Nieuw curriculum wiskunde in Engeland: visie op vakdidactiek wiskunde van National Centre for Excellence in the Teaching of Mathematics .
Ook in Vlaanderen wordt gewerkt aan een nieuw leerplan en aan nieuwe eindtermen. De visie van het 'National Centre..." komt vrij goed overeen met de visie die we in onze publicaties over wiskundeonderwijs propageerden: zie b.v Rekenen tot honderd, Uitgeverij Plantyn (Mechelen). We juichen ook toe dat men afstand neemt van de constructivistische aanpak.

Deel 1 : Mastery approaches to mathematics and the new national curriculum

1.‘Mastery’ in high performing countries

The content and principles underpinning the 2014 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. The OECD suggests that by age 15 students from these countries are on average up to three years ahead in maths compared to 15 year olds in England .
What underpins this success is the far higher proportion of pupils reaching a high standard and the relatively small gaps in attainment between pupils in comparison to England. Though there are many differences between the education systems of England and those of east and south-east Asia, we can learn from the ‘mastery’ approach to teaching commonly followed in these countries. Certain principles and features characterise this 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.
• 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.
• 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.The intention of these approaches is to provide all children with full access to the curriculum, enabling them to achieve confidence and competence – ‘mastery’ – in mathematics, rather than many failing to develop the maths skills they need for the future.

2. Curriculum changes

The 2014 national curriculum for mathematics has been designed to raise standards in maths, with the aim that the large majority of pupils will achieve mastery of the subject. Mathematics programmes of study state that:
• All pupils should become fluent in the fundamentals of mathematics, including through varied and frequent practice, so that pupils develop conceptual understanding and are able to recall and apply their knowledge rapidly and accurately to problems.
• The expectation is that the majority of pupils will move through the programmes of study at broadly the same pace. When to progress should always be based on the security of pupils’ understanding and their readiness to progress to the next stage.
• Pupils who grasp concepts rapidly should be challenged through rich and sophisticated problems before any acceleration through new content. Those pupils who are not sufficiently fluent with earlier material should consolidate their understanding, including through additional practice, before moving on.
For many schools and teachers the shift to this ‘mastery curriculum’ will be a significant one. It will require new approaches to lesson design, teaching, use of resources and support for pupils.

3.Key features of the mastery approach

3.1 Curriculum design
A detailed, structured curriculum is mapped out across all phases, ensuring continuity and supporting transition. Effective mastery curricula in mathematics are designed in relatively small carefully sequenced steps, which must each be mastered before pupils move to the next stage. Fundamental skills and knowledge are secured first. This often entails focusing on curriculum content in considerable depth at early stages.

3.2 Teaching resources
A coherent programme of high quality curriculum materials is used to support classroom teaching. Concrete and pictorial representations of mathematics are chosen carefully to help build procedural and conceptual knowledge together.
Exercises are structured with great care to build deep conceptual knowledge alongside developing procedural fluency. The focus is on the development of deep structural knowledge and the ability to make connections. Making connections in mathematics deepens knowledge of concepts and procedures, ensures what is learnt is sustained over time, and cuts down the time required to assimilate and master later concepts and techniques. One medium for coherent curriculum materials is high quality textbooks. These have the additional advantage that pupils also use them to return to topics studied, for consolidation and for revision. They represent an important link between school and home.

3.3 Lesson design
Lessons are crafted with similar care and are often perfected over time with input from other teachers, drawing on evidence from observations of pupils in class. Lesson designs set out in detail well-tested methods to teach a given mathematical topic. They include a variety of representations needed to introduce and explore a concept effectively and also set out related teacher explanations and questions to pupils.

3.4 Teaching methods
In highly successful systems, teachers are clear that their role is to teach in a precise way which makes it possible for all pupils to engage successfully with tasks at the expected level of challenge. Pupils work on the same tasks and engage in common discussions. Concepts are often explored together to make mathematical relationships explicit and strengthen pupils’ understanding of mathematical connectivity. Precise questioning during lessons ensures that pupils develop fluent technical proficiency and think deeply about the underpinning mathematical concepts. There is no prioritisation between technical proficiency and conceptual understanding; in successful classrooms these two key aspects of mathematical learning are developed in parallel.

3.5 Pupil support and differentiation

Taking a mastery approach, differentiation occurs in the support and intervention provided to different pupils, not in the topics taught, particularly at earlier stages. There is no differentiation in content taught, but the questioning and scaffolding individual pupils receive in class as they work through problems will differ, with higher attainers challenged through more demanding problems which deepen their knowledge of the same content. Pupils’ difficulties and misconceptions are identified through immediate formative assessment and addressed with rapid intervention – commonly through individual or small group support later the same day: there are very few “closing the gap” strategies, because there are very few gaps to close.

3.6 Productivity and practice

Fluency comes from deep knowledge and practice. Pupils work hard and are productive. At early stages, explicit learning of multiplication tables is important in the journey towards fluency and contributes to quick and efficient mental calculation. Practice leads to other number facts becoming second nature. The ability to recall facts from long term memory and manipulate them to work out other facts is also All tasks are chosen and sequenced carefully, offering appropriate variation in order to reveal the underlying mathematical structure to pupils. Both class work and homework provide this ‘intelligent practice’, which helps to develop deep and sustainable knowledge.

Deel 2: Bijlage over differentiatie

" I think 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. "
Charlie’s Angles: Approaches to differentiation; defining a ‘mastery’ approach
Thoughts on topical issues of mathematics education from the NCETM’s Director, Charlie Stripp

I’ll be controversial: I think it may well be the case that one of the most common ways we use differentiation in primary school mathematics, which is intended to help challenge the ‘more able’ pupils and to help the ‘weaker’ pupils to grasp the basics, has had, and continues to have, a very negative effect on the mathematical attainment of our children at primary school and throughout their education, and that this is one of the root causes of our low position in international comparisons of achievement in mathematics education.

If my suspicion about the damage caused by current practice in differentiation in many maths lessons is correct, we should do something about it. However, I do recognise that an individual school’s interpretation of differentiation is rarely as black and white as I paint it below, and I know that many primary teachers put a great deal of thought and effort into developing differentiation models for maths teaching. For that reason, we should examine the evidence very carefully and carry out serious trials to help determine whether a different approach will improve children’s mathematical learning.

Put crudely, standard approaches to differentiation commonly used in our 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. This approach is used with the best of intentions: to give extra help to those who are having difficulty with maths, so they can grasp key ideas, and to challenge those who seem to grasp ideas quickly. It sounds like common sense. However, in the light of international evidence from high performing jurisdictions in the Far East, and the ‘mindset’1 research I referred to in my last blog, I’m beginning to wonder whether such approaches to differentiation may be very damaging in several ways.

For the children identified as ‘mathematically weak’:
1.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.
2.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.
3.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’:
1.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.

2.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. A key finding from Carol Dweck’s work on mindsets1 is that you should not praise children for being clever when they succeed at something, but instead should praise them for working hard. That way, they will learn to associate achievement with effort (which is something they can influence themselves – by working hard!), not ‘cleverness’ (a trait perceived as absolute and that they cannot change).
I’m not going to address differentiation in secondary school maths teaching directly here as I plan to make that the subject of a future article in this blog, but I do think much of what I’m saying here also applies at secondary level.

Countries at the top of the table for attainment in mathematics education employ a mastery approach to teaching mathematics. Teachers in these countries do not differentiate their maths teaching by restricting the mathematics that ‘weaker’ children experience, whilst encouraging ‘able’ children to ‘get ahead’ through extension tasks (terms such as ‘weaker’ and ‘able’ are subjective, and imply that children’s ability in maths is fixed - I think they are very damaging and we should stop using them – many teachers already have, but many still use them). Instead, countries employing a mastery approach expose almost all of the children to the same curriculum content at the same pace, allowing them all full access to the curriculum by focusing on developing deep understanding and secure fluency with facts and procedures, and providing differentiation by offering rapid support and intervention to address each individual pupil’s needs. An approach based on mastery principles:makes use of mathematical representations that expose the underlying structure of the mathematics;helps children 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;blends whole class discussion and precise questioning with intelligent practice and, where necessary, individual support.

Colleagues at the NCETM and I have produced this short paper: ‘Mastery approaches to mathematics and the new National Curriculum’ , which defines what we mean by mastery, links it to the National Curriculum, and highlights its implications for the professional development of teachers. This work is supported by the Department for Education, which is keen to see how mastery teaching can raise achievement in schools. This video clip of an English year 2 primary class learning how to add fractions shows mastery teaching in action.

A major element of the NCETM’s leadership and development of mastery teaching is through the DfE-funded ‘England-China Mathematics Education Innovation Research Project’, involving more than 60 teachers from England shadowing primary mathematics teachers in Shanghai (the English teachers are in Shanghai as I write this) to observe mastery teaching in practice, followed by the Shanghai teachers coming to England to exemplify mastery teaching in our classrooms and to support the English teachers in making sense of and trying out a mastery approach to their maths teaching.

This project is being run through the NCETM’s Maths Hubs initiative. Testing out new ideas in the classroom to gather evidence of how effective they are, before advocating which should be adopted more widely, is a key role of the Maths Hubs. The English primary school teachers involved have embarked on this project with great enthusiasm. They have a strong desire to learn as much as they can about how maths is taught in Shanghai and want to use what they learn to develop their own teaching back in England to improve their pupils’ learning.
The project will help us to develop how we use the mastery approach to maths teaching in our primary schools, to improve maths education and the mathematical futures of our young people. It also provides a brilliant opportunity to develop close working relationships between the English and Chinese teachers involved, so that they can learn from each other, to the benefit of teachers and children in both England and Shanghai.

It might also lead us to start moving away from the practice of dividing primary maths classes into different tables, with harmless sounding names, but names which nevertheless don’t fool even the pupils on the ‘red’ table!
It will not be quick or straightforward to improve the learning of our lower attaining pupils, narrowing the wide gaps between pupils’ mathematical attainment that currently exist in our classrooms, but we must be committed to doing so. I believe that mastery teaching will – with time and effort – enable us to achieve this.