In December staff from ITE providers gathered in London for the ‘Teaching Mathematics for Mastery’ conference, jointly organised by the Universities’ Council for the Education of Teachers (UCET), the National Centre for Excellence in the Teaching of Mathematics (NCETM) and the National Association of School-Based Teacher Trainers (NASBTT).
Speakers from NCETM presented information on what trainee teachers need to know and understand about teaching for mastery, while ITE providers’ shared approaches for embedding teaching for mastery within ITE programmes. The role of Maths Hubs working in partnership with ITE providers was also presented.
The mastery of mathematics is the desired outcome for all pupils, so that learners develop a deep, long-term, secure and adaptable understanding of the subject.
This is in line with the vision of the 2014 national curriculum for mathematics with the aims that all pupils:
- become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately.
- reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language.
- can solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions.
The expectation is that the majority of pupils will move through the national curriculum programmes of study at broadly the same pace but those who are not sufficiently fluent with earlier material should consolidate their understanding, including through additional practice, before moving on.
The content and principles underpinning the new mathematics curriculum reflect those found in high performing education systems internationally, particularly those of east and south-east Asian countries such as Singapore, Japan, South Korea and
China. Though there are many differences between the education systems of England and those of east and south-east Asia, the suggestion is that we learn from the mastery approach to teaching commonly followed in these countries.
Principles of Teaching Maths for Mastery
The approach based on mastery is characterised by certain principles:
- The use of mathematical representations that expose the underlying structure of the mathematics;
- Children are helped to make sense of concepts and achieve fluency through carefully structured questions, exercises and problems that use conceptual and procedural variation to provide ‘intelligent practice’, which develops conceptual understanding and procedural fluency in parallel;
- Whole class discussion, precise questioning and intelligent practice, are blended, where necessary, with individual support.
Pupils will have developed mastery when they demonstrate:
- procedural fluency, factual knowledge and conceptual understanding (rapid and accurate recall and application of facts and concepts)
- a growing confidence to reason mathematically
- the ability to apply mathematics to solve problems, to conjecture and to test hypotheses.
There’s nothing particularly new about this but the widespread use of the word ‘mastery’ in relation to mathematics teaching and mathematics learning is relatively new. Some of the implications of implementing and embedding teaching for mastery approaches to teaching mathematics are also new and have required some schools and ITE providers to make changes to their practice.
Reviewing Practice – Meeting the needs of all pupils
One of the changes requires a shift away from labelling pupils as ‘high ability’ or ‘low ability’. NCETM’s Director, Charlie Stripp states, “it may well be the case that one of the most common ways we use differentiation in primary school mathematics… has had, and continues to have, a very negative effect on the mathematical attainment of our children at primary school and throughout their education.”
Standard approaches to differentiation commonly used in primary school maths lessons involve some children being identified as ‘mathematically weak’ and being taught a reduced curriculum with ‘easier’ work to do, whilst others are identified as ‘mathematically able’ and given extension tasks. Stripp argues that terms such as ‘weaker’ and ‘able’ are subjective, and imply that children’s ability in maths is fixed and this may be very damaging in several ways:
For the children identified as ‘mathematically weak’:
- They are aware that they are being given less-demanding tasks, and this helps to fix them in a negative ‘I’m no good at maths’ mindset that will blight their mathematical futures.
- Because they are missing out on some of the curriculum, their access to the knowledge and understanding they need to make progress is restricted, so they get further and further behind, which reinforces their negative view of maths and their sense of exclusion.
- Being challenged (at a level appropriate to the individual) is a vital part of learning. With low challenge, children can get used to not thinking hard about ideas and persevering to achieve success.
For the children identified as ‘mathematically able’:
- Extension work, unless very skilfully managed, can encourage the idea that success in maths is like a race, with a constant need to rush ahead, or it can involve unfocused investigative work that contributes little to pupils’ understanding. This means extension work can often result in superficial learning. Secure progress in learning maths is based on developing procedural fluency and a deep understanding of concepts in parallel, enabling connections to be made between mathematical ideas. Without deep learning that develops both of these aspects, progress cannot be sustained.
- Being identified as ‘able’ can limit pupils’ future progress by making them unwilling to tackle maths they find demanding because they don’t want to challenge their perception of themselves as being ‘clever’ and therefore finding maths easy.
In the mastery approach teachers reinforce an expectation that all pupils are capable of achieving high standards in mathematics. The large majority of pupils progress through the curriculum content at the same pace. Differentiation is achieved by emphasising deep knowledge and through individual support and intervention. The use of whole class teaching is a move away from giving pupils different tasks. Teachers who employ a mastery approach to teaching mathematics do not differentiate their maths teaching by restricting the mathematics that ‘weaker’ children experience, whilst encouraging ‘able’ children to ‘get ahead’ through extension tasks. Instead, teachers employing a mastery approach expose almost all of the children to the same curriculum content at the same pace, providing differentiation by offering rapid support and intervention to address each individual pupil’s needs. Teachers use precise questioning in class to test conceptual and procedural knowledge, and assess pupils regularly to identify those requiring intervention so that all pupils keep up. This requires time for the teacher to think carefully about the concepts – to choose questions for conceptual reasons and carefully prepare models and representations which support generalisation.
In the early primary years, the amount of mathematical topics handled in class is reduced, but more time is spent dealing with each topic, so that early understanding is cemented. Teaching is underpinned by methodical curriculum design and supported by carefully crafted lessons and resources to foster deep conceptual and procedural knowledge. Practice and consolidation play a central role. Carefully designed variation within this builds fluency and understanding of underlying mathematical concepts in tandem.
Implications for ITE Providers
The widespread implementation of mastery within ITE will bring with it challenges that providers will need to overcome. These include the need to develop providers’ and partnership schools’ understanding of the principles of mastery.
There exist influences within Primary Education that shape teachers’ current practice which may need to be challenged. The way in which the National Numeracy Strategy was interpreted by some led to many schools rigidly teaching one hour maths lessons, utilising a 3-part lesson structure. Often lessons consist of teacher explanation followed by pupil practice (completing worksheets containing routine problems). There is often a low level of teacher-pupil interaction within the lesson. A further challenge is that teachers are familiar with assessment through levels (and sub levels) and are not yet certain of assessment without levels.
Trainee teachers will need opportunities to embed the principles of mastery, but schools may resist this. The challenge is to change established attitudes held by teachers, to enable them and trainees to teach in a mastery way, even though many have not experienced mastery. Some school-based trainers may also have fixed ability thinking and practices. This mindset will also need to be challenged. Solutions to these challenges may include continuing professional development in the form of school-based training for mentors, ‘Teaching for Mastery’ events and involvement in Maths Hub projects. In the North East region, Maths Hubs include the Great North Maths Hub and the Archimedes NE Maths Hub.
For further information follow the link to:
A presentation to teachers on teaching for mastery by Debbie Morgan NCETM Director for Primary, December 2015