Delivering Excellence in Mathematics ALT 01 V3

DELIVERING EXCELLENCE IN MATHEMATICS
Guidance Document

This guidance report was written by Amanda Wilson (Waterton Academy Trust)

The author was supported by an advisory group which consisted of:
David Dickinson (WatertonAcademy Trust), Linsey Cavell (Waterton Academy Trust),
Head Teachers’ Board (Waterton Academy Trust), Darren Dickinson (Omnibus Educaton)
and Jonathan Sharp (Doncaster Research School).

The guidance is based on a review of the statutory documents and research evidence below:
• National Curriculum in England: Mathematics Programme of Study (DFE 2013)
• Improving Mathematics in Key Stages 2 and 3 (EEF Nov 2017)
• NCETM (National Centre for Excellence in the Teaching of Mathematics) Teaching for Mastery

https://www.ncetm.org.uk/resources/50819
• NRICH Project https://nrich.maths.org/
• Rosenshine’s Principles of Instruction (American Educator 2012)
• Metacognition and Self-regulated Learning (EEF 2018)
• OFSTED Education inspection framework Overview of research (Jan 2019)

The author would like to thank colleagues and practitioners from within and beyond the
Trust who provided support and feedback on drafts of this guidance.

Contents

4 FOREWORD

6 INTRODUCTION

7 ACTING ON GUIDANCE

8 SUMMARY OF RECOMMENDATIONS

RECOMMENDATIONS

11 RECOMMENDATION 1
MASTERY

12 RECOMMENDATION 2
REASONING

13 RECOMMENDATION 3
FLUENCY

14 RECOMMENDATION 4
PROBLEM SOLVING

16 RECOMMENDATION 5
ASSESSMENT AND MODERATION

18 RECOMMENDATION 6
MANIPULATIVES AND
REPRESENTATIONS

20 RECOMMENDATION 7
MATHEMATICAL LANGUAGE

22 RECOMMENDATION 8
PEDAGOGY

24 HOW WAS THIS GUIDANCE
COMPILED?

25 GLOSSARY

32 REFERENCES

Foreword

4

Research shows us that leaving school with Our school improvement team has worked
good educational qualifications is a tirelessly with colleagues from the Research
prerequisite for progressing into quality jobs, School Network, other trusts, external experts
apprenticeships, and further education. and independent educational companies to
create a number of reports that are designed
The knowledge and skills our young people learn to support our schools in their decision making
at school are vital to prepare them for process and signpost them to the best
employment and for a life of achievement and evidence possible
independence. Yet too many of our young people
across do not achieve their full potential in terms I do hope these reports will help to support the
of academic qualifications and as a result, risk a development of consistently excellent,
life of social and economic exclusion. evidence-informed teaching that creates great
opportunities for all our children. It is a starting
As a trust, we at Waterton believe the best way to point for a more evidence-informed approach to
ensure we provide the best education possible teaching for our organisation and one which I am
for the pupils in our charge and to give them the particularly excited about. As a trust, we will
life chances they deserve is through better use of continue to work with partners, particularly the
evidence: looking at what has—and has not— network of Research Schools, universities, local
worked in the past can put us in a much better authorities, independent advisors and other
place to judge what is likely to work in the future. trusts and will be producing a range of
But as we know, it can be difficult to know where supporting resources, tools, and training to
to start. There are thousands of studies and help you implement the recommendations in
research papers about teaching available to your classrooms.
teachers and trusts and schools are inundated
with information about programmes and training David Dickinson OBE
courses which make claims about impressive Chief Executive Officer Waterton Academy Trust.
evidence of impact.
How can anyone know which findings are the
most secure, reliable, and relevant to their school
and pupils?

This is why as a trust we have decided to produce
our own guidance reports. Each report will offer
practical, evidence based recommendations that
are relevant to all pupils and can be applied
across the trust in ways that best suits the many
different contexts we have. To develop the
recommendations, as a trust we have reviewed
the best available research and consulted with
leaders, teachers and external experts to arrive
at the key principles for effective teaching.

5

Introduction

What does this guidance cover? Who is this guidance for?

This guidance report focuses on the teaching of This guidance is aimed primarily at subject
mathematics from the start of Nursery to the end leaders, head teachers, and other staff with
of Year 6. The decision to focus on these Key responsibility for leading improvements in the
Stages was made after an initial consultation teaching of mathematics in Waterton schools.
period with our school improvement team, Classroom teachers and teaching assistants will
teachers and members of the advisory group. also find this guidance useful as a resource to aid
their day-to-day teaching.
It is not intended that this report will provide a
comprehensive guide to mathematics teaching. It may also be used by:
We have made recommendations where there
are research findings that schools can use to • governors to support and challenge
make a significant difference to pupils’ learning,
and have focused on the areas that appear to be school staff
most relevant to practitioners. There are aspects
of mathematics teaching not covered by this • parents to support their childs’s learning;
guidance. In these situations, teachers must draw
on their knowledge of mathematics, professional • school improvement professionals to inform
experience and judgement, and assessment of their development of both professional
their pupils’ knowledge and understanding. development for teachers and interventions
for pupils;

The focus is on improving the quality of teaching.
Excellent mathematics teaching requires good
content knowledge, but this is not sufficient.
Excellent teachers also know the ways in which
pupils learn mathematics, the difficulties they are
likely to encounter, and how mathematics can be
most effectively taught.

6

Acting on the guidance

We recognise that the effective 1 - Continuing Professional Development (CPD) 3 - It is important to consider the precise detail
implementation of these recommendations - will be an important component of provided beneath the headline
such that they make a real impact on children implementation and is key to raising the recommendations. Schools must consider
- is both critical and challenging. quality of teaching and teacher knowledge. As carefully if they have the capacity and
a trust we will be providing professional resources to effectively implement the
There are several key principles to consider when development for all our schools and this will recommendations.
acting on this guidance. be aligned to the guidance provided by the
Teacher Development Trust. 4 - Inevitably, change takes time, and we
recommend taking at least two terms to plan,
2 - These recommendations do not provide a ‘one develop, and pilot strategies on a small scale
size fits all’ solution. It is important to consider before rolling out new practices across the
the delicate balance between implementing school. Gather support for change across
the recommendations faithfully and applying the school and set aside regular time
them appropriately in a school’s particular throughout the year to focus on this project
context. Implementing the recommendations and review progress.
effectively will therefore require careful
consideration of context as well as sound
professional judgement.

IMPLEMENTATION PROCESS BEGINS

Treat scale-up as a new Identify a key priority that is
implementation process amenable to change

Continuously acknowledge Systematically explore
support and reward good programmes or practices
implementation practices to implement

Plan for sustaining and Examine the fit and
scaling the intervention feasibility with the
from the outset school context

STABLE USE SUSTAIN EXPLORE ADOPTION
OF APPROACH DELIVER PREPARE DECISION

Use implementation data Develop a clear, logical
to drive faithful adoption and well specified plan
and intelligent adaption
NOT READY
Reinforce initial training - ADAPT PLAN
with follow-on support
within the school Assess the readiness of
the school to deliver the
Support staff and solve implementation plan
problems using a flexible
leadership approach READY

Prepare practically e.g.
train staff, develop
infrastructure

DELIVERY BEGINS

Figure 1:
Implementation can be described as a series of stages relating to thinking about, preparing for, delivering, and sustaining change.
Education Endowment Foundation (2019) Putting Evidence to Work: A School’s Guide to Implementation

7

SUMMARY OF 1 2
RECOMMENDATIONS
MASTERY REASONING

Develop a mastery Reasoning underpins
approach so all children every maths lesson
develop confidence and
competence in maths

• Teachers expect all children • The ability to reason
are capable of achieving high mathematically is the most
standards in maths. important factor in a pupil’s
success in mathematics.
• Children move at broadly the
same pace. • Reasoning should underpin
every maths lesson.
• Questioning is precise; it tests
procedural and conceptual • The ability to reason does not
knowledge and identifies depend on arithmetic ability.
those pupils needing
intervention to keep up. • Very young children can
reason.
• Mastery uses methodical
curriculum design, carefully • Reasoning should be explicitly
crafted lessons and a small taught.
steps approach.
• Teachers should model their
• Resources foster deep own reasoning.
conceptual and procedural
knowledge.

• Practice and consolidation are
key. Variation is used to build
fluency and understanding
together.

• Emphasis is on pupils making
connections and developing
deep understanding.

11 12

8

3 4 56

FLUENCY PROBLEM ASSESSMENT MANIPULATIVES
SOLVING AND MODERATION AND
Children develop REPRESENTATIONS
efficiency, accuracy and Children are taught Develop accurate
flexibility effective strategies assessment and Use manipulatives and
moderation processes representations to
uncover mathematical
structures

• Fluency encompasses a • Problem solving strategies • Teachers start by determining • Manipulatives (physical
mixture of efficiency, begin in Early Years. what children already know objects used to teach maths)
accuracy and flexibility. and plan lessons accordingly. and representations (such as
• Problem solving refers to number lines or part whole
• Fluency begins in Early Years situations in which pupils do • Teachers use formative models) can help children of
with subitising and not have a readily-available assessment to adapt their all ages to engage with
partitioning. method that they can use. teaching on an ongoing basis. mathematical ideas.

• We need to move away from • Problem solving requires • Developmental progressions • Manipulatives should help
counting strategies as soon as direct teaching to expose starting from Early Years children to build abstract
possible; the ability to mathematical structures. inform decisions around what mental models.
partition single digit numbers a child should learn next.
in a variety of ways is crucial. • To have long term impact, • They need to be used
Children need to focus on the • Misconceptions and purposefully and
• Teachers should create daily problem solving process not difficulties are addressed with appropriately to have an
opportunities to develop just the answer. rapid intervention. impact.
fluency.
• Research suggests pupils • Knowledge of common • Pupils need to understand
• Fluency is more than just should be taught to: misconceptions is used to links between the
memorising facts. address errors before they manipulative and the ideas
- Use and compare different arise. they represent.
• The key to fluency is in making approaches.
connections. • Effective feedback is specific, • Manipulatives should act as a
- Interrogate and use existing clear, encourages and temporary scaffold.
• By offering children practice in mathematical knowledge. supports further effort and is
context we help them to make given sparingly. • They should expose the
links between the types of - Monitor, reflect on and underlying structure of the
situations that a strategy communicate their problem • Summative assessment tests maths.
might be used for. solving. are used to support teacher
assessments. • Manipulatives can support
• Use of the Concrete Pictorial mathematical discussion.
Abstract (CPA) model helps to • Moderation allows teachers to
build fluency. benchmark judgements, while
ensuring consistent standards
• Children need to talk about and reliable outcomes.
their maths to develop
fluency.

13 14 - 15 16 - 17 18 - 19

9

7 8

MATHEMATICAL MATHEMATICS
LANGUAGE PEDAGOGY

Use precise language to Teachers develop
support mathematical effective mathematics
thinking pedagogy

• Using the correct • Maths should be taught daily.
mathematical terminology
and using full sentences to • Teachers:
explain their thinking • Develop their own
improves children’s understanding of maths.
reasoning and conceptual • Teach in a precise way so all
understanding. pupils engage in the
learning.
• Mathematical talk reveals • Make rich connections across
pupils’ understanding and mathematical ideas.
misunderstandings. • Support children to build
abstract mental methods.
• It also supports language • Understand the sequence of
development, the learning.
development of social • Weave reasoning through all
skills/learning behaviours and learning.
supports learning by boosting • Teach problem solving
memory. strategies.
• Break learning into smaller
• Teachers need to model using steps.
precise mathematical • Model effectively.
language and verbalising their • Use guided practice and
own thoughts. spaced practice.
• Are aware of cognitive load.
• Teachers improve the quality • Question effectively.
of mathematical discussion by • Use the environment to
guiding and scaffolding pupils. support learning.

• Stem sentences support
children.

20 - 21 22 - 23

10

1 1

MASTERY

Develop a mastery approach so all children achieve confidence and competence in maths

Teaching for mastery, as advocated by the • Teaching is supported by methodical curriculum The aim of these approaches is to provide all
National Centre for Excellence in the Teaching design and by carefully crafted lessons and children with full access to the curriculum,
of Mathematics (NCETM), is largely based on resources that expose the structure of maths enabling them to achieve confidence and
the way maths has been successfully taught and develop an understanding of how and why competence – ‘mastery’ – in mathematics, rather
in East Asia, particularly in Shanghai, China. it works. than many failing to develop the maths skills
The Programme for International Student they need for the future.
Assessment (PISA) shows that East Asian Teachers can use the White Rose Scheme of
countries including China are consistently the Learning and NCETM PD resources to inform EVIDENCE SUMMARY
highest performing in mathematics planning. Exercises should be structured with
internationally.¹ great care to build deep conceptual knowledge • The content and principles
alongside developing procedural fluency. An underpinning the 2014 mathematics
Though there are many differences between the effective curriculum is designed in relatively small curriculum reflect those found in high
education systems of England and those of East carefully sequenced steps, which must each be performing education systems
and Southeast Asia, we can learn from the mastered before pupils move on to the next internationally, particularly those of
‘mastery’ approach to teaching, which is stage. Fundamental skills and knowledge are East and Southeast Asian countries
commonly followed in these countries. secured first. This often entails focusing on such as Singapore, Japan, South Korea
curriculum content in considerable depth at early and China.²
According to the NCETM², the following principles stages.
and features characterise the mastery approach: • Key features of mastery are
• Lessons include a variety of representations to Coherence, Variation, Representation
• Teachers reinforce an expectation that all pupils introduce and explore a concept effectively. and Structure, Mathematical Thinking
are capable of achieving high standards in and Fluency. (The 5 Big Ideas)
mathematics. • Number objectives are prioritised.
• Children move through the content at
• Positive mind-sets are promoted; we learn • Practice and consolidation play a central role. the same pace, they experience the
from mistakes. Evidence from cognitive science research³ same tasks although some children
suggests that learning key facts so they can be will go deeper.
• The large majority of pupils progress through recalled automatically ‘frees up’ working
the curriculum content at the same pace. memory. Working memory can then focus on • Teaching for mastery is in the early
There is no differentiation in the content taught, more complex problem solving, rather than stages of adoption in England. The
rather in the questioning and scaffolding reaching cognitive overload trying to calculate NCETM continues to monitor research
individual pupils receive in class as they work simple operations. and evidence of its effectiveness.⁵
through problems. Pupils work on the same
tasks and engage in common discussions. • Carefully designed variation or ‘intelligent
Concepts are often explored together to make practice’ builds fluency and understanding of
mathematical relationships explicit and underlying mathematical concepts in tandem.
strengthen pupils’ understanding of
mathematical connectivity. Higher attainers are • The focus of the mastery approach is on the
challenged through more demanding problems, development of deep structural knowledge and
which deepen their knowledge of the same the ability to make connections. Making
content. connections in mathematics deepens
knowledge of concepts and procedures,
• Teachers use precise questioning to test ensures what is learnt is sustained over time,
knowledge and identify misconceptions. and cuts down the time required to assimilate
They use formative assessment regularly to and master later concepts and techniques.
identify those requiring intervention, often
delivered through individual or small group Evidence shows that what children know at the
support the same day, so that all pupils keep up. start of Year 1 is a predictor of later success in
maths. Therefore, the aim of Early Years
• The accurate use of mathematical language practitioners is to close the gap by the end of
helps children to think mathematically. Reception.⁴

11

2 2

REASONING EVIDENCE SUMMARY

Reasoning underpins every maths lesson • The ability to reason mathematically is
the most important factor in a pupil’s
The National Curriculum aims for all pupils to success in maths.
reason mathematically by following a line of
enquiry, conjecturing (making statements • Reasoning should underpin every
about) relationships and generalisations, and maths lesson.
developing an argument, justification or proof
using mathematical language.¹ • Children need to be taught to
Development Matters² identifies one of the articulate explanations and justify
characteristics of effective learning as ‘creating their thinking.
and thinking critically’. Young children are able to
engage in mathematical reasoning from the Early
Years onwards.³
The ‘ability to reason mathematically is the most
important factor in a pupil’s success in
mathematics… Such skills support deep and
sustainable learning and enable pupils to make
connections in mathematics’.⁴
Reasoning is not only important in its own right
but also impacts on fluency and problem
solving.⁵ Therefore, it must underpin every maths
lesson. Children need to see patterns and make
generalisations in all areas of maths.
Mathematical reasoning is not dependent on
arithmetical ability. Maths lessons should not
follow the format of fluency, followed by
reasoning, then problem solving; reasoning can
happen at any time.
To develop reasoning, pupils should be taught to
articulate explanations and justify their thinking.
For example, pausing during problem solving to
ask children to explain why a particular approach
is used or, during fluency practice, asking how
they know they have the correct answer.
Teachers should model their own reasoning.⁶
They should scaffold children’s responses to
begin with, but need to be aware that reasoning
is a subjective process and allow scope for
individuality in approaches.

12

3 3

FLUENCY

Children develop efficiency, accuracy and flexibility

The National Curriculum aims for all pupils to Teachers should create daily opportunities to Teachers should also help children to make links
become fluent. develop fluency. Key facts such as addition facts between the types of situations that a particular
within ten and multiplication tables should be strategy might suit. Giving children practice in
Fluency encompasses a mixture of efficiency, learnt to automaticity to avoid cognitive overload context helps them to make the links, ensuring
accuracy and flexibility.¹ in the working memory and enable pupils to that important mathematical procedures cannot
focus on new concepts.³ Resources from the be forgotten because they are based in a web of
Efficiency Nottingham Number Fluency Project⁴ support connected ideas about fundamental
Fluency relies on quick and efficient recall of facts learning the additive facts to 10 + 10. mathematical relationships.⁸ It also means pupils
and procedures in order for learners to keep are able to reconstruct steps in a procedure that
track of sub-problems, think strategically and Lessons should begin with a short review. they have forgotten.⁹
solve problems. Research shows that daily review can strengthen
previous learning and can lead to fluent recall. Teachers should use the Concrete Pictorial
Accuracy Abstract (CPA)¹⁰ model to help build fluency.
Fluency requires accuracy, which depends on The research also shows that students need Children literally see the maths with concrete
several aspects of the problem-solving process, extensive, successful, independent practice in apparatus, they can then visualise the apparatus
including careful recording, knowledge of number order for skills and knowledge to become before moving on to using a pictorial
facts and other important number relationships, automatic.⁵ representation alongside the abstract. This then
and double-checking results. supports them to ‘see’ the maths abstractly.
Rather than simply memorising key facts and
Flexibility procedures, children need to see patterns, begin Children need to talk about their maths to
Fluency also demands the flexibility to move to use derived facts (facts we can work out from develop fluency, describing why and how it
between different contexts and representations known facts e.g. If I know 5 x 6, I also know 50 x 6) worked, and how their method is the same or
of mathematics. Learners need to recognise and make links and connections with previous different to those of others.¹¹
relationships, make connections and make learning. They need to understand why they are
appropriate choices from a whole toolkit of doing what they are doing and know when it is
methods, strategies and approaches. appropriate to use different methods.

From Early Years onwards, children need to be The key to fluency is in making connections, and EVIDENCE SUMMARY
secure in each ‘building block’ in becoming fluent making them at the right time in a child’s learning.⁶
before moving on. • Fluency involves efficiency, accuracy
When learning multiplication facts, identifying and flexibility.
Subitising (the ability to instantaneously patterns and relationships between the tables
recognise the number of objects in a small group (for example, that the products in the 6× table • Each small step or ‘building block’
without the need to count them) and partitioning are double the products in the 3× table) helps needs to be consolidated before
single digit numbers in a variety of ways are children develop a strong sense of number moving on.
developed by regular practice beginning in Early relationship; this is an important prerequisite for
Years. Building on this ability, children need to procedural fluency.⁷
move on from counting strategies as quickly as
possible.²

• Extensive successful practice is
needed to build automaticity.

CONCRETE PICTORIAL ABSTRACT • Teachers should create daily
opportunities to develop fluency.

• Making connections in maths is key

2+1=3+ to fluency.

Note that this is not a strictly linear model

13

4 4

PROBLEM SOLVING

Children are taught effective strategies

Problem solving strategies begin in Early Problem solving involves the application of
Years¹; very young children are natural mathematics skills and understanding. It may not
problem solvers.² As well as teaching problem be present in every maths lesson, but may be the
solving, quality provision should encourage result of a sequence of lessons.
children to pose their own problems with a
range of possible solutions. Teachers must be aware that it is possible for
children to engage in problem solving activities
The National Curriculum aims to ensure that all without any meaningful long-term learning, for
pupils can solve problems by applying their example, children can have a random guess and
mathematics to a variety of routine and happen upon the correct answer. To have
non-routine problems with increasing long-term impact, teachers need to ensure that
sophistication. These include breaking down children focus on the process, not just the
problems into a series of smaller steps and answer.⁴
persevering in seeking solutions.
Direct teaching should be used to expose
Problem solving generally refers to situations in mathematical structures when problem solving.
which pupils do not have a readily available Using ‘numberless problems’ and teaching
method that they can use. Instead, they have to children to predict what a question could be
approach the problem flexibly and work out a asking are effective strategies to reveal
solution for themselves. Pupils need to draw on a underlying structures.⁵
variety of problem-solving strategies, which
enable them to make sense of unfamiliar Teachers should model problem solving,
situations and tackle them intelligently.³ articulating their thought processes as they
tackle problems. They need to scaffold problem
solving to begin with and ensure that the context
is meaningful to the pupils and relates to their
direct experiences.

Teachers also need to reduce cognitive load –
avoid presenting difficult calculations as well as
problems with a deep structure to begin with;
gradually build the level of challenge.⁶

Research from the Education Endowment
Foundation (EEF)⁷ suggests teachers should
use the following strategies when teaching
problem solving:

• Select genuine problem-solving tasks that
pupils do not have well-rehearsed, ready-made
methods to solve. Problem solving does not
mean routine questions set in context, or ‘word
problems’, designed to illustrate the use of a
specific method.

14

4

• Organise teaching so that problems with similar • Require pupils to monitor, reflect on, and
structures and different contexts are presented communicate their reasoning and choice of
together, and, likewise, that problems with the strategy (developing metacognition):
same context but different structures are
presented together. While working on a problem, encourage pupils to
ask questions like,
• Teach pupils to use and compare different
approaches. There are often multiple ways to • ‘What am I trying to work out?’,
approach a problem. Make the most of • ‘How am I going about it?’,
examining different solutions to the same • ‘Is the approach that I’m taking working?’, and
problem and looking for similarities in solution • ‘What other approaches could I try?’
approaches to different problems.
When the problem is completed, encourage
• Teach pupils to interrogate and use their pupils to ask questions like,
existing mathematical knowledge to solve
problems. Pupils should be encouraged to • ‘What worked well when solving this problem?’,
search their knowledge of similar problems they
have encountered for strategies that were • ‘What didn’t work well?’,
successful and for facts and concepts that
might be relevant. • ‘What other problems could be solved by a
similar approach?’, and
• Encourage pupils to use visual representations
that provide insight into the structure of a • ‘What similar problems to this one have I
problem and into its mathematical formulation. solved in the past?’

• Use worked examples to enable pupils to Paying attention to underlying mathematical
analyse the use of different strategies. Worked structure helps pupils make connections
examples, or ‘solved problems’, present the between problems, solution strategies, and
problem and a correct solution together, they representations that may, on the surface,
remove the need to carry out the procedures appear different, but are actually
required to reach the solution and enable pupils mathematically equivalent.⁷
to focus on the reasoning and strategies
involved. Worked examples may be complete, EVIDENCE SUMMARY
incomplete, or incorrect, deliberately containing
common errors and misconceptions for • Problem solving strategies need to be
learners to uncover. Analysing and discussing taught.
worked examples helps students develop a
deeper understanding of the logical processes • Teachers need to expose the
used to solve problems. underlying mathematical structure of
problems.

• Grouping similar problems together
will support children.

• Children need to learn to draw on
their existing mathematical
knowledge.

• Research supports the use of visual
representations, worked examples
and metacognitive approaches.

15

55

ASSESSMENT AND MODERATION

Develop accurate assessment and moderation processes to inform teaching and support pupils’ needs.

Teachers start by determining what children
already know and plan lessons accordingly.

Formative assessment is a crucial part of every
lesson. Teachers continually adapt their teaching
using information from assessment so it builds
on pupils’ existing knowledge, addresses their
weaknesses, and focuses on the next steps that
they need in order to make progress.¹

Assessment can be based on evidence from
low-stakes testing, informal observation of
pupils, or discussions with them about
mathematics.

Low-stakes testing involves the frequent use of ”One characteristic of effective teachers is their ability to anticipate students’ errors
assessments that are not reported. They can take and warn them about possible errors some of them are likely to make.” ⁷
the form of short tests (typically 3 to 5 questions)
at the start of every lesson based on content The order in which skills and concepts build on Teachers must uncover misconceptions and
previously taught. Pupils mark them themselves, one another as children develop knowledge is address them immediately within the lesson
the marks are not recorded anywhere and the called a developmental progression.⁵ Teachers when possible. Established practice
teacher goes through the answers straight away should use knowledge of these to inform demonstrates that Same Day Intervention is
so students have immediate feedback. The idea decisions about how to assess understanding, highly effective in addressing gaps in
of the test being ‘low stakes’ is essential, it what a child should learn next and then how to understanding before the next day’s lesson.⁸
shouldn’t be a pass or fail assessment and should choose activities at the correct level.⁶
avoid creating unnecessary anxiety and stress.² Diagnostic tests are a useful tool for exposing
Constant assessment, using precise questioning, misconceptions.⁹
In Early Years, assessment is through should be used to expose misconceptions and
observation. (see Delivering Excellence in Early inform next planning steps. Teachers should Teachers should make use of common
Years, Recommendation 4: Observations). When check the responses of all students.⁷ Techniques misconceptions¹⁰ and errors when teaching and
observing pupils, practitioners should ensure such as ‘show me’, asking who agrees/has a planning; predicting the difficulties learners are
they check what children know in a variety of different answer, listening-in to talk pairs or likely to encounter in advance.
contexts. It will take time to consolidate learning scanning the room are effective.
and transfer that learning to different contexts.³

Even if a child appears to be engaging
successfully in mathematical activities (for
example, reciting the count sequence), they may
not have a full grasp of the underlying concepts
(for example, the meaning of numbers in the
count sequence). Children may also appear to
have grasped an idea in one context but then fail
to show that knowledge in a different context.⁴

16

5

Potential misconceptions can be prevented by Summative Assessment Tests should
comparing examples to non-examples when supplement teacher assessments. They are
teaching new concepts.¹¹ beneficial for:

E.g. A non-example could be used when • Providing a baseline
introducing shape, if a child suggests “A square is • Supporting question level analysis
a four sided shape,” then alternative 4-sided • Summarising and comparing attainment
shapes could be presented, leading to a
definition that includes 4 equal sides and 4 right nationally, within cohorts and against
angles. age-related expectations

Specific feedback from Teachers, TAs and peers is • Guiding next steps in teaching and learning.
generally found to have large effects on
learning.¹² It should not lead to onerous marking Moderation
or heavy workload. The EEF¹³ state that effective Moderation allows teachers to benchmark
feedback (oral or written) should: judgements, while ensuring consistent standards
and reliable outcomes.
• be specific, accurate, and clear (for example,
‘You are now factorising numbers efficiently, by Internal moderation is conducted within and
taking out larger factors earlier on’, rather than, across schools throughout the academic year. It
‘Your factorising is getting better’); supports the quality assurance of TA judgements
and provides a valuable opportunity for
• be given sparingly so that it is meaningful (for professional development.
example, ‘One of the angles you calculated in
this problem is incorrect – find and correct it.); External moderation is conducted by Local
Authorities to validate teacher assessment
• compare what a pupil is doing right now with judgements and ensure that they are consistent
what they have done wrong before (for with national standards.¹⁴
example, ‘Your rounding of your final answers is
much more accurate than it used to be’);

• encourage and support further effort by helping
pupils identify things that are hard and require
extra attention (for example, ‘You need to put
extra effort into checking that your final answer
makes sense and is a reasonable size’);

• provide guidance to pupils on how to respond
to teachers’ comments, and give them time to
do so; and

• provide specific guidance on how to improve
rather than just telling pupils when they are
incorrect (for example, ‘When you are unsure
about adding and subtracting numbers, try
placing them on a number line.’)

EVIDENCE SUMMARY

• The best intervention ensures
children keep up rather than catch up.

• Feedback studies typically show very
high effects on learning.

17

66

MANIPULATIVES AND REPRESENTATIONS

Teachers and children use manipulatives and pictorial representations to uncover mathematical structure

Manipulatives are physical objects that you Pupils need to understand links between the of a novelty manipulative which can take away
can handle and move. These range from manipulative and the ideas they represent³. This from the intended learning aim. Research
everyday items such as buttons and shells, to requires practitioners to explicitly help children suggests that older children can also be
resources designed specifically for teaching to link the materials (and the actions performed distracted by surface features.⁹
arithmetic, such as Dienes or Cuisenaire rods. on or with them) to the mathematics of the
situation. Manipulatives and representations can be used
A ‘representation’ refers to a particular form in to encourage discussion about mathematics.
which mathematics is presented. Manipulatives should help children to build Children can work in pairs and small groups from
Representations include informal drawings, abstract mental models. The aim is always for Early Years onwards using manipulatives to solve
mathematical symbols, and more formal children to be able to do the maths without the problems and to encourage questions about
diagrams, such as a number line or graph. manipulative. other children’s strategies and reasoning. This
can prompt the sharing and comparison of
A key skill of any teacher is to be able to represent Manipulatives should be temporary. They should different approaches. Manipulatives can also be
the maths in ways that expose the underlying act as a scaffold that can be removed once used by children to communicate what they
mathematical structure and develop an independence has been achieved. The decision know.
understanding of how and why the maths works. to remove a manipulative should be made in
Manipulatives and representations can be response to the pupils’ improved knowledge and Stem sentences are sentences given by the
powerful tools for supporting pupils to engage understanding, not their age. teacher for children to repeat and/or complete so
with mathematical ideas and to understand and they can communicate their ideas with
use mathematics independently.¹ When moving away from manipulatives, pupils mathematical precision and clarity. They are used
may find it helpful to draw diagrams or imagine to describe a representation and help pupils
Children of all ages benefit from using using the manipulatives. As children’s move to working in the abstract (for example “Ten
manipulatives and representations. understanding of mathematical ideas develops, tenths is equivalent to one whole.”); they can
practitioners should encourage children to use themselves be seen as representations.
They can help children to develop visual images, pictures, symbols and more abstract diagrams to
increase engagement and enjoyment, help represent and communicate ideas and concepts. There is promising evidence that comparison and
practitioners see what children understand and discussion of different representations can help
provide a bridge to abstract thinking.² Children should be encouraged to represent pupils develop conceptual understanding.¹⁰
problems in their own way⁵ and should be free to Teachers should purposefully select different
How they are used is important. They need to be invent and explore their own representations to representations of key mathematical ideas to
used purposefully and appropriately in order to record their thinking and communicate their discuss and compare, with the aim of supporting
have an impact.³ Teachers should ensure that understanding. pupils to develop more abstract, diagrammatic
there is a clear rationale for using a particular representations. However, it is important to note
manipulative or representation to teach a specific Practitioners should encourage children’s use of that using too many representations at one time
mathematical concept. They should consider fingers, which can be important manipulatives for may cause confusion¹¹, so teacher judgement is
carefully how the manipulative will be used to children.⁶·⁷ Fingers can be useful for supporting key.
build on existing understanding, and help counting and later on for counting in groups
children to develop increasingly sophisticated (unitising). As well as scaffolding learning, manipulatives are
approaches and ideas.⁴ used as a tool for supporting higher level
It is important that young children have mathematical thinking and reasoning.¹²
opportunities to engage in both free and
structured play with manipulatives. Using a given Settings should plan their use of manipulatives
manipulative regularly, or introducing it through and representations to ensure a consistent
play to gain familiarity can be beneficial⁸ as young approach.¹³
children can be distracted by the surface features

18

6

EVIDENCE SUMMARY

• Research shows that use of
manipulatives can have an impact
particularly on retention, but how
they are used is also important.³

• Manipulatives should help children to
build abstract mental models.

• Manipulatives should be temporary.
• Children should be encouraged to

represent maths in their own way.

159

7 7

MATHEMATICAL LANGUAGE

Teachers and children use precise language to support mathematical thinking

Development Matters and the National The quality of children’s mathematical reasoning
Curriculum for Mathematics both reflect the and conceptual understanding is significantly
importance of spoken language in pupils’ enhanced if they are consistently expected to use
development across the whole curriculum – the correct mathematical terminology and to
cognitively, socially and linguistically. explain their mathematical thinking in complete
sentences.²
The quality and variety of language that pupils
hear and speak are key factors in developing their Mathematical talk reveals pupils’ understanding
mathematical vocabulary and presenting a as well as misunderstandings. It also supports
mathematical justification, argument or proof. deeper reasoning, language development, the
Pupils must be assisted in making their thinking development of social skills and supports
clear to themselves as well as others and learning by boosting memory (for example, by
teachers should ensure that pupils build secure having to retrieve facts).³
foundations by using discussion to probe and
remedy their misconceptions.¹ Children need to talk about their maths,
describing why and how it worked, and how their
method is the same or different to those of
others. In other words, they need opportunities
to use the higher-level skills of comparing,
explaining and justifying.

Teachers need to model using precise
mathematical language. From Early Years
onwards, children should be exposed to
mathematical language throughout the day.⁴
Practitioners should seize opportunities to
reinforce mathematical vocabulary, as well as
create opportunities for extended discussions. In
Early Years and Key Stage 1, using storybooks
and discussing ideas and strategies as you play
games can be particularly effective.

Teachers should support children by verbalising
their own thoughts and encouraging the pupils to
do the same,⁵ asking probing questions to refine
their ideas, and ensuring that they have time to
rehearse, explain, compare suggestions and
refine their answers.

20

7

Teaching children precise mathematical communicate their ideas with mathematical
language and insisting upon its use supports precision and clarity. They are used to describe a
children's ability to think mathematically. representation and help pupils move to working
Having the language and using it empowers in the abstract (for example “ten tenths is
children’s ability to think about the concept.⁶ equivalent to one whole”).

Teachers improve the quality of mathematical Established practice for using stem sentences is:
discussion by scaffolding, for example by giving • I say - The teacher says the sentence first,
prompts or frames for talk pairs. Teachers • You say - Children repeat the sentence to
develop learning behaviours so children actively themselves,
listen to one another, build on or challenge one • We say - Everyone (the teacher and whole
another’s statements and further improve their class) repeat the sentence together.
own responses.

Research suggests that teachers develop
mathematical thinking through:

• guiding class discussions,
• listening-in to peer talk,
• selecting who will share their ideas and in

which order,
• pre-warning those selected so they can

prepare,
• allowing peers to reflect on or build on

answers.⁷

Teachers should support pupils to use language
that reflects mathematical structure, for example
by rephrasing pupils’ responses that use vague,
non-mathematical language.⁸

Using stem sentences supports children in EVIDENCE SUMMARY
understanding mathematical structures,
articulating reasoning and in using precise • The accurate use of mathematical
mathematical vocabulary accurately. Stem language helps children to think
sentences are sentences given by the teacher for mathematically.
children to repeat and/or complete so they can
• Using the correct terminology and
speaking in full sentences improves
the quality of children’s reasoning and
conceptual understanding.

• Teachers should model verbalising
their own thoughts and scaffold
children to talk about their maths.

21

8 8

MATHEMATICS PEDAGOGY

Teachers develop their own understanding of maths as well as how children typically learn,
and how this relates to effective pedagogy

There is an expectation that maths is taught Tasks should not be over-scaffolded, children
daily in all year groups. need to experience desirable difficulties. These
difficulties improve learning because they
To be effective in the teaching of maths, it is prompt encoding and retrieval processes that
imperative that teachers develop an expert support comprehension, and remembering; that
subject knowledge and an understanding of how is what makes them desirable.⁴
children learn.¹ Schools should ensure that
teaching assistants have access to relevant CPD Teachers should understand the sequence of
so they too have the necessary knowledge and learning and adapt tasks according to the needs
understanding. of their class. They should be able to explain the
choice of task at that particular time. For
Being aware of mathematical structures, laws example, at the beginning of a sequence of
and principles (such as inverse relationships, learning, the pupils are novices and direct
commutative and associative laws and instruction is most effective; however, as children
compensation properties) means teachers are become more expert, tasks which are more open
able to create and exploit opportunities to teach ended may be appropriate.⁵
them.
The gradual release of responsibility model⁶
Lesson design should take into account children’s purposefully shifts the cognitive load from
prior knowledge and common misconceptions. teacher-as-model, to joint responsibility of
Teachers should use variation and intelligent teacher and learner, to independent practice and
practice in creating tasks. application by the learner.

Teaching needs to be precise so that all pupils For children to experience independent success,
can engage successfully with tasks at the guided practice⁷ or working through examples
expected level of challenge. Clear and detailed together is important:
instructions and explanations should be used to
support this.² • I do - Teacher models first
• We do - then children should have a chance to
The many connections between different
mathematical facts, procedures and concepts work together/with the teacher
should be emphasised to help children to create
a rich network of skills and knowledge.³ • You do - before working independently

Teachers need to support all children in building Reasoning should underpin all maths lessons.
abstract mental models. To develop deep Both reasoning and problem solving need to be
understanding, all children will need to work with modelled effectively.
concrete resources to begin with. Some children
may require more scaffold (more direct teaching Children should be taught a range of strategies
or spending longer with concrete apparatus), but for solving problems. Teachers should help pupils
the aim is for all children to build a deep to compare and choose between them, deciding
understanding and no longer need the support. when different methods are appropriate and
efficient.

22

8

Cognitive load theory needs to be considered. To
ensure success, learning should be broken into
small steps and the amount of material students
receive at one time should be limited, for
example, breaking formal column methods down
so children only have to complete one small part
at a time, such as completing the remainder in
short division. Teachers should also use spaced
practice to ensure that new learning is committed
to long-term memory.⁸

Effective teachers ask a large number of
questions to check for understanding. Questions
allow a teacher to determine how well the
material has been learned and whether there is a
need for additional instruction. The most
effective teachers ask students to explain the
process they used to answer the question, to
explain how the answer was found. They also
ensure active participation of all students.⁷

The classroom environment should support
learning, including the use of vocabulary and
worked examples.

“The most successful teachers spent EVIDENCE SUMMARY
more time in guided practice, more
time asking questions, more time • Teachers should have expert
checking for understanding and knowledge of the subjects they
more time correcting errors.” ⁷ teach.⁹

• Lesson design should include
variation and intelligent practice.

• All children should be supported to
build abstract mental models.

• Teacher modelling should be precise.
• Cognitive load theory should be

considered.
• Questioning should be used to check

understanding.

23

How was this guidence compiled

This guidance report draws on the best The guidance report was created over three
available evidence regarding the teaching of stages.
mathematics from the start of Nursery to the
end of Year 6. 1 - Scoping. The process began with consultation
period to gather evidence of best practice in
The primary sources of evidence for the Waterton Academy Trust schools.
recommendations are:
2 - Evidence review. The Waterton School
• Improving Mathematics in Key Stages 2 and 3 Improvement Team reviewed a body of
(EEF Nov 2017) research pertinent to their areas of
responsibility.
• Improving Mathematics in the Early Years and
Key Stage1 (EEF Jan 2020) 3 - Writing recommendations. The individual
report teams worked collaboratively to draft
• Metacognition and Self regulated Learning (EEF the recommendations.
2018)
4 - Review recommendations. The Waterton
• NCETM Teaching for Mastery School Improvement Team worked with the
Teaching for Mastery | NCETM support of the Advisory Group to draft and
finalise the recommendations.
• Rosenshine, B. Principles of Instruction;
Research-based Strategies that all teachers
should know. American Educator. (2012)

24

Glossary

Abstract Abstract is the ‘symbolic’ stage, where children are able to use abstract
Addend symbols to model and solve maths problems.
Associative Law
Cognitive load A number which is added to another.
Commutative Law 3+3=6
Common misconceptions addend + addend = sum

Compensation properties Numbers can be grouped in any way and the answer remains the same.
Addition is associative, e.g. 1 + (2 + 3) = (1 + 2) + 3.
Conceptual understanding Multiplication is also associative e.g. 1 x (2 x 3) = (1 x 2) x 3.
Subtraction and division are not associative.

The load on working memory during problem solving, thinking and
reasoning. The amount of information that working memory can hold at
one time.

The order of the numbers in an operation can be changed without
changing the answer.
Addition is commutative, for example 2 + 3 = 3 + 2.
Multiplication is also commutative, for example 3 x 2 = 2 x 3.
Subtraction and division are not commutative.

A widely held idea that is not correct, it may be built on a partial truth.
See also misconception

If one addend is increased and the other is decreased by the same
amount, the sum stays the same. (same sum) e.g. 5 + 7 = 6 + 6
If one addend is increased (or decreased) and the other is kept the same,
the sum increases (or decreases) by the same amount.
If the minuend and subtrahend are changed by the same amount, the
difference stays the same. (same difference) e.g. 32- 15 = 29 - 12
If the minuend is increased (or decreased) and the subtrahend is kept the
same, the difference increases (or decreases) by the same amount.
If the minuend is kept the same and the subtrahend is increased (or
decreased), the difference decreases (or increases) by the same amount.

Understanding abstract ideas such as numbers, operations and
relations. It also includes developing a precise definition e.g. of what a
square is.

25

Glossary

Concrete Concrete is the ‘doing’ stage, using concrete objects to solve problems. It
brings concepts to life by allowing children to handle physical objects
themselves.

Concrete Pictorial Abstract The Concrete > Pictorial > Abstract (CPA) builds on children’s existing
(CPA) knowledge by introducing abstract concepts in a concrete and tangible
way. It involves moving from concrete materials, to pictorial
representations, to abstract symbols and problems. Although CPA has
three distinct stages, a skilled teacher will go back and forth between
them to reinforce concepts.

Conjecture A conjecture is a mathematical statement that has not yet been
rigorously proved. Conjectures arise when one notices a pattern that
Desirable/deliberate holds true for many cases.
difficulty
A desirable difficulty is a learning task that requires a considerable but
desirable amount of effort, thereby improving long-term performance.

Developmental progression A description of the typical path children tend to follow in developing
understanding of a mathematical topic.

Derived facts Derived facts are ones we can work out from known facts e.g. If I know 5
Diagnostic questions x 6, I also know 50 x 6.
Difference
A set of questions which help identify and understand students’ mistakes
and misconceptions. Diagnostic Questions is also a website with
resources to support teachers.

The result of subtracting one number from another. How much one
number differs from another. For Example: The difference between 8 and
3 is 5.

26

Glossary

Expert Someone with deep understanding.
Efficacy trial
Formative assessment Efficacy trials test whether an intervention can work under developer-led
Guided practice conditions in a number of schools.
Intelligent practice
Inverse relationships Formative assessment (also known as assessment for learning) is using
evidence to decide what to do next. It takes place on a day-to-day basis
Low stakes testing during teaching and learning, allowing teachers and pupils to assess
attainment and progress more frequently. It can include targeted
Manipulatives questioning, recap starter activities, or peer and self-assessment.
Guided practice is a teaching practice pioneered by Barbara Rogoff. It
consists of three steps for practicing new skills in the classroom. First,
the teacher models how to do a task to the student (I do). Second, the
student does the task with guidance from the teacher (We do). Third, the
student practices the task independently (You do).
A key feature of teaching for mastery. The precise designing of pupil
activities and practice questions, so that, rather than pupils repeating
lots of similar examples which can be done mechanically with no deep
understanding, the tasks use procedural variation to reveal the
underlying structure.

An opposite relationship , e.g. addition and subtraction are inverse
relationships, as are multiplication and division.

Low-stakes testing involves the frequent use of assessments that are not
reported. They can take the form of short tests (typically 3 to 5
questions) at the start of every lesson based on content previously
taught. Pupils mark them themselves, the marks are not recorded
anywhere and the teacher goes through the answers straight away so
students have immediate feedback. The idea of the test being ‘low
stakes’ is essential, it shouldn’t be a pass or fail assessment and should
avoid creating unnecessary anxiety and stress.

A physical object that pupils or teachers can touch and move, used to
support the teaching and learning of mathematics. Popular
manipulatives include Cuisenaire rods and Dienes blocks.

27

Glossary

Metacognition Metacognition is an awareness and understanding of one’s own thought
Memory processes and how to monitor and purposefully direct one’s own
(working memory) learning.
Minuend
Misconception The structures and processes used for temporarily storing and
Non examples manipulating information.
Non-routine
Novice A quantity or number from which another is to be subtracted.
Numberless problems 7–5=2
Partition Minuend – subtrahend = difference

An idea which is not correct. Often misconceptions are formed when
knowledge has been applied outside of the context in which it is useful.
E.g. when you multiply by 10 you add a zero (this only applies to whole
numbers). See also common misconceptions

A non-example is something that is not an example of the concept. It is
useful to include when learning about a concept.
E.g. when introducing shape, if a child suggests “A square is a four sided
shape,” then alternative 4-sided shapes could be presented, leading to a
definition which includes 4 equal sides and 4 right angles.

A non-routine problem is any complex problem that requires some
degree of creativity or originality to solve. Non-routine problems
typically do not have an immediately apparent strategy for solving them.
Often, these problems can be solved in multiple ways.

A beginner, someone who is not very familiar or experienced in a
particular area or skill.

A strategy introduced by Brian Bushart for tackling word problems.
The problems are initially presented without numbers so children can
understand the context.

Split numbers into smaller addends or factors. 25 can be partitioned into
20 and 5 or 10 and 15 or 18 and 7 or… Single digit numbers can also be
partitioned e.g. 5 can be partitioned into 1 and 4 or 2 and 3.

28

Glossary

Pedagogy The method and practice of teaching.
Pictorial
Representation Pictorial is the ‘seeing’ stage, using representations of the objects
Routine involved in maths problems. This stage encourages children to make a
mental connection between the physical object and abstract levels of
Same day intervention understanding, by drawing or looking at pictures, circles, diagrams or
models which represent the objects in the problem.
Scaffold(ing)
Spaced practice A representation is a way of presenting maths e.g. informal sketches,
graphs, number lines or bar models.

A routine problem uses clear procedures.

Same Day Intervention (SDI) is an approach to teaching maths. Teachers
adapt their classroom style based on Shanghai methods, for example
using frequent modelling and an ‘I do, you do’ approach in initial class
teaching. After a 30-minute lesson, pupils answer some questions
independently and then have 15 minutes away from their teacher
(attending assembly or a teaching-assistant-led activity) while the
teacher marks their answers using a rapid marking code. The remaining
30 minutes of the lesson is an intervention session, where the teacher
groups children together based on how they answered the questions so
that they can efficiently address common misconceptions. The aim is to
use the additional support to ensure that all children reach a certain level
of understanding by the end of the day, preventing an achievement gap
from forming.

Provide (providing) temporary support which is incrementally removed
when it is no longer needed. Scaffolding can be through directed
teaching, questioning, use of apparatus or breaking a task into smaller
steps.

Spaced practice (or 'distributed practice') involves repeatedly coming
back to the information that we are learning in various short sessions,
spaced out over time.
E.g. revisiting the previous week’s learning every Monday.

29

Glossary

Subtrahend A quantity or number to be subtracted from another.
Sub-problem 7–5=2
Summative assessment Minuend – subtrahend = difference

Stem sentences A problem that is part of a larger problem.
e.g. to complete the long multiplication 287 x 34 you would have to
complete several other multiplication calculations and keep track of the
process.

Summative assessments assess pupil learning at the end of a period of
learning (often at the end of a topic, term, year or key stage). They can be
used to compare pupil performance to national expectations. They are
high stakes.

Stem sentences are sentences given by the teacher for children to repeat
and/or complete so they can communicate their ideas with mathematical
precision and clarity. They are used to describe a representation and
help pupils move to working in the abstract (for example “Ten tenths is
equivalent to one whole.” or “The whole has been divide into __ equal
parts. Each part is one ___.”).

Subitising Subitising is the ability to instantaneously recognise the number of
Sum objects in a small group without the need to count them.
Unitising
The result of adding two or more numbers together.

A way of counting a large group by counting equal smaller groups within
it e.g. counting in 2s, counting in 5p pieces, counting in multiples.

30

Glossary Conceptual variation means representing a concept in different ways to
highlight the essential features. When giving examples of a mathematical
Variation concept, it is useful to add variation to emphasise:
Worked examples a. What it is (as varied as possible);
b. What it is not.
Procedural variation is used to highlight mathematical structures. When
constructing a set of activities / questions it is important to consider
what connects the examples; what mathematical structures are being
highlighted?

Worked examples present the problem and a correct solution together.

31

References

Recommendation 1 – Mastery
1 - references to data: 2009 https://www.oecd.org/pisa/pisaproducts/46619703.pdf PISA webpage – all

data - https://www.oecd.org/pisa/ Pisa 2012 – new report
2 - https://www.ncetm.org.uk/home
3 - Willingham, Daniel T. "Is it true that some people just can’t do math?" American Educator 33.4

(2009): 14-19.
4 - Bold Beginnings: The Reception Curriculum in a sample of good and outstanding Primary Schools

Ofsted (2017)
ht tps: //asset s.publishing.ser vice.gov.uk /government /uploads/s y s tem/uploads/at tachment _ data/
file/663560/28933 _Ofs ted _-_ Early_Years _Curriculum _ Repor t _-_ Accessible.pdf
5 - https://www.ncetm.org.uk/teaching-for-mastery/mastery-explained/supporting-research-
evidence-and-argument/
Further Reading:
http://mikeaskew.net/page3/page4/files/EffectiveTeachersofNumeracy.pdf
Recommendation 2 – Reasoning
1 - https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/
file/335158/PRIMARY_national_curriculum_-_Mathematics_220714.pdf
2 - https://www.foundationyears.org.uk/files/2012/03/Development-Matters-FINAL-PRINT-
AMENDED.pdf
3 - Askew M. Reasoning as a mathematical habit of mind (2020) The Mathematical Gazette 104
(559)1-11 Retrieved from
https://www.researchgate.net/publication/339636620_Reasoning_as_a_mathematical_habit_of_
mind
4 - Nunes et al. Development of Maths Capabilities and Confidence in Primary School (2009)
ht tps: //dera.ioe.ac.uk /1115 4 /1/DCSF -RR118.pdf
5 - https://www.ncetm.org.uk/classroom-resources/pm-reasoning-skills/
6 - EEF Improving Mathematics in Key Stages Two and Three (Nov 2017) Guidance Report
Recommendation 5
ht tps: //educationendowment foundation.org.uk /tools/guidance-repor t s/maths-k s-2-3/
Recommendation 3 – Fluency
1 - Russell, Susan. Developing Computational Fluency with Whole Numbers in the Elementary Grades.
New England Math Journal (2000)
https://investigations.terc.edu/inv2/wp-content/uploads/2017/10/Developing-Computational-
Fluency-with-Whole-Numbers-in-the-Elementary-Grades.pdf
2 - Gray, E and Tall, E. Duality, Ambiguity and Flexibility: A Proceptual View of Simple Arithmetic.
The Journal for Research in Mathematics Education, 26 (2), 115–141 (1994)
3 - https://www.ncetm.org.uk/teaching-for-mastery/mastery-explained/supporting-research-evidence
-and-argument/

32

References

4 - http://www.nottinghamschools.org.uk/media/1170651/number-fact-fluency-programme.pdf
5 - Rosenshine, B. Principles of Instruction; Research-based Strategies that all teachers should know.

American Educator. (2012) pp12-20
https://files.eric.ed.gov/fulltext/EJ971753.pdf
6 - https://nrich.maths.org/10624
7 - McClure, M. Developing Number Fluency - What, Why and How. nrich.maths.org/10624 Published
April 2014
https://nrich.maths.org/content/id/10624/Developing%20Number%20Fluency%20-%20What%2C
%20Why%20and%20How.pdf
8 - Russell, Susan Jo. (May, 2000). Developing Computational Fluency with Whole Numbers in the
Elementary Grades. In Ferrucci, Beverly J. and Heid, M. Kathleen (eds). Millenium Focus Issue:
Perspectives on Principles and Standards. The New England Math Journal. Volume XXXII, Number 2.
Keene, NH: Association of Teachers of Mathematics in New England. Pages 40-54. cited in NRICH
https://nrich.maths.org/10624
9 - EEF Improving Mathematics in Key Stages Two and Three (Nov 2017) Guidance Report rec 4 p17 -
reference Brown, J. S., & Van Lehan, K. (1982). Towards a generative theory of ‘bugs’. In T. P.
Carpenter, J. M. Moser, & T. A. Romberg (Eds.), Addition and Subtraction: A cognitive perspective.
Hillsdale, NJ: Lawrence Erlbaum
10 - Leong, Y. H., Ho, W. K., & Cheng, L. P. (2015). Concrete-Pictorial-Abstract: Surveying its origins and
charting its future. The Mathematics Educator, 16(1), 1-18. Retrieved from
ht tp: //math.nie.edu.sg /ame/matheduc /tme/tmeV16 _1/ TME16 _1.pdf
11 - https://www.nctm.org/Publications/Teaching-Children-Mathematics/Blog/Using-Talk-to-Make-
Sense-of-Mathematics/
Further Reading:
https://nrich.maths.org/10624
NCETM Calculation Guidance
ht tps: //w w w.ncetm.org.uk /media/k 20boquz /ncetm- calculation-guidance- oc tober-2015.pdf
Willingham, Daniel T. "Is it true that some people just can’t do math?" American Educator 33.4 (2009):
14-19.This paper offers evidence from cognitive science of cognitive overload. Willingham also argues
that procedural fluency and conceptual understanding should be taught in tandem.
Cambridge Mathematics Espresso 10 asks Why is working memory important for
mathematics learning?
https://www.cambridgemaths.org/espresso/view/working-memory-for-mathematics-learning/
Baroody, Arthur J. "Mastering the basic number combinations." Teaching Children Mathematics 23
(2006): 22-31. This article argues that basic number fluency is best achieved by teaching it alongside
conceptual understanding. Includes classroom suggestions and examples.
Dahlin, Bo, and David Watkins. "The role of repetition in the processes of memorising and
understanding: A comparison of the views of German and Chinese secondary school students in
Hong Kong." British Journal of Educational Psychology 70.1 (2000): 65-84 (full text requires payment).
I See Reasoning – free resources linked to x tables
http://www.iseemaths.com/tables/
Sayer, M. Kids Who Count with Their Fingers Are Smarter, Researchers Say (2017)
https://www.thehealthy.com/family/childrens-health/counting-with-fingers/

33

References

Recommendation 4 – Problem Solving
1 - https://nrich.maths.org/12166
2 - https://nrich.maths.org/11113
3 - EEF Improving Mathematics in Key Stages Two and Three (Nov 2017) Guidance Report

Recommendation 5
ht tps: //educationendowment foundation.org.uk /tools/guidance-repor t s/maths-k s-2-3/
4 - Mathematics as a Complex Problem-Solving Activity by Jacob Klerlein and Sheena Hervey,
Generation Ready accessed at
https://www.generationready.com/mathematics-as-a-complex-problem-solving-activity/
5 - Blog on numberless problems by Brian Bushart (2014)
https://bstockus.wordpress.com/2014/10/06/numberless-word-problems/
6 - Metacognition and Self-regulated Learning (EEF 2018) recommendation 4
ht tps: //educationendowment foundation.org.uk /tools/guidance-repor t s/metacognition-and-self-
regulated-learning/
7 - EEF Improving Mathematics in Key Stages Two and Three (Nov 2017) Guidance Report
recommendation 3
ht tps: //educationendowment foundation.org.uk /tools/guidance-repor t s/maths-k s-2-3/
Further Reading:
Deliberate difficulty - Craig Barton Podcast –
http://www.mrbartonmaths.com/blog/robert-and-elizabeth-bjork-memory-forgetting-testing-
desirable-difficulties/
Early Years Problem Solving ideas
ht tps: //w w w.barrscour tprimar y school.co.uk /wp - content /uploads/2015/12 /E YFS -reasoning.pdf
I See Problem Solving
http://www.iseemaths.com/early-number/
Rosenshine, B. Principles of Instruction; Research-based Strategies that all teachers should know.
American Educator. (2012) pp12-20
https://files.eric.ed.gov/fulltext/EJ971753.pdf
Proven Practices from Teach Like a Champion by Dr. Doug Lemov
ht tps: //dese.mo.gov/sites/default / files/11-Research-ProvenPrac ticesTL AC.pdf
John Hattie. Visible learning for Mathematics
Dylan William. Problem Solving

Recommendation 5 – Assessment and Moderation
1 - EEF Improving Mathematics in Key Stages Two and Three (Nov 2017) Guidance Report

ht tps: //educationendowment foundation.org.uk /tools/guidance-repor t s/maths-k s-2-3/
2 - https://ukedchat.com/2020/03/16/low-stakes-testing/
3 - EEF Improving Mathematics in Key Stage One and Early Years ( Jan 2020) Guidance Report

https://educationendowmentfoundation.org.uk/tools/guidance-reports/early-maths/
4 - Pirie, S. E. B. and Kieren, T. E. (1994) ‘Growth in Mathematical Understanding: How Can We

Characterise It and How Can We Represent It? Educational Studies in Mathematics, 26, pp.
165–190.

34

References

5 - Clements, Douglas H. and Sarama, Julie (2014). Learning and Teaching Early Math: The Learning
Trajectories Approach (2nd edn) (Studies in Mathematical Thinking and Learning Series, London:
Routledge).
Also see https://www.learningtrajectories.org.

6 - RFrye, D., Baroody, A. J., Burchinal, M., Carver, S. M., Jordan, N. C., & McDowell, J. (2013). Teaching
math to young children: A practice guide (NCEE 2014-4005). Washington, DC: National Center for
Education Evaluation and Regional Assistance (NCEE), Institute of Education Sciences, U.S.
Department of Education. Retrieved from the NCEE website: http://whatworks.ed.gov

7 - Rosenshine, B. Principles of Instruction; Research-based Strategies that all teachers should know.
American Educator. (2012) pp12-20
https://files.eric.ed.gov/fulltext/EJ971753.pdf

8 - EEF Improving Mathematics in Key Stages Two and Three (Nov 2017) Guidance Report
ht tps: //educationendowment foundation.org.uk /tools/guidance-repor t s/maths-k s-2-3/

9 - http://www.mrbartonmaths.com/blog/diagnostic-questions/
10 - Hansen, A. (Ed.) (2017). Children’s Errors in Mathematics (4th ed.). London: Sage.
11 - http://mrbartonmaths.com/teachers/research/examples.html
12 - https://educationendowmentfoundation.org.uk/evidence-summaries/teaching-learning-

toolkit/feedback/
13 - EEF Improving Mathematics in Key Stages Two and Three (EEF Nov 2017)

Recommendation 1 page 9
ht tps: //educationendowment foundation.org.uk /tools/guidance-repor t s/maths-k s-2-3/
14 - Standards and Testing Agency (Oct 2019) Key Stage 1 Teacher Assessment Guidance
ht tps: //w w w.gov.uk /government /publications/key-s tage-1-teacher-assessment-guidance
Further Reading:
Common Misconceptions
https://www.resourceaholic.com/p/topics-in-depth.html
http://www.counton.org/resources/misconceptions/
Ryan, J., & Williams, J. (2007). Children’s mathematics 4-15: Learning from errors and misconceptions.
McGrawHill Education.
Hart, K. M., Brown, M. L., Kuchemann, D. E., Kerslake, D., Ruddock, G., & McCartney, M. (1981). Children’s
understanding of mathematics: 11-16. London: John Murray.
Low stakes testing:
http://mrbartonmaths.com/teachers/research/testing.html
https://www.tes.com/teaching-resources/blog/tes-maths-pedagogy-place-low-stakes-testing
Non-examples generator: https://nonexamples.com/
More guidance on how to conduct useful and accurate assessment is available in the EEF’s guidance on
Assessing and Monitoring Pupil Progress, available online
ht tps: //educationendowment foundation.org.uk /tools/assessing-and-monitoring-pupil-progress/

35

References

Recommendation 6 – Manipulatives and Representations
1 - EEF Improving Mathematics in Key Stages Two and Three (Nov 2017) Guidance Report

Recommendation 2
ht tps: //educationendowment foundation.org.uk /tools/guidance-repor t s/maths-k s-2-3/
2 - Griffiths, R., Back, J. and Gifford, S. (2016) Making Numbers: Using Manipulatives to Teach Arithmetic,
Oxford: Oxford University Press.
3 - Carbonneau, K. J., Marley, S. C., & Selig, J. P. (2013). A meta-analysis of the efficacy of teaching
mathematics with concrete manipulatives. Journal of Educational Psychology, 105(2), 380.
4 - Cross, C. T., Woods , T. A. and Schweingruber, H. (2009) Mathematics Learning in Early Childhood:
Paths Towards Excellence and Equity, Washington DC: National Academies Press.
https://doi.org/10.17226/12519
5 - Thomas, N. D., Mulligan, J. T. and Goldin, G. A. (2002) ‘Children’s Representation and Structural
Development of the Counting Sequence 1–100’, Journal of Mathematical Behavior, 21, pp. 117–133.
https://doi.org/10.1016/s0732-3123(02)00106-2
6 - Deans for Impact (2019) ‘The Science of Early Learning: How Young Children Develop Agency,
Numeracy, and Literacy’, Austin, TX: Deans for Impact.
ht tps: //deans forimpac t.org /wp - content /uploads/2017/01/ The_ Science_of_ Early_ Learning.pdf
7 - Boaler, J, Chen, L. Why Kids Should Use Their Fingers in Math class. (2016) Accessed at
https://www.theatlantic.com/education/archive/2016/04/why-kids-should-use-their-fingers-in-
math-class/478053/
8 - Griffiths, R., Back, J. and Gifford, S. (2016) Making Numbers: Using Manipulatives to Teach Arithmetic,
Oxford: Oxford University Press.
9 - https://www.ncetm.org.uk/features/a-grey-area-the-importance-of-colour-in-representations/
10 - Clements, D., Baroody, A. J. and Sarama, J. (2013) ‘Background Research on Early Mathematics’,
background research for the National Governor’s Association (NGA) Center Project on Early
Mathematics. https://www.du.edu/marsicoinstitute/media/documents/
dc_background_research_early_math.pdf
11 - Ainsworth, S. (2006). DeFT: A conceptual framework for considering learning with multiple
representations. Learning and Instruction, 16(3), 183-198. doi: 10.1016/j. learninstruc.2006.03.001
12 - Griffiths, R., Gifford, S., and Back, J. (2016). Making Numbers: Using Manipulatives to Teach
Arithmetic. Oxford University Press.
13 - EEF Improving Mathematics in Key Stage one and Early Years ( Jan 2020) Guidance Report
Improving Mathematics in the Early Years and Key Stage 1 | Education Endowment Foundation |
EEF
Further Reading:
Using manipulatives in the foundations of arithmetic | Nuffield Foundation

36

References

Recommendation 7 – Mathematical Language
1 - Mathematics programmes of study: key stages 1 and 2 National Curriculum in England

ht tps: //asset s.publishing.ser vice.gov.uk /government /uploads/s y s tem/uploads/at tachment _ data/
file/335158/PRIMARY_national_curriculum_-_Mathematics_220714.pdf
2 - https://www.ncetm.org.uk/media/k20boquz/ncetm-calculation-guidance-october-2015.pdf
3 - Using Math Talk to Help Students Learn, Grades K-6 (2009) Suzanne H. Chapin, Catherine O'Connor,
Mary Catherine O'Connor, Nancy Canavan Anderson
4 - EEF Improving Mathematics in Key Stage one and Early Years ( Jan 2020) Guidance Report
ht tps: //educationendowment foundation.org.uk /tools/guidance-repor t s/early-maths/
5 - Metacognition and Self regulated Learning (EEF 2018) recommendation 3
ht tps: //educationendowment foundation.org.uk /tools/guidance-repor t s/metacognition-and-self-
regulated-learning/
6 - Deans for Impact (2019) ‘The Science of Early Learning: How Young Children Develop Agency,
Numeracy, and Literacy’, Austin, TX: Deans for Impact.
ht tps: //deans forimpac t.org /wp - content /uploads/2017/01/ The_ Science_of_ Early_ Learning.pdf
7 - Mary Kay Stein 5 Practices for Orchestrating Productive Mathematics Discussions
https://www.researchgate.net/profile/Elizabeth_Hughes8/publication/250890079_Orchestrating_
Productive_Mathematical_Discussions_Five_Practices_for_Helping_Teachers_Move_Beyond_Show_
and_Tell/links/5501ac410cf2d60c0e5fcb54.pdf
8 - EEF Improving Mathematics in Key Stages Two and Three (Nov 2017) Guidance Report
ht tps: //educationendowment foundation.org.uk /tools/guidance-repor t s/maths-k s-2-3/
Further Reading:
Mary Stein
Recommendation 8 – Mathematics Pedagogy
1 - EEF Improving Mathematics in Key Stage one and Early Years ( Jan 2020) Guidance Report
ht tps: //educationendowment foundation.org.uk /tools/guidance-repor t s/early-maths/
2 - Rosenshine, B. Principles of Instruction; Research-based Strategies that all teachers should know.
American Educator. (2012) pp12-20
https://files.eric.ed.gov/fulltext/EJ971753.pdf
3 - EEF Improving Mathematics in Key Stages Two and Three (Nov 2017) Guidance Report
recommendation 4
ht tps: //educationendowment foundation.org.uk /tools/guidance-repor t s/maths-k s-2-3/
4 - Metacognition and Self regulated Learning (EEF 2018) recommendation 3
ht tps: //educationendowment foundation.org.uk /tools/guidance-repor t s/metacognition-and-self-
regulated-learning/
5 - Metacognition and Self-regulated Learning (EEF 2018) recommendation 3
ht tps: //educationendowment foundation.org.uk /tools/guidance-repor t s/metacognition-and-self-
regulated-learning/

37

References

6 - Frey Nancy and Fisher Douglas (2013) ASCD Gradual release of responsibility
modelformativeassessmentandccswithelaliteracymod_3-reading3.pdf (ascd.org)

7 - Rosenshine, B. Principles of Instruction; Research-based Strategies that all teachers should know.
American Educator. (2012) pp12-20
https://files.eric.ed.gov/fulltext/EJ971753.pdf - principles of instruction

8 - Metacognition and Self-regulated Learning (EEF 2018)
ht tps: //educationendowment foundation.org.uk /tools/guidance-repor t s/metacognition-and-self-
regulated-learning/

9 - https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_
data/ file/8 43108/School _ inspec tion _ handbook _-_ sec tion _ 5.pdf

Further Reading:
Desirable difficulties
Making Things Hard on Yourself, But in a Good Way: Creating Desirable Difficulties to Enhance Learning
Author(s): Elizabeth L. Bjork and Robert Bjork Accessed at
https://www.researchgate.net/publication/284097727_Making_th-
ings_hard_on_yourself_but_in_a_good_way_Creating_desirable_difficulties_to_enhance_learning
Novice to expert - based on ideas/research such as Mark McCourt Teaching for Mastery (2019) John
Catt publishing p32-34
Guided Practice
Craig Barton How I wish I’d Taught Maths (2018) John Catt Publishing
Willingham, D.T. (2002) Ask the cognitive Scientist. Allocating study time: massed versus distributed
practice American educator 26 (2) pp37-39
National Research Council (2000) How people learn: Brain Mind experience and school: expanded
edition Washington DC: National Academies Press
Dylan William effective questioning
https://www.teachermagazine.com/sea_en/articles/teacher-podcast-dylan-wiliam-on-effective-
questioning-in-the-classroom
New Documents 2020:
DfE Mathematics guidance: key stages 1 and 2 Non-statutory guidance for the national curriculum in
England (2020)
ht tps: //w w w.gov.uk /government /publications/teaching-mathematic s-in-primar y-schools
EBE (2020) Great Teaching toolkit evidence review
https://www.greatteaching.com/

38



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