter.

This being the case, I wonder about what happens at the start of education. In my (admittedly limited) experience of working in KS1, gaps in number sense, spatial awareness, attention span, etc are often vast, to the point where addressing them before a new idea can be learned is impossible without delaying the teaching of the new idea for a very long time. This is the case despite the excellent work undertaken in reception. (I'd go as far as to say that I found a mastery approach harder to implement in KS1 than in upper KS2, despite the greater absolute gap in maths attainment in the latter). I suspect that for a mastery approach to mathematics to be successful in KS1, the prime areas (and number sense)focused upon in reception would need to be prioritised for significantly longer than they are, with carefully judged focus on those children who need more support in these areas and a concomitant delay in formal instruction, perhaps until the latter stages of Y1. I wonder whether there are any other mastery advocates who would disagree with this assessment.

If so, how might the practical issues that I have described be addressed?

Thank you for your question, Christopher.

It is not unusual to find implementing a mastery approach somewhat easier in KS2 than KS1 in the current system (which has been a product ofmany years of a conveyor belt approach to curriculum). I’d suggest this isbecause KS2 pupils – even those at the lowest level of attainment – have abetter developed schema of knowledge than KS1 pupils. Making sense of new ideas is only possible through constructing that sense from ideas already understood – so the older pupils have more stories, metaphors and images on which to call. The vast majority of what children learn is beyond the content of the school curriculum, so it is often a happy surprise to find that older pupils can construct new meaning in unexpected ways.

There is a fear (and I don’t use that word lightly) amongst many primary teachers that there will be professional consequence to them if they, themselves rather than the pupils, do not ‘keep up’ with the curriculum. I’ve written many times about this conveyor belt approach problem, so won’t labour the point here, save to say that it is understandable that teachers in KS1 feel they are unable to work on an idea such as number sense or place value for a very prolonged period of time. Teachers know (and will tell you privately if they feel they are not at risk of admonishment) that moving on through the curriculum content when pupils have not yet gripped obviously prerequisite ideas is a reckless and idiotic thing to do. But they often feel they have no choice.

I think we make a huge error in England by starting formal mathematics early and racing towards a view of successful mathematics learning that holds written algorithms up as the way of identifying whether or not a 4-, 5-, 6- or 7-year-old child is doing well. I suggest we would perhaps have a great deal more success in our aim for all pupils to become mathematically literate if, instead of the current fetish for standardisation of written responses, we provided an early years (up to age 7) education that focussed on truly understanding numerosity, place value, proportional reasoning and relationships between quantities – none of which is best achieved through a worship of written algorithm.

A prolonged view of early years education (as is not uncommon around the world) would, in my view, give a much stronger foundation for all pupils to construct a successful understanding of mathematics as they continue through school.

All of this can be achieved through a mastery approach, but it requires a shift in policy that takes the aim of all pupils having secure foundational knowledge in place before embarking on a more formal process of using that knowledge to develop the ability to communicate through mathematical symbolism and convention. Sadly, there is little appetite forsuch an approach in the UK.

Nilam Patel (@NILAMPA04557349) asked:

What do you do when some students finish an independent task before others?

Thank you for your question, Nilam. It’s a perennial problem, isn’t it? We all know that moment when a pupil says they are finished long before we expected them to be. Even the most careful planning and the most diligently selected tasks that take into account everything we know about the pupils in order to get the level of difficulty just right can sometimes leave us surprised by the speed at which an individual grips and overcomes the problems. This is, of course, also a pretty lovely moment –it shows the pupil has really applied themselves and worked determinedly to nail whatever task they have been set. I think there are two important responses that should come next.

Firstly, we should appreciate that some pupils absolutely love to completetheir work quickly, but that this might not always align with completing their work carefully. So, teachers take the time to ensure they have presented their thinking elegantly and with mathematical precision – we should always be encouraging pupils to treat mathematical communication with the attention it requires.

Assuming they have indeed completed the task to the highest level of accuracy and mathematical sophistication they can, then, secondly, I think it is important that all teachers have up their sleeves a range of challenging prompts that extend the task, engage the pupil in serious thought, and keep them working at the limits of their comfort zone. Thereare lots of stock responses we can quickly use that have huge impact (perhaps it would be a good idea if teacher training included dozens of these stock responses, so we all know them before facing the situation you describe?)

Some of my favourites include:

  • Can you generalise your solution?
  • What if this approach was applied to *insert new scenario*?
  • What if the questions were posed in a different base?
  • Under what conditions would your solution break down? What are the boundary conditions of the idea we are learning about today?
  • Now that you have gripped this new idea, what ideas did you once hold to be true are now probably false?

I hope that is a useful starter for the list... but would love to hear other people’s favourites. Perhaps add in the comments below?

Andy Waters (@MrAJWaters) asked:

Do you think mixed attainment classes are essential to teach maths via mastery?

Thank you for your question, Andy. This has been addressed in the previous questions, so I’ll just briefly reiterate: absolutely not. Mixed attainment (where the gap is large) and a mastery approach are not compatible models.

We know that learning takes place at the boundaries of our comfort zone, so let’s take the education of all pupils seriously and ensure that the new ideas we are asking them to grip are at the right level for them.

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.

Comments (3)

  • Robert J Smith

    Robert J Smith

    • 27 April 2021 at 12:36
    • #

    Let the pupil become the teacher. How would they explain it to their peers? What misconceptions can they think of? How might these impact the questions they choose? How do they help their peers avoid them?

    Context matters. Pupils (sometimes) need a sense of "where would I use this in 'real life'. Extend a task by asking pupils how they might apply the mathematics. For example, considering angles in parallel lines when considering a page being put through a scanner. What angle do you need to rotate it by to straighten it up? How might you be able to do this more efficiently? Exploration of the relationship between mathematics and ‘real life’ is a great way to extend a task. When we start to generalise, we can further think about the models and representations we use. Which ones work best? Why? Which do not help?

    As a teacher, sometimes by providing. a simple model with its solution is one way to lead on to a more complex situation.

    Teachers need to find ways to provide instructions or information to a problem in a way that allows pupils to think and focus their attention on different aspects.

    For example, they might want to consider the following to extend a task.

    i) Change the problem's starting conditions

    ii) Get the pupils to "break" the problem to its component parts. What is essential? What is not? Do pupils find it more or less helpful with more or less information?

    I am sure others have lots of ideas to add. I look forward to reading them in the comments below.

    reply

  • Stuart Welsh

    Stuart Welsh

    • 27 April 2021 at 12:45
    • #

    Extending a task is not about making it harder by changing the numbers, or by giving pupils more of the same. Rather it should be about trying to give the pupil an opportunity to engage at a deeper level with the original "idea" in the task. 

    The best extending prompts will allow some level of generalisation and abstraction. This doesn't mean we expect pupils to produce beautiful mathematical proofs. Just writing down their thinking/reasons/justifications in words is a worthwhile process.

    Some prompts:

    • Have you found all the possible answer for this task? How can you be sure?
    • Is it possible to represent your solution in another way, e.g. in a diagram, a table, in words, or on a graph?
    • Will your method work for different types of numbers, fractions, decimals, negatives, etc? How do you know? Prove it.
    • What is the minimum amount of information you would need to solve a task like this?

    Sports coaches are experts at adding conditions during practice sessions, e.g. A football coach might demand that players only use their "weaker" foot during a passing drill, or that players strike the ball "first time" when it is passed to them. We can do a similar thing with tasks.

    • Can you repeat this task but only use negatives, fractions, expressions with four terms, etc.
    • Create your own question/task of this type where the solution has to be/have ___

    It is also worth getting pupils to reflect on the way they "tackled" the task. Were they systematic and organised? Did they make tables and diagrams? How clearly do they communicate their reasoning? Reflection of this kind should help pupils when they meet new tasks in the future.

    reply

  • Dave Taylor

    Dave Taylor

    • 27 April 2021 at 18:55
    • #

    When students finish an independent task before others in the class, I'm a big fan of asking them to explain how their answers are related. I use activities that have features of variation theory, and those from my Increasingly Difficult Questions web site, so questions such as:

    - "How does your answer for A change to give your answer for B? Look at questions A and B. What is the same and what is different? Why does the change between A and B that you identified happen?"
    "Continue the activity. What question would you write next and why?"

    - "What makes question 4 more difficult than question 3?"
    "Write question 13. Where could you take this idea next?"

    Sometimes, it might be appropriate to simply ask "Did you go about this the most efficient way? Can you find the most efficient way?"

    I've found that a good choice of activities are key for extending your fastest working students. Some activities are easier to prompt students to further develop their understanding than others.

    reply

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