This paper is suitable for teaching staff involved in setting and teaching the mathematics curriculum in all schools, including academies, and across all Key Stages. It is intended to provide information on mathematics teaching for mastery; school leaders should decide based on their context whether a mastery approach is appropriate for their school.
‘Mastery’ is the new buzzword in mathematics education. The National Centre for Excellence in the Teaching of Mathematics (NCETM) has promoted this approach widely, and it has become the dominant topic of discussion for schools looking to improve their mathematics teaching.
Is it a revolution in curriculum design or a soon-to-be-forgotten fad?
The idea of developing mastery in a curriculum subject has existed for some time. In the 1980s, a key principle of mastery learning was the frequent use of small feedback loops to periodically check whether a pupil had learnt the current content before moving on. The defining principle of a mastery curriculum was to fix learning and allow the time that took to flex. This contrasted with traditional curricula, which tended to fix the time spent on each topic, whether or not the learning was secure.
Recent interest in mastery learning may have stemmed from John Hattie’s successful 2008 synthesis of over 800 meta-analyses, Visible Learning. Out of 130 possible interventions, mastery learning was listed as ‘one of the winners,’ ranked 29th, with an average effect size of 0.58. This is roughly equivalent to seven months’ additional progress over pupils not following a mastery curriculum.
Four years later, mastery made an appearance in the Sutton Trust’s 2012 Teaching and Learning Toolkit, this time being awarded joint third place, with similar gains to those identified by Hattie being reported. Mastery learning, commented the authors of the toolkit, is one of the least expensive changes a school could make. Both Hattie and the Sutton Trust describe mastery learning as a curriculum design that features every child learning one topic before moving on to the next.
Unhelpfully, there is no single, universally agreed definition of what teaching mathematics for mastery (or mastery learning more generally) actually is. There are four main ways in which the term is currently used:
To describe a curriculum in which a topic is studied until everyone has learnt it, before moving on.
To describe a curriculum which is designed to encourage pupils to spend ‘more time on fewer things,’ as one of several strategies to maximise the chances that every child ‘masters’ that which is to be learnt.
To describe a curriculum which focusses on problem solving, and requires pupils to demonstrate their understanding by being able to give a variety of explanations.
The NCETM uses ‘teaching mathematics for mastery’ to describe a mathematics curriculum which focuses on both procedural/factual fluency and conceptual understanding. Longer time is spent on key topics to develop deep learning which is sustained over time and can be built upon – this is especially important in mathematics.
Traditional curricula, at least since the 1960s, are usually built on a spiral model (as promoted by Jerome Bruner in The Process of Education). The idea is for content to be revisited over time, and studied at a higher level as the child develops.
Advocates of mastery mathematics argue that this approach has become interpreted as ‘try to cover everything a bit in one year, and then everything a bit more the next year.’ In the second year, pupils should be studying a higher level of the previous content. Mathematics is so broad, though, that this leads to each unit being only a week or two long; that’s roughly three to eight lessons on a topic before moving on to something completely different. The topic might then not be seen again for up to a year. Supporters of mastery mathematics argue that this results in children having forgotten almost everything by the following year.
In primary schools, the argument goes, this can mean children are presented with work on fractions before they can fluently recall their times table facts. This inevitably presents problems and a teacher might reasonably allow the child to use a calculator to handle the multiplication since it doesn’t appear to be times table recall that is practised anymore, but rather the procedures for fraction arithmetic.
A ‘mastery’ approach, however, would argue that the fraction arithmetic is in fact an opportunity for pupils to practise both the new content they just learnt about fractions, and the times tables they learnt previously. Removing the opportunity to practise times table recall, and the insistence that it be ‘mastered,’ would embed the gap in knowledge which would then follow the pupil up into secondary school.
Likewise, in secondary schools a spiral curriculum might result in pupils in Year 8 needing to be retaught everything from Year 7 again. This then repeats in Year 9, and Year 10, up to Year 11; the higher content is never visited.
Why is teaching for mastery in mathematics currently such a hot topic? The success of jurisdictions following mastery learning models in the PISA league tables – most notably Shanghai and Singapore – has led to increased interest in the approach among UK policy makers. The new National Curriculum is built around the idea of ‘fewer things in greater depth’, and schools minister Nick Gibb has stated an explicit desire for schools to adopt a mastery style mathematics curriculum. In a speech to the London Thames Maths Hub Primary Conference in spring 2015, for example, he described mastery mathematics as “[one] of the evidence-based approaches we have put at the heart of our education reforms”.
The different ways in which the concept of mastery is being interpreted mean that there is no single way to ‘do’ mastery mathematics. Current models, however, include the following:
The NCETM, working with the Maths Hubs and the Department for Education, has defined an interpretation of mathematics teaching for mastery developed from the England-China Exchange programme. Key characteristics include keeping the class working together, as outlined in the National Curriculum (“The expectation is that the majority of pupils will move through the programmes of study at broadly the same pace.”). More time is spent teaching key topics, such as addition, subtraction, multiplication, division and fractions in order to enable pupils to develop the depth and detail required for learning to be sustained over time. Assessment is focused on depth of conceptual understanding and factual and procedural fluency.
The King Solomon curriculum structure was co-developed with three other academies within the Ark network. This has developed into Mathematics Mastery, a not-for-profit, professional development programme. As well as developing skills to apply procedures accurately, the Mathematics Mastery approach includes a focus on problem-solving activities and developing deep conceptual understanding through looking at different concrete and visual representations. They define ‘mastery’ as being demonstrated when a child can think mathematically with the concept so that they can independently apply it to a totally new problem in an unfamiliar situation.
One aspect of this approach is described in the book Make it Stick. As the authors note, “it’s one thing to feel confident about your knowledge; it’s something else to demonstrate mastery.” In this context, some could argue that the authors are using the word ‘mastery’ to describe the idea of consistently fluent and accurate recall of factual and procedural knowledge, rather than being able to demonstrate ‘understanding’ of where those facts come from, or why those procedures work.
The Ark Academies network has adapted the approach introduced at King Solomon, using it to create a branded curriculum package called Mathematics Mastery, which is now sold to schools. This iteration continued with much of the same structure, but was premised on a new design principle: a focus on problem-solving activities. Rather than a focus on mastering procedural strategies, the goal in the Ark approach is that ‘mastery’ would be demonstrated when a child could explain the ‘why’ behind a procedure, or calculate a result through a variety of methods, or represent an idea visually, and deal comfortably with novel contexts.
Michaela Community School has adapted the model to work with streams, rather than mixed ability groups. Pupils are in the same group for all their subjects – as in KSA’s mixed ability model – but those groups have been broadly set-based on prior attainment in core subjects. MCS has kept the idea of mastery as procedural fluency – with ‘problem solving’ and ‘explanation’ building on top of that fundamental mastery – but has adapted the sequence of content. Higher-attaining groups are able to move through the fundamentals more quickly. They then develop their mathematical reasoning through work on problems created by Tony Gardiner, Don Steward, the UKMT, and others. In some cases they will gain access to more advanced content earlier in their development, for example learning to simplify surds in Year 7, having completed work with roots. Lower-attaining groups are afforded more time to master their procedural fluency, which is seen as necessary to engage with complex problems later in their school careers.
A number of barriers currently prevent schools from adopting mastery learning models:
The conflation of different ideas, expressed under the same umbrella term of ‘mastery,’ has made it difficult for people to understand what exactly ‘mastery learning’ really is.
There is currently a dearth of training available to teachers and school leaders and, where it is available, the provider is unlikely to distinguish what they mean by mastery in contrast to the alternatives.
The curriculum re-design required to move from a spiral model to a mastery model is significant.
The change in curriculum can mean re-inventing resources which can cause a substantial amount of work for teachers.
At least at secondary level, pre-existing textbooks that follow a mastery model are difficult to find.
The change in the philosophy underpinning a curriculum’s design is likely also to require a different approach to assessment.
There may also be cultural issues:
For teachers whose professional experience has taught them that, essentially, ‘pupils forget almost everything,’ the idea of focussing so tightly on only a few things each year is an emotionally difficult barrier to overcome. A leap of faith may be required to move to a curriculum model that focuses on fewer things in greater depth.
Teachers who are accustomed to teaching units that last only a week or two can become dependent on the novelty of the change in topic as a means of behaviour management; this is especially so in schools where discipline is poor. They can resist a move to much longer units from fear that children will become bored, and that even they will become bored teaching the same topic for so long.
School leaders or teachers wanting to find out more about mastery mathematics could consider the following:
Contact local Maths Hubs to ask what training they are able to offer. www.mathshubs.org.uk
Look for information being gathered by teachers visiting Shanghai as part of the national exchange programme, usually published on the Maths Hubs website.
Review the NCETM website on mastery, which outlines a range of things that they are doing to support schools. In particular the national working group run in conjunction with the Department for Education is aiming to produce guidance and support for school leaders.
Look for mathematics conferences involving the NCETM, organisations who have specific expertise or schools that offer training in mastery curriculum design. www.ncetm.org.uk
Contact schools who have adopted a teaching mathematics for mastery approach; many are happy to enter into exchanges that involve knowledge-sharing. For information on the King Solomon/Ark approach see www.mathematicsmastery.org
Consider investing substantially in at least one senior leader’s knowledge of assessment theory. Dylan Wiliam at the Institute of Education and Daniel Koretz, an American author and professor of educational assessment and testing, have both produced advanced theoretical work that can help a school re-evaluate its assessment structures to support a mastery curriculum.
ASCL Primary and Governance Specialist Julie McCulloch