What is mastery in mathematics?
The term ‘mastery’ means pupils acquiring a profound, secure, lasting and adaptable knowledge and skillset in mathematics.
You will sometimes also hear the phrase “teaching for mastery” which refers to schoolwide policies and classroom practices that come together to allow children the best opportunities to achieve mastery in mathematics.
For teachers, teaching assistants and school leaders to decide whether children have achieved mastery in mathematics means finding consistently that the children have been given a solid grounding in the fundamentals of mathematics, such as times tables, number bonds and other key facts and mental methods, to allow children to progress fluently within deeper levels of mathematical thinking.
Teaching for mastery applies a similar principle to Carol Dweck’s ‘Growth Mindset’: given ample opportunity and support, if not all then most children can become competent mathematicians. If you are seeking to establish a mastery-principle led classroom, then you must nurture a culture that – with hard work – the children you are teaching can succeed in mathematics.
How mastery in mathematics relates to the National Curriculum
- [Pupils are expected to] 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
- The expectation is that the majority of pupils will move through the programmes of study at broadly the same pace. However, decisions about 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 being offered rich and sophisticated problems before any acceleration through new content. Those who are not sufficiently fluent with earlier material should consolidate their understanding, including through additional practice, before moving on.
Source: National curriculum in England: mathematics programmes of study – GOV.UK
The NCETM’S ‘Five Big Ideas’ and how they have influenced teaching for mastery in mathematics
The ‘Five Big Ideas’ were developed by the NCETM and its Maths Hubs to provide a focus for their Mastery Specialist development programmes. It draws upon the research that instigated and ongoing research into the efficacy of the NCETM’s work in mathematics mastery.
The diagram below has become synonymous with how best to synthesise and express what mastery mathematics is at its core:
- Uses of representations and structure
- Incorporation of procedural and conceptual variation
- Focus on fluency: number bonds, times table etc.
- Mathematical thinking: regular opportunities for reasoning and problem solving
A summary of each of the ‘Five Big Ideas’
In essence, coherence refers to the way that mastery mathematics curricula have lessons that each focus on a small step, with each step following on from the previous lesson and leading onto the next lesson.
With MathShed, each lesson is a small step of its own and within each lesson we provide even smaller steps, providing micro-differentiation where each question or activity is marginally more difficult or complex than the last, allowing children to assume and process new concepts over time and across multiple contexts.
Representation and structure
It is key that the representations and structures used for each lesson are in line with the stage children are at in that moment of exploring or consolidating their understanding of a new concept.
So, for example, when teaching place value, providing children with Base 10 (Dienes) equipment, then moving on to using place value counters within place value charts, to then having children work with digits in place value charts and finally working through place value entirely in the abstract with children no longer needing the prior equipment and representations as a scaffold.
To ensure children have a deep understanding of key facts and new concepts in mathematics, it is essential that they have a diversity of opportunities to think through and discuss their learning. They must regularly be offered the chance to think deeply, reason about and discuss their queries and solutions with adults and peers.
It is for this reason that within MathShed resources you will find plenty of Talking Time questions, plenty of reasoning tasks and reflective Evaluation questions in our lessons.
In simple terms, fluency refers to children being able to recall key facts (number bonds, times tables) with automaticity and procedures (mental methods, such as adding two-digit numbers) at speed and with efficiency. Children should also be able to do so when presented with similar facts or procedures with different surface structures: an ability to switch between different mathematical equipment or pictorial representations with ease.
We know well that to achieve fluency, children need regular varied arithmetic practice which is why MathShed provides multiple resource lines (Quick Maths – our daily arithmetic fluency scheme; our abstract fluency web game/app; and our varied fluency question sets) to support children recall facts and procedures with automaticity.
Well-applied variation comes in twin forms.
First, it refers to the different ways teachers might represent the new concept at hand, doing so in as many ways as is helpfully possible, highlighting key characteristics, which helps in developing a sure, profound understanding of mathematics.
Second, it refers to how well each small step of learning is sequenced, not just lesson-to-lesson or day-to-day but within lessons and week-to-week too, allowing opportunities for recall practice, highlighting what is the same from previous learning and what has changed or become more advanced. In doing so, it also means drawing children’s attention to the importance of mathematical structures and the relationship between each to the others.
One of the ways we have achieved the latter in our MathShed teaching sequences is by providing Starter activities that tend to follow on from previous learning but also signpost where learning is headed next or, sometimes, to highlight a point of contract between these two episodes or concepts.
Teaching for mastery in mathematics with MathShed: the essentials
The key features of classroom teaching for mastery mathematics is based on how lessons are structured in high-performing cities in East and Southeast Asia, namely Shanghai and Singapore. Pupils are taught as a whole class by teachers who provide plenty of direct instruction, coupled with highly-interactive and Socratic tasks.
We try to exemplify this throughout MathShed resources, particularly in our provision of many engaging examples for teachers to model and children to practice or debate within the Talking Time phases of our lessons. By teaching in a whole-class setting and providing plenty of modelling as well as group or peer-to-peer discussion all children are offered the opportunity to succeed in their maths learning with no child or group of children being left behind.
Another key feature of mastery mathematics teaching, seen widely in Shanghai and Singapore, is early or same-day interventions for pupils who have not grasped the concepts covered in the main lesson earlier in the day.
We appreciate that staffing levels and timetables in British schools are not always conducive to being able to provide specialised support with such consistency which is what inspired our digital online MathShed lessons. The lessons can be set by teachers for pupils to complete later in the school day on a tablet or computer (such as a Chromebook) or set as home learning to allow children to be caught up before the next day’s learning.
The structure for a mastery lesson focuses on the small step of learning that it hinges upon, what the key facts or procedures are in achieving the lesson’s goal and any difficulties or misconceptions that may arise, such that a thoughtfully sequenced progression has been mapped out. Most mastery lessons will be set up so all of the children are sat facing the teacher, with the teacher directing the lesson’s interactions, including modelling, questioning, instructions for group, paired or individual activities, as well as feedback and any marking that takes place.
Due to this best practice set-up for mastery teaching and learning, MathShed lessons are multi-phase with a Starter activity, Talking Time slides for modelling, discussion and practice, then independent activities which are followed by mini-plenaries for further discussion, feedback and in-lesson marking. Each lesson also ends with a final reflective ‘Evaluation’ task.
Another key principle of mastery mathematics is that procedural, generally arithmetic, fluency must be introduced hand-in-hand with tasks or representations that provide a deep conceptual understanding of the procedure too.
We have carefully planned how concepts and procedures are introduced, which are the relevant structures or equipment to be used, and then carry these ideas forward from initial conceptualization through to procedural learning and beyond.
For example, when teaching adding fractions with the same denominator it is clear how bar model or strip diagrams are the best-fit representation for introducing the concept and how it leads to abstract fluency.
As well as providing whole-class teaching materials for the introduction of new concepts that encourage intelligent practice, MathShed’s lesson slides and accompanying printable and digital resources offer further opportunities for intelligent practice over time too. By providing example questions with a range of representations and surface structures, children are enabled to couple procedural fluency practice with a deepening of their conceptual understanding.
Again, it is a key feature of mastery mathematics teaching that a lot of teaching time is used in the development of a profound knowledge of the fundamental concepts and procedures that are the foundation of mathematics learning later on at school and beyond. Within mastery mathematics, it is imperative that structures and connections between various mathematical structures are highlighted and explored in detail such that deep learning happens consistently over the course of their mathematical schooling.
To reinforce an earlier point, a keystone of mastery in mathematics is children’s ability to recall fundamental facts such as times tables and number bonds to 10 (then 20 and 100) to limit the occasions of cognitive overload when children have to apply working memory to the exploration and processing of new mathematical concepts. It is for this reason that MathShed has the standalone abstract fluency games to focus on these key facts, which leads to and includes our Year 4 Multiplication Tables Check practice web game.
“If you would like any further support in using MathShed to support your children, groups, classes, school or trust in a mastery journey, don’t hesitate to contact us via email: firstname.lastname@example.org”