Elements of Teaching for Mastery

In this page we try to gather examples of what Teaching for Mastery might mean. The resources here have been developed by a number of different parties including the National Maths Hubs Programme. They are provided as is, but as with most CPD, much of the learning stems from the quality discussions with other professionals. 

Schools may want to start with the self evaluations documents below to reflect on their current practice.
Primary Teaching for Mastery Self Evaluation - Document (based on the work by www.glowmathshub.com)
Secondary Teaching for Mastery Self Evaluation - Document (based on the work by www.booleanmathshub.org.uk)
Fractions Sheet 1sheet 2sheet 3: A method of curriculum design that explores the "journey" through a topic and focuses on key aspects, difficult points, representations, variation and stem sentences
Developing Number Sense - A video by Jo Boaler on what we mean by number sense.
NCETM Essence of Teaching for Mastery - This document defines the fundamental elements of primary mathematics teaching for mastery.
NCETM Guidance on Marking (Secondary): The guidance suggests that the widespread practice of teachers giving individual and unique written tips and targets to every child in a class after every piece of work is a bad use of time.
Assessment Guidance from Kent and Medway Maths Hub

Teaching for Mastery can be seen to be broken down into the following elements:

 Feature         Exemplification Guidance/Examples/Links/Stimuli
There is a change of ethos within the establishment away from rapid acceleration through material to a deep conceptual understanding of mathematics.

This includes a belief that all pupils are capable of understanding and doing mathematics, given sufficient time. Pupils are neither ‘born with the maths gene’ nor ‘just no good at maths’.

With good teaching, appropriate resources, effort and a ‘can do’ attitude all children can achieve in and enjoy mathematics.

All pupils are encouraged to develop a growth mindset.

The class work together on the same key point, whilst at the same time challenging and supporting pupils to gain depth of understanding and proficiency. Acceleration to higher content is avoided.

High expectations for all.
NCETM Essence of Teaching for MasteryThis document defines the fundamental elements of primary mathematics teaching for mastery.
ACME - Raising the Bar: This Dec 2012 report from ACME (The Advisory Committee on Mathematics Education) argues for a curriculum emphasising depth and connections rather than acceleration to new content.
Dweck 2008 - Mindsets: 
The work of Carol Dweck (Stanford University) is the seminal work on growth mindsets and Jo Boaler’s (Stanford University) work (below) has given a specific mathematical dimension to these important ideas.
Jo Boaler - Positive Norms
Jo Boaler - Fluency Without Fear
NCETM Mastery: Microsite
What is Mastery - Debbie Morgan NCETM: A video presentation given by Dr Debbie Morgan (NCETM's Primary Director) to teachers at a conference in Lincolnshire.
What is Teaching for Mastery - An article by Charlie Stripp (NCETM Director) in SchoolsWeek.
ACSL Guidance: Guidance from the ACSL
Mastery in Action (Secondary): An account from greatmathsteachingideas.com of King Solomon Academy's approach to teaching for mastery.
Ofsted Q&A
A set of questions generated by maths teachers and responses given by Ofsted’s Sean Harford and Jane Jones from an #OfstedMaths Twitter chat hosted by Bruno Reddy and held on June 18th 2015.
Jane Jones HMI
A guest blog from Jane Jones (Ofsted National Lead for Mathematics ) outlining Ofsted’s thoughts on mastery, differentiation, etc.
Ofsted and Mastery: Series of slides by Jane Jones HMI given in a workshop at a Maths hub Conference
Five Myths of Mastery: A piece by the National Association of Mathematics Advisors.
NCETM Guidance on Marking: The guidance suggests that the widespread practice of teachers giving individual and unique written tips and targets to every child in a class after every piece of work is a bad use of time.
NCETM Guidance on Marking (Secondary): The guidance suggests that the widespread practice of teachers giving individual and unique written tips and targets to every child in a class after every piece of work is a bad use of time.
Early Years MathsThis article by Sue Gifford (University of Roehampton) was originally published on the Advisory Committee on Mathematics Education (ACME) blog and now appears on the NRICH site. She argues that if we want to create more positive attitudes and higher achievement in mathematics, what better place to start than in the early years?
Role of Mastery in Nurturing Young Mathematicians: NRICH article on problem solving.
Catching Up: An Australian report by Ben Jensen (2012) looking at how the success in Hong Kong, Singapore, Shanghai and Korea might be explained and emulated.
Family Maths Toolkit - Website for encouraging families to engage with maths.
#YesUCan: Work and resources from the GLOW Maths Hub on creating a 'can do' attitude towards mathematics.
NCETM Case Studies - Supporting Teaching for Mastery
NCETM Primary Videos - Videos showing elements of Mastery
How Everyone Can Succeed: TES article from White Rose Maths Hub
Maths: Everyone Can: Mind set video from White Rose Maths Hub
Research Documents: A document with a collection of relevant articles and links, some of which are on this site separately, which support teaching for mastery.
Curriculum Design

All pupils need access to these concepts and ideas and to the rich connections between them. There is no such thing as ‘special needs mathematics’ or ‘gifted and talented mathematics’. Mathematics is mathematics and the key ideas and building blocks are important for everyone.

A detailed curriculum is mapped out across all phases, ensuring continuity and supporting transition.

Considerable time is spent on securing fundamental skills and knowledge in the early stages.

There are clear minimum expectations for each year that all children (except those on individualised curricula) will leave the year group having mastered.

It is not expected that concepts and procedures will be re-taught later in the year or the following year; instead they will be built on.

The structure of the day / timetable allows for same day intervention to ensure all children master the key concepts and skills before moving on to the next lesson.

Exemplar SOW Primary [Including Mixed Age Classes]These freely available set of mastery overviews and schemes of learning have been developed by primary practitioners in conjunction with the White Rose Maths Hub to provide a curriculum plan that will support ‘Teaching for Mastery’.
Exemplar SOW SecondaryThis is the outline of the Mathematics Mastery secondary curriculum plan, designed for use by school leaders and teachers in schools in the ‘Mathematics Mastery’ partnership. Mathematics Mastery is happy to share this programme of study with schools beyond this community in order to support preparation for and implementation of the new National Curriculum, but please bear in mind that it is designed to be used in conjunction with the detailed Mathematics Mastery unit guides and resources.
Exemplar SOW Y7-9: Freely available overviews and schemes from the White Rose Maths Hub.
Exemplar SOW Y7-Y11A freely available collection of schemes of work from Kangaroo Maths for Key Stages 1 to 5.
Exemplar Y7-Y11 ProgressionA freely available collection of progression maps from Kangaroo Maths Key Stages 1 to 5.
Plymouth CIMT Resources (Primary)A comprehensive set of plans produced by the Centre for Innovation in Mathematics Teaching (University of Plymouth) with links to lesson plans, resources etc.
NRICH EYFS Guidance: An Article by Jenni Back 
explores the basic foundations of number sense and outlines relevant research in this area.
Teaching Mixed Age Classes: A report by Devon LA and Babcock into the teaching of students in mixed age classes.
Teacher Research Groups: Information about TRGs from NCETM.
EEF Toolkit - An evidenced based impact on education
Fractions Sheet 1, sheet 2, sheet 3: A method of curriculum design that explores the "journey" through a topic and focuses on key aspects, difficult points, representations, variation and stem sentences
Shanghai Curriculum (Primary & Junior): Example of the Shanghai curriculum collected during the exchange
Shanghai Curriculum (Secondary): Example of the Shanghai curriculum collected during the exchange
Lesson Design & Resources Lessons are carefully designed and structured to develop the necessary small conceptual steps for mastery.
Examples are chosen carefully to highlight the important conceptual ideas and tasks are chosen to provide pupils with intelligent practice
NCETM Calculation GuidanceThis document summarises and synthesises the discussions that took place between teachers who were part of the Shanghai/England primary exchange in 2014. It is not intended to be a calculation policy as such; rather, it could sit alongside a school’s existing policy, and the ideas captured here (which are indicative and not exhaustive) could inform and enhance teaching across all primary key stages.
Calculation Policy: By White Rose Maths Hub
NCETM Videos (Primary): A collection of videos from NCETM on Youtube.
NCETM Videos to Support Mastery: From Dr Debbie Morgan and a Y3 class focusing on the 6 times table at Two Mile Ash School.
Developing Number Sense - A video by Jo Boaler on what we mean by number sense.
Example Shanghai Powerpoint filesA series of slides from visiting Shanghai teachers showing how examples were carefully crafted for different lessons. Collection of lessons on multiplication and division with 2-digit numbers fractions and decimals as well as addition and subtraction of fractions.
Y2 and Y6 Problems from @WRMathshub. Series of reasoning problems published throughout March 2016.
KS1 Reasoning Problems from @WRMathsHub
KS2 Reasoning Problems from @WRMathsHub
NCETM National Curriculum Resource Tool
The National Curriculum for Mathematics Resource Tool by NCETM gives guidance on progression and glossaries.
GCSE Problem Solving PPT: By @EJMaths based on the AQA GCSE Problem Solving Booklet.
NCETM Multiplicative Reasoning Materials:
suggested programme of professional development (PD) using the materials in this microsite. It is designed as an integrated series of workshops for KS3 teachers with associated lessons for KS3 classes.
Key Understanding in Mathematics Learning: A series of articles by the Nuffield Foundation on learning and maths.
Fractions Topic Breakdown: An example of how a topic can be broken down into a sequence of lessons by Surrey Plus Maths Hub.
Progression in U+U addition facts: A table of addition facts up to 10+10, with suggested teaching sequence. With explanation.

Classroom Practice

Good use of concrete, pictorial and abstract (CPA) representations are provided and pupils are encouraged to connect and see relations between these different representations.

The CPA approach is based on Jerome Bruner’s stages of representation.

Using technical and precise language.

Bruner's Stages of RepresentationJerome Bruner is a psychologist who focused much of his research on the cognitive development of children and how it relates to education.
Concrete-Pictorial-Abstract: Surveying its origins and charting its futureAn article by Leong Yew Hoong, Ho Weng Kin, Cheng Lu Pien of the National Institute of Education, Nanyang Technological University, Singapore.
Improving Learning in Mathematics - Challenges and Strategies: Standards Unit by Malcolm Swam, University of Nottingham.
The Bar Model: An overview from NCETM of use of the Bar Model for visualisations.
Bar Model (ppt): A powerpoint from Highworth Grammar School used of in house CPD.
www.thesingaporemaths.com: Links from their website to modelling problems.
Thinking Blocks: Interactive tutorials for use with the Bar model.
Numicon: Youtube video about using Numicon.
Cuisenaire: Cuisenaire website.
Primary Shanghai Exchange Feedback: Mid Exchange report from the NCETM Shanghai Primary Exchange.
Nrich Manipulatives: Report by Jenni Back on the use of manipulatives in the classroom.
NRICH Use of Cuisenaire: List of resources by NRICH on using Cuisenaire.
CPA Approach: Video by Maths no Problem by 
Dr. Yeap Ban Har on the concrete, pictorial, abstract approach.

CPA:Article by mathteachingstrategies.wordpress.com in the CPA approach.
Multiplicative Reasoning: Resources for CPD by NCETM on developing multiplicative reasoning within the curriculum aimed at KS3.
Instructional Guidance: An article by Jinfa Cai et al (2013) looking at coherence, in particular instructional coherence in the classroom.
Support & Differentiation

The whole class are taught together ensuring that all pupils have access to the important mathematical ideas, skills and concepts.

Differentiation is achieved through questioning and scaffolding rather than by offering different tasks.

All pupils are regularly challenged through more demanding questions which deepen their understanding

Planning Different Types of Questions: NCETM guidance on planning different types of questions to support and challenge learners.
Questions Than Can Extend: Guidance from the Medway Numeracy team on strategies for questioning.
Charlie Stripp's Blog on Differentiation: One of the 'Charlie's Angels' blogs on topical issues by Charlie Stripp, Director of NCETM.
Maths Mastery Differentiation: Advice from Mathematics Mastery on how to teach the same topic to a class of students with different levels of attainment.
How to Extend a Task: Video by Maths No Problem from Dr Yeap Ban Yar explaining how a lesson can be used for enrichment.
Making Best Use of TAs: An EEF guidance report on the effective use of TAs
Same Day Intervention: Summary of outcomes from same day interventions from a school from the Outwood Grange Academies Trust
Questioning Template: A questioning grid based on the work by EEF to encourage deeper questioning.
Productivity & Practice

It is important for pupils to develop their procedural fluency alongside conceptual understanding and practice is an important part of achieving this.

However, mechanical practice should be avoided. Pupils need to be supplied with tasks which require ‘intelligent practice’ and have been designed using the principles of variation theory.

Regular homework is set and pupils are encouraged to monitor their own achievements and understanding.
Variation: An article by Lai and Murray.
Lo Variation Theory (2012): A short book on Variation Theory and the improvement of Teaching and Learning by Mun Ling Lo.
Variation Example: An example of variation taken from Mike Askew's book entitled 'Transforming Primary Mathematics, with notes.
Column Addition - Variation: Extracts from NCETM's Primary Magazine's New NC in Focus.
Column Subtraction - Variation: Extracts from NCETM's Primary Magazine's New NC in Focus.
Short Multiplication - Variation~: Extracts from NCETM's Primary Magazine's New NC in Focus.
Short Division - Variation: Extracts from NCETM's Primary Magazine's New NC in Focus.
Variation: A video of a talk given by Dr Debbie Morgan about variation theory.
Assessment     Assessment values depth of learning, knowing 'why' as well as knowing 'that' and knowing 'how'

NCETM Assessment MaterialsProduced by NCETM, working in conjunction with the Maths Hubs programme and Oxford University Press, these materials are designed to support teachers in assessing pupils’ depth of understanding.
They are divided into six separate documents, one for each of Years 1 to 6 inclusive and each document starts with an introduction to the principles of teaching for mastery and the implications for assessment in mathematics in the context of the new curriculum.
Exemplar Primary AssessmentsFreely available assessment items from the White Rose Maths Hub linked to the mastery overviews and schemes of learning described earlier in this document.
Exemplar +/- and Fractions: Examples of assessment materials provided by Chafford Hundred Primary at the National Forum Shanghai Primary Exchange 2016 with ppt.
Exemplar Y7 Assessments: Freely available assessments for Y7 from the White Rose Maths Hub
Assessing without Levels - A stage based approach by Kangaroo Maths
Assessment Guidance from Kent and Medway Maths Hub
Assessing Addition & Subtraction
Assessing Fractions
Assessing Multiplication & Division
Assessing Place Value
Assessing Reasoning

Classroom Displays
A link here to a number of freely available displays for classrooms.

We would welcome your feedback on the materials and ideas above. If you have any comments please email j.dodd@highworth.kent.sch.uk

Subpages (2): Core Maths NewGCSE