1. Understanding the mastery curriculum in Mathematics
Maths Evening 2017
Understanding the mastery curriculum in Mathematics

2. What does mastery mean in the primary maths classroom?
Inspired by teaching approaches developed in Singapore and Shanghai, mastery is an inclusive way of teaching that is grounded in the belief that all pupils can achieve in maths. According to the NCETM(National Centre of Excellence in Teaching Mathematics), a concept is deemed mastered when learners can represent it in multiple ways, can communicate solutions using mathematical language and can independently apply the concept to new problems in unfamiliar contexts.

3. What does it mean to master mathematics?
If you see a page of 20 calculations answered correctly, how would this make you feel?

4. What does it mean to master mathematics?
More explicitly mastery of a mathematical concept involves being able to:
explain it to someone else
make up own examples and non-examples
recognise the concept in new situations
describe the concept in his or her own words
represent the concept in a variety of ways (e.g. using CPA)
make use of the concept in various ways, including in new situations.
This is crucial to our vision for mathematics as a school and is what we will endeavour to deliver in our day-to-day teaching.

5. One curriculum
 It is not the case that some pupils can do mathematics and others cannot.
Pupils should have the opportunity to stay together and work through new content as a whole group. While mastery schemes of work may be challenging for some, the vast majority should be aiming for this standard.
It is important that high-attaining pupils fully understand key number concepts, rather than simply memorise a process. This will reap its rewards in the future at KS3, GCSE and A-level. Teachers can extend high-attaining students through depth, as opposed to acceleration onto new content.

6. Deeper thinking not harder sums
Depth NOT acceleration
The old national curriculum, measured in terms of levels, encouraged undue pace. Children were accelerated onto more complex concepts before really mastering earlier ones.

7. Imagine the cubes to the left represent the building blocks of maths
Imagine the cubes to the left represent the building blocks of maths. As you move on swiftly from one topic to the next the tower gets taller and taller, until eventually it comes crashing down.
By moving on before a child is ready, before they fully understand the concept, you are helping them build a tall tower, which at some point will become unstable.

8. Curriculum: Spiral vs Linear
Many primary schools are still following a spiral curriculum where topics get revisited each term. The nature of this is that you have to move on quite quickly from one topic to the next, whilst not really having time to ensure that the children have properly and deeply understood the concepts being taught. We are now following a linear curriculum in which topics are blocked together and covered in one go. This allows children to gain a deeper understanding of the key topics as there is less pressure to move on before they are ready. Evidence shows that children who have a deeper understanding of number and an enhanced ability to reason mathematically can progress through other topics more quickly.

9. Depth is achieved through variation
There are three elements that are critical on the journey towards mastery in maths. Fluency, reasoning and problem solving. Without one, the next cannot follow and it is only by developing these three skills that children can move towards mastery.
Children must first gain fluency in whatever topic they are studying before they move on to reason mathematically and then begin to solve a variety of different problems which probe and challenge their depth of understanding.

10. Fluency, reasoning and problem solving
Problem solving - Mathematical problem solving is at the heart of our approach. Pupils are encouraged to identify, understand and apply relevant mathematical principles and make connections between different ideas. This builds the skills needed to tackle new problems, rather than simply repeating routines without a secure understanding. Mathematical concepts are explored in a variety of representations and problem-solving contexts to give pupils a richer and deeper learning experience. Pupils combine different concepts to solve complex problems, and apply knowledge to real-life situations.
Reasoning - The way pupils speak and write about mathematics transforms their learning. Mastery approaches use a carefully sequenced, structured approach to introduce and reinforce mathematical vocabulary. Pupils explain the mathematics in full sentences. They should be able to say not just what the answer is, but how they know it’s right. This is key to building mathematical language and reasoning skills.
Fluency - Pupils should be able to recall and apply mathematical knowledge both rapidly and accurately.  However, it is important to stress that fluency often gets confused for just memorisation – it is far more than this. As well as fluency of facts and procedures, pupils should be able to move confidently between contexts and representations, recognise relationships and make connections in mathematics. This should help pupils develop a deep conceptual understanding of the subject. Frequent, carefully designed, intelligent practice will help them to achieve a high level of fluency.
Secure the fundamentals - A large proportion of time is spent reinforcing number to build competency and fluency. Number is usually at the heart of any primary mastery scheme of learning, with more time devoted to this than other areas of mathematics. It is important that pupils secure these key foundations of maths before being introduced to more difficult concepts. This increased focus on number will allow pupils to explore the concepts in more detail and secure a deeper understanding. Key number skills are fed through the rest of the scheme so that students become increasingly fluent. 

11. What does the National Curriculum say?
Aims:
“The national curriculum for mathematics aims to ensure 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
solve problems by applying their mathematics to a variety of routine and nonroutine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions.”

12. Example 1: Fluency Reasoning Problem solving
Reasoning – I know that I need to multiply the numbers on the outside, if the answe is 20 and one number I am multiplying is 2, the other number must be 10. The numbers on the outside seem to be going up/down sequentially so I can see if that would fit.
Problem solving – I can only make 7 with 7 X 1. I can make 18 with 2 X 9 and 3 X 6. I can’t use 4 X 4 to make 16 because I can only use each digit just once, so I will need to do 2 X 8. Therefore I will use 3 X 6 for 18.
Problem solving

13. Fluency
Example 2
Reasoning
Problem solving
Can you see how each task builds on the previous one? The skills required are the same but the depth of understanding required is very different. When these activities are put in the context of whole-class, mixed-ability teaching where the aim is for all children to develop the skills of fluency, reasoning and problem solving at the same time, the slow progression through these three elements is even more critical. 

14. Variation – Intelligent Practice
Variation is NOT:
- more of the same thing but a bit harder.
- the same as variety!
When using intelligent practice, all tasks are selected and sequenced carefully with purpose, offering appropriate variation so that when viewed together they reveal something about the underlying mathematical structure, concept or process. Put simply, variation reveals concepts.

16. Multiple representations for all
Concrete, pictorial, abstract
Objects, pictures, words, numbers and symbols are everywhere. The mastery approach incorporates all of these to help pupils explore and demonstrate mathematical ideas, enrich their learning experience and deepen understanding. Together, these elements help cement knowledge so pupils truly understand what they’ve learnt.
All pupils, when introduced to a key new concept, should have the opportunity to build competency in this topic by taking this approach. Pupils are encouraged to physically represent mathematical concepts. Objects and pictures are used to demonstrate and visualise abstract ideas, alongside numbers and symbols.

17. Concrete
In this stage a student is first introduced to an idea or a skill by acting it out with real objects. In division, for example, this might be done by separating balls into groups of red ones and green ones or by sharing 10 biscuits among 5 children. This is a ‘hands on’ component using real objects and it is the foundation for conceptual understanding.
Pictorial
In this stage a student has sufficiently understood the hands-on experiences performed and can now relate them to representations, such as a diagram or picture of the problem. In the case of a division exercise this could be the action of circling objects.
Abstract
In this symbolic stage a student is now capable of representing problems by using mathematical notation, for example: 10 ÷ 2 = 5 Students only use abstract numbers and figures when they have enough context to understand what they mean This is the ‘final’ and most challenging of the three stages.

18. Example:
The following example demonstrates the principles of CPA. It starts with a pictorial representation of the problem using a clear diagram (the use of counters is important for any child still at the concrete stage), but it then builds the abstract alongside, so that the child can make that leap between the pictorial and the abstract as and when they are comfortable in doing so.
Concrete – Before the pictorial stage, children should have plenty of opportunities to practise physically sharing out or grouping different manipulatives to gain a solid understanding of the concept

19. Example:
In this second example the concrete is used as a starting point for the question. Two oranges are divided up into quarters, three of them are circled from each which leaves six quarters which are then put back together to make one and a half oranges. The pictorial stage sees the introduction of circles instead of oranges and then the final stage is the introduction of the abstract, the numerical calculation. By putting all three alongside each other students can see the progression and when they are ready, they will naturally make the transition from concrete to pictorial to abstract.

20. In Summary… Teach fewer concepts in greater depth
Intelligent practice through secure teacher subject knowledge and higher order questioning should give children a rich appreciation of the concepts and foster inquisitive and rounded mathematicians.
If we can help children unlock this depth of learning then, over time, whole class, mixed ability teaching will become even more successful.