Recognising mastery

Old master

If people knew how hard I worked to get my mastery, it wouldn’t seem so wonderful at all. Michelangelo.

Clive Davies looks at the importance of “mastery” and what it means in terms of mathematical understanding.

Fewer ideas in greater depth

We are in the midst of a great change to classroom practice, particularly in mathematics. Racing learning through multiple  sub levels of understanding is being abandoned in favour of achieving – and demonstrating –  deeper conceptual understanding of any given idea. The idea of ‘Mastery’ is therefore now at the heart of mathematics, and teachers must get to grips with what that means and how it can be identified and what students have to do to achieve it.

Kevin Gessner
The Last Judgement –
Fresco by Michelangelo

Michelangelo really did get it right – the ‘mastery’ of a concept can only be achieved with work and time. If time is limited – as it so often is in a school year, it seems wise to cut the number of objectives down in order to teach fewer things in greater depth.

Using ideas in new contexts

‘Mastery’ has no single definition, but it may be linked with the concepts of shallow and deep learning. If shallow learning is superficial and temporary, then deep learning can be recalled and used appropriately.

Mastery takes this one level further, and is demonstrated when learning can be transferred and applied to different contexts.

Reaching effortless

The “Four Stages of Competence” is a useful model for illustrating mastery:

For a child to have reached that ‘effortless’ level of understanding a mathematical idea, they should:

  • Be fluent in the fundamentals of mathematics, which requires varied, frequent practice with increasingly complex problems.
  • Reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language.
  • Be able to solve problems by applying mathematics to a variety of routine and non-routine problems with increasing sophistication.
Independent and consistent

Mastery requires independent, consistent work and involves the ability to explain one’s own understanding. Some practical indicators that a child has reached this level could be:

  • being able to describe the concept to somebody else
  • recognising real-life instances of problems (i.e. noticing that triangles are used in the environment for their strength, such as in pylons) or
  • making up their own examples of the problem
Thinking harder

But how can we ensure a child really will master a new idea? Firstly, we must be wary of accelerating able pupils through new material once a concept appears to be grasped; we should also recognise that a child’s ability to answer questions with more difficult numbers does not necessarily equate to mastery. The increase in challenge should come from thinking harder about the concept or topic being taught, and being able to respond to an appropriately harder question. An indicative question to judge a Year 6 pupil’s mastery of multiplication and its meaning might be: “My age this year is a multiple of 8. Next year it will be a multiple of 7. How old am I?”

Formative assessment  not summative testing

This means building in the important stage of checking if pupils if have ‘got it’ through formative “assessment” rather than summative “testing”, using well thought-out activities related to the actual objective being taught. If we do this regularly then it may provide a more reliable indicator of mastery than than testing, while also giving students the all important opportunity to “work at” their mastery. Just like Michelangelo.


Clive Davies, OBE, is the founder of Focus Education, a family-run organisation providing advice and educational support to schools in the UK and internationally. In a career spanning 40 years, he has served as a head teacher, Ofsted inspector, Local Authority adviser, trainer and consultant.

 You can meet members of the Focus Education team at the GESS Dubai Conference, from 27th February – 1st March 2018 at the Dubai World Trade Center. To find out more about Focus Education visit:


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