The Essence of Maths Teaching for Mastery

• Maths teaching for mastery rejects the idea that a large proportion of people ‘just can’t do maths’.
• All pupils are encouraged by the belief that by working hard at maths they can succeed.
• If a pupil fails to grasp a concept or procedure, this is identified quickly and early intervention ensures the pupil is ready to move forward with the whole class in the next lesson.
• Lesson design identifies the new mathematics that is to be taught, the key points, the difficult points and a carefully sequenced journey through the learning.
• Procedural fluency and conceptual understanding are developed in tandem because each supports the development of the other.
• It is recognised that practice is a vital part of learning, but the practice used is intelligent practice that both reinforces pupils’ procedural fluency and develops their conceptual understanding.
• Significant time is spent developing deep knowledge of the key ideas that are needed to underpin future learning. The structure and connections within the mathematics are emphasised, so that pupils develop deep learning that can be sustained.
• Key facts such as multiplication tables and addition facts within 10 are learnt to automaticity to avoid cognitive overload

What does it mean to master mathematics?

A mathematical concept or skill has been mastered when a pupil can represent it in multiple ways, has the mathematical language to communicate related ideas, and can independently apply the concept to new problems in unfamiliar situations.

Mastery is a journey and long-term goal, achieved through exploration, clarification, practice and application over time. At each stage of learning, pupils should be able to demonstrate a deep, conceptual understanding of the topic and be able to build on this over time.

This is not about just being able to memorise key facts and procedures, which tends to lead to superficial understanding that can easily be forgotten. Pupils should be able to select which mathematical approach is most effective in different scenarios.

All pupils can achieve in mathematics

A positive teacher mindset and strong subject knowledge are key to student success in mathematics. It is not the case that some pupils can do mathematics and others cannot.

By making high expectations clear and emphasising the value of mathematics education, pupils are encouraged to build confidence and resilience. A positive teacher mindset in maths encourages a love of learning and resilience that enables everyone to achieve.

Focus on depth - Deepen understanding before accelerating content coverage

All pupils benefit from deepening their conceptual understanding of mathematics. Pupils must be given time to fully understand, explore and apply ideas, rather than accelerate through new topics. This approach enables pupils to truly grasp a concept, and the challenge comes from investigating it in new, alternative and more complex ways.

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.

Concrete – Pupils should have the opportunity to use concrete objects and manipulatives to help them understand and explain what they are doing.

Pictorial – Pupils should then build on this concrete approach by using pictorial representations. These representations can then be used to reason and solve problems.

Abstract – With the foundations firmly laid, students should be able to move to an abstract approach using numbers and key concepts with confidence.

Fluency, reasoning and problem solving – teaching supports the aims of the National Curriculum

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.

Number at the heart - Secure the fundamentals

A large proportion of time is spent reinforcing number to build competency and fluency, 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.

What will maths look like in these classrooms?

In lessons you will see teachers slowing down and lessons will have a whole class focus, but where they seek to ensure depth due to the use of representations to identify structures and a clear focus on the small steps required for future learning.

Lesson design identifies new mathematics being taught, the key points, the difficult points and a carefully sequenced journey though the learning. In a typical lesson the teacher leads back and forth interaction, including questioning, short tasks, explanation, demonstration and discussion.

Significant time is spent developing the deep knowledge of the key ideas and mathematical structures that are needed to underpin future learning. The structure and connections within the mathematics are emphasised, so that pupils develop deep learning that can be sustained.

Procedural fluency and conceptual understanding are developed in tandem and intelligent practice reinforces this.

Key number facts such as multiplication tables and addition facts to 10 are given sustained focus so that they are learnt to automaticity in order to avoid cognitive overload in the working memory and enable the pupil to focus on new concepts.