Our National Curriculum Key Aims:

To become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils have conceptual understanding and are able to recall and apply their knowledge rapidly and accurately to problems

To reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language
To 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.

A central component in the NCETM/Maths Hubs programmes to develop Mastery Specialists has been discussion of Five Big Ideas, drawn from research evidence, underpinning teaching for mastery.
The 5 Big Ideas

Fluency
Fluency demands more of learners than memorisation of a single procedure or collection of facts. It encompassed a mixture of efficiency, accuracy and flexibility.
Quick and efficient recall of facts and procedures is important in order for learners’ to keep track of sub problems, think strategically and solve problems.
Fluency also demands the flexibility to move between different contexts and representations of mathematics, to recognise relationships and make connections and to make appropriate choices from a whole toolkit of methods, strategies and approaches.

Variation
Provide opportunities for intelligence practice where pupils can
focus on relationship not just procedure
explain what is on and make connections
use one problem to work out the next
create their own examples

Coherence

Small steps are easier to take
Focusing on one key point each lesson allows for deep and sustainable learning
Certain images, techniques and concepts are important re-cursors to later ideas. Getting the sequencing of these right is an important skill in planning and teaching for mastery.
When something has been deeply understood and mastered, it can and should be used in the next steps of learning

Structures and Representation
Representations used in lessons expose the mathematical structure being taught, the aim being that students can do the maths without recourse to the representation.

Mathematical Thinking
Mathematical thinking is central to deep and sustainable learning of mathematics.
Taught ideas that are understood deeply are not just “received” passively but worked on by the learner. They need to be though about, reasoned with and discussed.
Mathematical thinking involves:
Looking for pattern in order to discern structure;
looking for relationships and connecting ideas;
reasoning logically, explaining, conjecturing and proving.