# data[].attributes.body
Manipulatives - MathsBot.com

# data[].attributes.body
https://bit.ly/2YaNPbx

# data[].attributes.attachment_link.description
Education Scotland is a Scottish Government executive agency responsible for supporting quality and improvement in Scottish education.

# data[].attributes.attachment_link.readable_text
Explore this resource

This package of professional learning resources aims to enhance knowledge and understanding of effective learning and teaching progression, whilst impacting positively on numeracy attainment. It includes the following resources:

Guidance document

Number and Number Processes

Fractions, Decimal Fractions and Percentages

Time

Data and Analysis

Ideas of Chance and Uncertainty

Money (New 2022)

Glossary

These resources, produced as part of the National Improvement Framework, aim to support schools and establishments to identify opportunities for targeted learning and enhance the effective teaching of numeracy concepts and associated skills. They also aim to help practitioners to:

identify methods for enhancing learners' progression and skills in numeracy and support application;

plan how to develop their own practice to ensure the progression of skills when teaching numeracy.

Within each document there is progressive guidance for each level from early through to fourth. Each level includes the following:

Effective learning and teaching approaches;

Things to consider;

Links to other curricular areas;

Reflective questions.

In addition to supporting practitioners to review and reflect on the current progress of their learners, this resource also supports practitioners to identify development themes required for the reinforcement of learning. Furthermore, this resource can support practitioners to review and reflect strengths and improvement areas within their own professional learning.

School leaders can use this resource to support curriculum development and improvement. Local authority leaders can use this resource to support consideration and enhancement of centrally provided professional learning offers to practitioners.

Downloads

Guidance documents
PDF file: Guidance document (510 KB)PDF file: Guidance document (Gaelic version)(480 KB)
Professional Learning resource documents
PDF file: Number and Number Processes (2.7 MB)PDF file: Number and Number Processes (Gaelic version) (3.2 MB)PDF file: Fractions, Decimal Fractions and Percentages (1.9 MB)PDF file: Fractions, Decimal Fractions and Percentages (Gaelic version) (2.2 MB)PDF file: Time (2.1 MB)PDF file: Time (Gaelic version)(2.2 MB)PDF file: Data and Analysis (1.8 MB)PDF file: Data and Analysis (Gaelic version)(2 MB)PDF file: Ideas of Chance and Uncertainty (652 KB)PDF file: Ideas of Chance and Uncertainty (Gaelic version) (900 KB)PDF file: Money (2.6 MB)
Money - Gaelic Version coming soon.

Sketchnotes
PDF file: Early Level Sketchnote - Number and Number Processes (267 KB)
Glossaries
PDF file: Glossary (436 KB)PDF file: Glossary (Gaelic version) (480 KB)PDF file: Numeracy Glossary (Gaelic version - alphabetical) (479 KB)

# data[].attributes.attachment_link.description
This package of professional learning resources aims to enhance knowledge and understanding of effective learning and teaching progression, whilst impacting positively on numeracy attainment.

# data[].attributes.body
https://bit.ly/3DqPOst

# data[].attributes.attachment_link.title
Transforming Scotland into a maths positive nation: final report of the Making Maths Count group

# data[].attributes.attachment_link.description
Final report of the Scottish Government's Making Maths Count group, which identifies three main areas to be improved.

# data[].attributes.attachment_link.readable_text
Curriculum documents
Curriculum for Excellence - An Overview of Key Documents
National Thematic Inspection: Numeracy and Mathematics
Word file: National thematic inspection: numeracy and mathematics - Self-evaluation form - ELC settings (148 KB)Word file: National thematic inspection: numeracy and mathematics - Self-evaluation form - Schools (148 KB)National Numeracy and Mathematics Progression Framework
Transforming Scotland into a Maths Positive Nation: Final Report of the Making Maths Count Group Principles and Practice
Higher Order Thinking Skills in Maths
An Overview of Financial Education Resources
Family learning
Interdisciplinary learning
Are Boys and Girls Equally Prepared for Life?
Parentzone Scotland: Supporting Numeracy at Home
Parentzone Scotland: An Overview of the Numeracy and Mathematics Curriculum
Parentzone Scotland: Numeracy and Mathematics Glossary
Parentzone Scotland: I am a Mathematician
Parentzone Scotland: I am a Mathematician - Gaelic version
Scotland Learns: Parent activities for numeracy and mathematics Differentiated Learning in Numeracy and Mathematics Kennel Club - In the Dog House A Family Learning Approach to Maths - Pizza Maths in Renfrewshire
Big Chef, Little Chef - A Family Learning Programme
Clackmannanshire Numeracy and Mathematics Progression Framework
Engaging Families in Supporting their Child's Learning
Family Learning through STEM - Glasgow Clyde College
Increasing Parental Engagement through Opportunities for Families to Learn Together
Warout Family Cooking Programme - Friday Foodies Money Talks- Family Finances
On the Money Numeracy across learning
Numeracy in Social Studies
A Summary of STEM Resources
Scotland Learns: Practitioner activities for numeracy and mathematics Experiences and Outcomes
Curriculum for Excellence Benchmarks
Milestones to Support Learners with Complex Additional Support Needs
Key publications

Self-evaluation tools

Numeracy Professional Learning Resources

This suite of five professional learning papers, together with an accompanying guidance document and glossary has been designed to support improved learning and teaching of numeracy in the Broad General Education.

Professional Learning Community for Numeracy and Mathematics

The Numeracy and Mathematics PLC formally known as the National Numeracy and Mathematics Hub (Glow login required) has a range of professional learning modules, research papers and support materials. Sections include:

Additional support and resources

Financial education

Practice exemplars

Working collaboratively to support improvement
A Constructivist Approach to Improving Numeracy at Hareleeshill Primary School
Developing number talks and promoting mathematical mindsets in South Ayrshire
Closing the Poverty Related Attainment Gap in Numeracy - A Scottish Borders Cluster Approach
Numeracy in Inverclyde
Using a Lesson Study Approach to Improve Learning and Teaching in Numeracy in Glasgow
What is working in Numeracy and Mathematics - The Scottish Attainment Challenge

Research

Supporting learning at home

# data[].attributes.attachment_link.title
A summary of resources to support learning, teaching and assessment within numeracy and mathematics

# data[].attributes.attachment_link.description
Curriculum for Excellence - An Overview of Key Documents National Thematic Inspection: Numeracy and Mathematics Word file: National thematic inspection: numeracy and mathematics - Self-evaluation form - ELC settings (148 KB) Word file: National thematic inspection: numeracy and mathematics - Self-evaluation form - Schools (148 KB) National Numeracy and Mathematics Progression Framework Transforming Scotland into a Maths Positive Nation: Final Report of the Making Maths Count Group Principles and Practice Higher Order Thinking Skills in Maths An Overview of Financial Education Resources Family learning Interdisciplinary learning Are Boys and Girls Equally Prepared for Life?

# data[].attributes.body
Virtual Manipulatives | Mathematical Modelling | Toy Theater

# data[].attributes.attachment_link.readable_text
Online manipulatives bring all of the learning and fun without the mess. Our math virtual manipulatives help students grasp abstract concepts through modeling. Literacy virtual manipulatives create a fun space for practicing spelling, rhyming, and more!

# data[].attributes.attachment_link.title
Virtual Manipulatives | Mathematical Modelling | Toy Theater

# data[].attributes.attachment_link.description
Original teaching tools aimed at supporting teachers in the elementary school classroom. Two color counters, 3D dice, probability spinners, graph builders, fraction bars, base ten blocks, and more!

# data[].attributes.body
Virtual Learning Spaces (sharepoint.com) (Glow login required)

# data[].attributes.body
Numeracy and mathematics activities | Scotland Learns | National Improvement Hub (education.gov.scot)

# data[].attributes.body
https://bit.ly/3gKzSY4

# data[].attributes.body
Mathematics and numeracy | Curriculum areas | Learning in Scotland | Parent Zone (education.gov.scot)

# data[].attributes.attachment_link.readable_text
​​​Learning mathematics gives children and young people access to the wider curriculum and the opportunity to pursue further studies which may lead to careers in fields including science, technology and engineering. It equips us with the skills we need to interpret and analyse information, simplify and solve problems, assess risk and make informed decisions.

Mathematics is rich and stimulating and offers learners of all ages the opportunity to solve problems, develop logical reasoning and think creatively, skills which are not only sought after by employers in today's global economy, but are also essential life skills.

Numeracy

Numeracy is a skill which permeates and supports all areas of learning, allowing young people access to the wider curriculum and develop key skills for life and work. In our everyday lives we will regularly be required to estimate, use number processes, work with fractions and percentages, manage money and time, make measurements and read information from charts and tables.

Being numerate helps us to function responsibly in everyday life and contribute effectively to society. It increases our opportunities within the world of work and establishes foundations which can be built upon through lifelong learning.

What will my children learn?

Each area of the curriculum is broken down into experiences and outcomes. These are clear and concise statements about children's learning and progression from pre-school to S3. This stage of your child's education is known as the broad general education (BGE).

You can download and read the experiences and outcomes for numeracy and mathematics.

You can also access the National Numeracy and Mathematics Progression Framework. This resource has been created to deepen knowledge and understanding of progression within the experiences and outcomes for numeracy and mathematics.

The National Numeracy and Mathematics Progression Framework aims to support planning by identifying key milestones that learners should know before moving on to the next stage of learning.

How can I support my child's learning?

Supporting numeracy at home provides simple ideas to build numeracy skills into everyday activities with your child.

Numeracy and Mathematics Glossary

This glossary has been developed for parents and carers of children and young people in the broad general education in Scottish schools. It provides clear definitions of some of the commonly used terms in numeracy and mathematics.

Further information

Learn about the different mathematical areas and how they are used in everyday activities.
PDF file: The importance of numeracy and mathematics (175 KB)
Related links

A series of summaries of the National 4 and National 5 qualifications from the National Parent Forum of Scotland.

Information and links for parents and learners studying for National Qualifications in mathematics.

# data[].attributes.attachment_link.description
​​​Learning mathematics gives children and young people access to the wider curriculum and the opportunity to pursue further studies which may lead to careers in fields including science, technology and engineering. It equips us with the skills we need to interpret and analyse information, simplify and solve problems, assess risk and make informed decisions.

# data[].attributes.body
Numeracy and mathematics glossary | Supporting numeracy at home | Learning at home | Parent Zone (education.gov.scot)

# data[].attributes.attachment_link.readable_text
This glossary has been developed for parents and carers of children and young people in the broad general education in Scottish schools. This includes nursery through to 4th year in secondary schools. It provides clear definitions of some of the commonly used terms in numeracy and mathematics. The glossary may also be useful to any others who have an interest in the learning and teaching of numeracy and mathematics, including teachers and learners.

The glossary has been divided into the 15 'organisers' within the numeracy and mathematics curriculum. To find out more about each organiser, and how it relates to real life contexts, read the document below:
PDF file: The importance of numeracy and mathematics (175 KB) PDF file: The importance of numeracy and mathematics (Gaelic version) (196 KB)
The different organisers are:
PDF file: Estimation and rounding (174 KB) PDF file: Estimation and rounding (Gaelic version) (278 KB)
It has been organised in this way to make it easier to find the definition of the term you are looking for. For example, if looking for information on angles, you would select the organiser 'Angle, symmetry and transformation' by clicking on the link above. The document provides a list of all related terms, with a definition and in most cases, a supporting image. The terms are listed in alphabetical order to ensure easy access. Some terms appear in more than one organiser.

Words highlighted in blue are linked to further information available on other relevant websites.

Education Scotland understands that this glossary is not exhaustive and terms may be added.

# data[].attributes.attachment_link.description
Education Scotland is a Scottish Government executive agency responsible for supporting quality and improvement in Scottish education.

# data[].attributes.body
@edscot_maths

# data[].attributes.attachment_link.readable_text
In this two-part challenge, mathematical problems are given to reveal the issues and consequences of climate change. We're challenging students across Scotland to see how you can use maths to better the world. Whether it's by calculating the water flow in the world's rivers or comparing the biggest CO2 emitters, we want to see what you've got! The solution to these problems can empower you to make green choices in your own life and sort fact from misinformation, so give it a go!

Created by Institute and Faculty of Actuaries

# data[].attributes.attachment_link.description
In this two-part challenge, mathematical problems are given to reveal the issues and consequences of climate change. We're challenging students across Scotland to see how you can use maths to better the world. Whether it's by calculating the water flow in the world's rivers or comparing the biggest CO2 emitters, we want to see what you've got!

# data[].attributes.body
Talk Money Week | Join the conversation | The Money and Pensions Service (maps.org.uk)

# data[].attributes.attachment_link.readable_text
If you find it hard to talk about money, you're not alone. That's why every year we encourage people to open up about their finances. By having a conversation, you can improve your physical, mental and financial wellbeing.

Download your participation pack
Many of us keep unwanted money secrets from the people closest to us. Don't suffer in silence.

During COVID-19 we may be physically closer than ever to our immediate families, but financially we are still distant from some of our loved ones.

The effect of the pandemic has made it more important than ever to start conversations about money.

Research shows that people who talk about money:

make better and less risky financial decisions

have stronger personal relationships

help their children form good lifetime money habits

feel less stressed or anxious and more in control.

Building money conversations into our everyday lives also helps us build financial confidence and resilience to face whatever the future throws at us.

Use our online guides to help you start conversations about money with your:

If your partner or family controls access to your money, or runs up debts in your name, it's financial abuse. But there's no need to struggle on alone. Here are some of the things you can do and where to go for help and support.

Looking for money or pensions help?

MoneyHelper is here to make your money and pension choices clearer.
Here to put you in control with impartial help that's on your side, backed by government and free to use.

# data[].attributes.attachment_link.title
Talk Money Week | Join the conversation | The Money and Pensions Service

# data[].attributes.attachment_link.description
If you find it hard to talk about money, you're not alone. That's why every year we encourage people to open up about their finances. By having a conversation, you can improve your physical, mental and financial wellbeing. Download your participation pack Many of us keep unwanted money secrets from the people closest to us.

# data[].attributes.body
Numeracy and Mathematics PLC - Home (sharepoint.com)

# data[].attributes.attachment_link.readable_text
Background to the CPA framework

Children (and adults!) can find maths difficult because it is abstract. The CPA approach builds on children's existing 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. The CPA framework is so established in Singapore maths teaching that the Ministry of Education will not approve any teaching materials that do not use the approach.

Concrete step of CPA

Concrete is the "doing" stage. During this stage, students use concrete objects to model problems. Unlike traditional maths teaching methods where teachers demonstrate how to solve a problem, the CPA approach brings concepts to life by allowing children to experience and handle physical (concrete) objects. With the CPA framework, every abstract concept is first introduced using physical, interactive concrete materials.

For example, if a problem involves adding pieces of fruit, children can first handle actual fruit. From there, they can progress to handling abstract counters or cubes which represent the fruit.

Pictorial step of CPA

Pictorial is the "seeing" stage. Here, visual representations of concrete objects are used to model problems. This stage encourages children to make a mental connection between the physical object they just handled and the abstract pictures, diagrams or models that represent the objects from the problem.

Building or drawing a model makes it easier for children to grasp difficult abstract concepts (for example, fractions). Simply put, it helps students visualise abstract problems and make them more accessible.

Abstract step of CPA

Abstract is the "symbolic" stage, where children use abstract symbols to model problems. Students will not progress to this stage until they have demonstrated that they have a solid understanding of the concrete and pictorial stages of the problem. The abstract stage involves the teacher introducing abstract concepts (for example, mathematical symbols). Children are introduced to the concept at a symbolic level, using only numbers, notation, and mathematical symbols (for example, +, -, x, /) to indicate addition, multiplication or division.

# data[].attributes.attachment_link.title
CPA Approach Explained | Learn the Concrete, Pictorial, Abstract Method

# data[].attributes.attachment_link.description
Everything to get you started with Jerome Bruner's concrete, pictorial, abstract framework. Also referred to as concrete, representational, abstract, this maths mastery strategy progresses students through mathematical concepts in an intuitive way. Here's what CPA is, how to structure a lesson plan, and why it works.

# data[].attributes.attachment_link.readable_text
The Concrete Pictorial Abstract approach is now an essential tool in teaching maths at KS1 and KS2, so here we explain what it is, why its use is so widespread, what misconceptions there may be around using concrete resources throughout a child's primary maths education, and how best to use the CPA approach yourself in your KS1 and KS2 maths lessons.

The maths curriculum is far too broad to cover in one blog, so the focus here will be on specifically how the CPA approach can be used to support the teaching and learning of the four written calculation methods.

What is the Concrete Pictorial Abstract in Maths?

The Concrete Pictorial Abstract (CPA) approach is a system of learning that uses physical and visual aids to build a child's understanding of abstract topics.

Pupils are introduced to a new mathematical concept through the use of concrete resources (e.g. fruit, Dienes blocks etc). When they are comfortable solving problems with physical aids, they are given problems with pictures - usually pictorial representations of the concrete objects they were using.

Then they are asked to solve problems where they only have the abstract i.e. numbers or other symbols. Building these steps across a lesson can help pupils better understand the relationship between numbers and the real world, and therefore helps secure their understanding of the mathematical concept they are learning.

Origins of Concrete Pictorial Abstract Approach

Anyone working in primary mathematics education can't fail to have noticed that the word 'maths' is rarely heard these days without a mention of the term 'mastery' alongside it.

This is no surprise, with 'mastery' being the Government's flagship policy for improving mathematics and with millions of pounds being injected into the Teaching for Mastery programme; a programme involving thousands of schools across the country.

Prior to 2015, the term 'mastery' was rarely used. With the constant references to high achieving Asian-style Maths from East Asian countries including Singapore and Shanghai (and the much publicised Shanghai Teacher Exchange Programme), a teacher could be forgiven for believing 'teaching for mastery' to be something which was imported directly from these countries..

The fact that the CPA approach is a key component in maths teaching in these countries only added to the misconception.

The CPA maths model in Teaching for Mastery

To find the origins of the mastery maths approach, we need to go much further back in time and look much closer to home.

The concept of 'mastery' was first proposed in 1968 by Benjamin Bloom. At this time the phrase 'learning for mastery' was used instead. Bloom believed students must achieve mastery in prerequisite knowledge before moving forward to learn subsequent information.

Bloom suggested that if learners don't get something the first time, then they should be taught again and in different ways until they do.

Jerome Bruner and Concrete Pictorial Abstract

Looking more specifically at the origins of the CPA approach, we again need to go back to the teaching methods of the 1960s, when American psychologist Jerome Bruner proposed this approach as a means of scaffolding learning.

He believed the abstract nature of learning (which is especially true in maths) to be a 'mystery' to many children. It therefore needs to be scaffolded by the use of effective representations and maths manipulatives.

He found that when pupils used the CPA approach as part of their mathematics education, they were able to build on each stage towards a greater mathematical understanding of the concepts being learned, which in turn led to information and knowledge being internalised to a greater degree.

Many teachers mistakenly believe mastery, and specifically the CPA approach, to have been a method imported from Singapore.

In actual fact, the Singapore Maths curriculum has been heavily influenced by a combination of Bruner's ideas about learning and recommendations from the 1982 Cockcroft Report (a report by the HMI in England, which suggested that computational skills should be related to practical situations and applied to problems).

To get a better handle on the concept of maths mastery as a whole, take a look at our Ultimate Maths Mastery guide.

Why use the Concrete Pictorial Abstract approach in Maths?

Pupils achieve a much deeper understanding if they don't have to resort to rote learning and are able to solve problems without having to memorise.

When teaching reading to young children, we accept that children need to have seen what the word is to understand it. Putting together the letters c- a- t would be meaningless and abstract if children had no idea what a cat was or had never seen a picture.

People often don't think of this when it comes to maths, but to children many mathematical concepts can be equally meaningless without a concrete resource or picture to go with it. This applies equally to mathematics teaching at KS1 or at KS2.

What is a 'Concrete' representation in the CPA approach?

As part of the CPA approach, new concepts are introduced through the use of physical objects or practical equipment. These can be physically handled, enabling children to explore different mathematical concepts. These are sometimes referred to as maths manipulatives and can include ordinary household items such as straws or dice, or specific mathematical resources such as dienes or Numicon.

The abstract nature of maths can be confusing for children, but through the use of concrete materials they are able to 'see' and make sense of what is actually happening.

Previously, there has been the misconception that concrete resources are only for learners who find maths difficult. In fact concrete resources can be used in a great variety of ways at every level. All children, regardless of ability, benefit from the use of practical resources in ensuring understanding goes beyond the learning of a procedure.

Practical resources promote reasoning and discussion, enabling children to articulate and explain a concept. Teachers are also able to observe the children to gain a greater understanding of where misconceptions lie and to establish the depth of their understanding.

What is a 'Pictorial' representation in the CPA approach?

Once children are confident with a concept using concrete resources, they progress to drawing pictorial representations or quick sketches of the objects. By doing this, they are no longer manipulating the physical resources, but still benefit from the visual support the resources provide.

Some teachers choose to leave this stage out, but pictorial recording is key to ensuring that children can make the link between a concrete resource and abstract notation. Without it, children can find actually visualising a problem difficult.

One of the most common methods of representing the pictorial stage is through the bar model which is often used in more complex multi step problem solving.
Read more: What is a bar model

What is an 'Abstract' representation in the CPA approach?

Once children have a secure understanding of the concept through the use of concrete resources and visual images, they are then able to move on to the abstract stage. Here, children are using abstract symbols to model problems - usually numerals. To be able to access this stage effectively, children need access to the previous two stages alongside it.

For the most effective learning to take place, children need to constantly go back and forth between each of the stages. This ensures concepts are reinforced and understood.

How to teach using the Concrete Pictorial Abstract method at primary school

A common misconception with this CPA model is that you teach the concrete, then the pictorial and finally the abstract. But all stages should be taught simultaneously whenever a new concept is introduced and when the teacher wants to build further on the concept.

When concrete resources, pictorial representations and abstract recordings are all used within the same activity, it ensures pupils are able to make strong links between each stage.

If you're concerned about differentiating effectively using the CPA approach, have a look at our differentiation strategies guide for ideas to get you started. Or if you're short on time, our White Rose Maths aligned lesson slides incorporate the CPA approach into them and some are free to download and use.

Using 'The Four Operations' to model how CPA works

In the following section I will be looking at the 'four operations' and how the CPA approach can be used at different stages of teaching them. In particular, I will examine how the 3 parts of the CPA approach should be intertwined rather than taught as 3 separate things.

As this blog is to share ideas rather than say how the calculation methods should be taught, I am only going to cover the four operations briefly.

It's important to take your school's Calculation Policy into account when determining how the CPA approach can work best for you.

Teaching addition using the Concrete Pictorial Abstract approach

In the early stages of learning column addition, it is helpful for children to use familiar objects. For example, straws or lollipop sticks can be bundled into groups of ten and used individually to represent the tens and ones.

Once children are familiar making 2-digit numbers using these resources, they can set the resources out on a baseboard to represent the two numbers in a column addition calculation.

Initially children complete calculations where the units do not add to more than 9, before progressing to calculations involving exchanging/ regrouping. Alongside the concrete resources, children can annotate the baseboard to show the digits being used, which helps to build a link towards the abstract formal method.

Once children are confident using the concrete resources they can then record them pictorially, again recording the digits alongside to ensure links are constantly being made between the concrete, pictorial and abstract stages.

The next step is for children to progress to using more formal mathematical equipment. Dienes base ten should be introduced alongside the straws, to enable children to see what is the same and what is different. As confidence grows using the Dienes, children can be introduced to the hundreds column for column addition, adding together 3-digit and 2-digit numbers

Alongside the concrete resources children should be recording the numbers on the baseboard, and again have the opportunity to record pictorial representations.

Once children are completely secure with the value of digits and the base ten nature of our number system, Dienes equipment can be replaced with place value counters. These help children as they progress towards the abstract, as unlike the dienes they are all the same size.

These should be introduced in the same way as the other resources, with children making use of a baseboard without regrouping initially, then progressing to calculations which do involve regrouping.

As with the other equipment, children should have the opportunity to record the digits alongside the concrete resources and to progress to recording pictorially once they are secure. The place value counters can be used to introduce children to larger numbers, calculating column addition involving the thousands and then the ten thousands column.

Read also: How To Teach Addition For KS2 Interventions In Year 5 and Year 6

Teaching subtraction using the Concrete Pictorial Abstract

The method for teaching column subtraction is very similar to the method for column addition. Children should start by using familiar objects (such as straws) to make the 2-digit numbers, set out on a baseboard as column subtraction.

To begin with, ensure the ones being subtracted don't exceed those in the first number. Once children are confident with this concept, they can progress to calculations which require exchanging. As with addition, the digits should be recorded alongside the concrete resources to ensure links are being built between the concrete and abstract.

Once secure with using the concrete resources, children should have the opportunity to record pictorially, again recording the digits alongside.

As with addition, children should eventually progress to using formal mathematical equipment, such as Dienes.

These should be introduced alongside the straws so pupils will make the link between the two resource types. Digits are noted down alongside the concrete resources and once secure in their understanding children can record the Dienes pictorially, to ensure links are built between the concrete and abstract.

Once secure with the value of the digits using Dienes, children progress to using place value counters.

Again, the counters enable children to work concretely with larger numbers, as well as bridging the gap from the use of Dienes to the abstract. Along with the counters, children should be recording the digits and they should have the opportunity to record pictorially once confident with the method using concrete resources.

Read also: How to Teach Subtraction for KS2 Interventions in Year 5 and Year 6

Teaching multiplication using the Concrete Pictorial Abstract approach

The grid method is an important step in the teaching of multiplication, as it helps children to understand the concept of partitioning to multiply each digit separately.

Pupils can begin by drawing out the grid and representing the number being multiplied concretely. For example, 23 x 3 can be shown using straws, setting out 2 tens and 3 ones three times. This way, children can actually see what is happening when they multiply the tens and the ones.

As with addition and subtraction, children should be recording the digits alongside the concrete apparatus, and recording pictorially once they are confident with the concrete resources.

Children are then able to progress to representing the numbers in a grid, using place value counters.

The video above is a great example of how this might be done.

Once children are confident using the counters, they can again record them pictorially, ensuring they are writing the digits alongside both the concrete apparatus and the visual representations.

Progressing to the expanded method and then the short method of column multiplication is much easier for children if these are introduced alongside the grid method, to enable them to see the link.

Read also: How to Teach Multiplication for KS2 Interventions in Year 5 and Year 6

Teaching division using the Concrete Pictorial Abstract (CPA) Approach

As children work towards the formal written method for division, it is important they understand what is meant by both division as grouping and division as sharing.

Concrete resources are invaluable for representing this concept. Pupils need to understand how numbers can be partitioned and that each digit can be divided by both grouping and sharing.

Once confident using concrete resources (such bundles of ten and individual straws, or Dienes blocks), children can record them pictorially, before progressing to more formal short division.

As children work towards understanding short division (also known as the bus stop method), concrete resources can be used to help them understand that 2-digit numbers can be partitioned and divided by both sharing and grouping. This can be through the use of bundles of ten straws and individual straws or dienes blocks to represent the tens and ones.

As with the other operations, it's important that children are recording the digits alongside the concrete resources and are having the opportunity to draw visual representations. As children grow in confidence and once they are ready to progress to larger numbers, place value counters can replace the dienes.

Read also: How to Teach Division for KS2 Interventions in Year 5 and Year 6

The Concrete Pictorial Abstract Approach is here to stay

It may have taken many years for CPA to reach the level of popularity it has today, but it is definitely here to stay. I'm not one to jump on the bandwagon when it comes to the latest teaching 'fad', however this has been one I've been happy to jump on.
Read more:
I have seen first-hand how successful it can be when children have the opportunity to work in this way and I love the fact that children are now starting to have the conceptual understanding in maths that I never had as a child.

Do you have pupils who need extra support in maths?
Every week Third Space Learning's maths specialist tutors support thousands of pupils across hundreds of schools with weekly online 1-to-1 lessons and maths interventions designed to plug gaps and boost progress.

Since 2013 we've helped over 150,000 primary and secondary school pupils become more confident, able mathematicians. Learn more or request a personalised quote for your school to speak to us about your school's needs and how we can help.

# data[].attributes.attachment_link.title
The CPA Approach: What It Is & How to Use It In Primary Maths

# data[].attributes.attachment_link.description
The Concrete Pictorial Abstract (CPA) approach helps pupils develop a deeper, more secure understanding of how to solve maths problems.

# data[].attributes.attachment_link.readable_text
Article by NRICH team

What do we mean by "Low Threshold High Ceiling"?

Here at NRICH, we've been using the phrase Low Threshold High Ceiling (LTHC) for many years to describe attributes shared by lots of our tasks. In the context of learning technologies, the phrase LTHC (also sometimes referred to as 'low-threshold-no-ceiling', or 'low-floor high-ceiling') originates in Seymour Papert's description of the central design principle of the Logo programming language. The 'low threshold' means that new users, including those who have never programmed before, should find it easy to get started, and 'no ceiling' (or 'high-ceiling') means the programming language shouldn't be limiting for advanced users. We use the term in a similar way when talking about our resources. It can be summed up as follows: a Low Threshold High Ceiling task means everyone can get started, and everyone can get stuck.

Everyone can get started
If an activity is considered to be LTHC for a particular group of learners, pretty much everyone in the group should be able to make a start on it. The threshold needs to be mathematically accessible for all the students in the group, that is, everyone needs to have the prior mathematical knowledge required to start working on the problem. This will of course vary depending on the age and prior attainment of the learners in the group - a task that has a low threshold for 17 year olds might be far out of the reach of most 7 year olds!

As well as the mathematical threshold, it's important to consider the psychological threshold of the task - how resourceful and resilient will students need to be? Some problems may require a very basic understanding of the mathematical content to solve, and yet still be extremely challenging (see for example the task Shopping Basket). When faced with such a task, learners might struggle to know where to start, or to find any small step that they can take towards solving the problem. This places a high level of demand on learners at the start of the task. This doesn't mean such tasks are without value! They can still be rich tasks, worthwhile for students to spend time struggling with, but they are not LTHC. On NRICH, the Challenge Level of a task indicated by one, two or three stars might give an indication of the mathematical and psychological threshold. LTHC tasks are likely to have a challenge level of one star - you can see some examples of such tasks in the article Creating a Low Threshold High Ceiling Classroom. (Shopping Basket has a challenge level of three stars!)

Everyone can get stuck
Part of becoming a resilient mathematician is learning to recognise what it feels like to be stuck, and what strategies can be useful in getting yourself unstuck. Some undergraduate mathematics students report that they found maths lessons at school very straightforward until the very end of school or the beginning of university, and the culture shock of finding maths difficult for the first time was very daunting! One way to mitigate against that is to make sure that all students experience mathematical struggle throughout their time in school, and that being stuck is normalised and recognised as an important part of mathematical problem solving. Low Threshold High Ceiling tasks are designed to have lots of built-in extension opportunities, so that there are harder questions to be asked and more challenging problems to solve. This means that all learners can potentially reach a point where they don't immediately know what to do next, and they can start to develop their resilience and learn fruitful strategies for making progress when they feel as if they've come up against a brick wall.

Why do we like Low Threshold High Ceiling tasks?
At NRICH, we promote the use of Low Threshold High Ceiling tasks because we believe that working on such tasks helps learners develop into competent, confident mathematicians. The Low Threshold may mitigate against the development of maths anxiety by making sure that learners do not fail at the first hurdle. The High Ceiling offers everyone the opportunity to develop their resilience. Furthermore, LTHC tasks give learners a sense of agency as there is more than one pathway available to them. Rather than a classroom situation where different children are given different activities to work on, the whole class is working together on the same task. There is no predetermination as to who will work on extension work and who will stick to the support material, and everyone has a sense of what is going on, with plenaries giving everyone a chance to hear how others in the classroom have worked on the same activity in different ways. LTHC tasks allow learners to demonstrate what they can do, rather than what they can't. As teachers it's very easy to predict how well our learners will cope with a particular piece of mathematics, and sometimes that prediction can be a self-fulfilling prophecy. When the ceiling is raised it can be surprising what heights learners can achieve.

To find out more about effective teaching using Low Threshold High Ceiling tasks, read the article Creating a Low Threshold High Ceiling Classroom.

# data[].attributes.attachment_link.title
Low Threshold High Ceiling - an Introduction

# data[].attributes.attachment_link.description
In this article for teachers, we explain what is meant by Low Threshold High Ceiling tasks, and why we like them.

# data[].attributes.headline
Session One Slides - Introduction to N&M

# data[].attributes.subject
Session One Slides - Introduction to N&M

# data[].attributes.attachment_link.title
NQT ECT Session 1 - Introductory Session (1)

# data[].attributes.attachment_link.readable_text
Realising the Ambition: Being Me builds upon the original principles and philosophy of Pre-Birth to 3 and Building the Ambition. The new guidance retains the relevant content from the previous guidance which it replaces, extending and strengthening it in line with current research and evidence about how children develop and learn.

This refreshed early years national practice guidance for Scotland presents key information about the characteristics of child development based on research and evidence.

In essence, 'Realising the Ambition: Being Me' increases expectations of high quality but still provides the necessary support for all who work in the early years sector and beyond.

Explore this resource

High quality provision consists of many different interconnected aspects. The imagery of tartan has been used in the guidance to remind us of the uniqueness of our Scottish context - with each section seen as an interconnected thread contributing to the whole piece. Although each section has been developed so it can be used in any order to meet different needs and different contexts, we must always be mindful of this bigger picture.

The terms of 'when I am a baby', 'when I am a toddler' and 'when I am a young child' have been used to illustrate developmental needs over time - but these should not be seen as rigid lines of progression. All babies and young children develop their individual skills, knowledge and attributes at different rates according to who they are. The guidance can be used to support children with more complex needs by taking this into account.

The guidance can be used individually, or in short informal group settings where practitioners have time to come together, or as part of a more substantial CPL programme. We would recommend that at points of transition between ELC and primary school, the guidance can be used by designated staff working together to promote a clear understanding of curriculum progression and continuity in children's learning and development.

Please note that the link to the Leuven scales on page 50 is no longer live. Alternative links are provided in the 'Related links' section below.

Challenge questions

The Challenge Questions PowerPoint contains suggested challenge questions for each section of Realising the ambition: Being me.

Webinars

Our live Realising the Ambition sessions have already been delivered to thousands of early years educators across Scotland. We now have a recorded version available to enable you to share or recap on the themes.

1. Realising the Ambition overview part 1 discusses the guiding principles and the importance of the imagery used in the guidance.

2. Realising the Ambition overview part 2 looks at pedagogy in practice and transitions.

3. Realising the Ambition Overview for Childminders was produced in partnership with the Scottish Childminding Association.

# data[].attributes.attachment_link.description
Education Scotland is a Scottish Government executive agency responsible for supporting quality and improvement in Scottish education.

# data[].attributes.headline
Scottish Book Trust - Song and Rhyme Library

# data[].attributes.subject
Scottish Book Trust - Song and Rhyme Library

# data[].attributes.attachment_link.description
You can find songs and rhymes to suit different moods or times of day. Try a range of songs and rhymes and discover your child's favourites!

# data[].attributes.attachment_link.readable_text
Article by Alf Coles

This article follows on from Journeys on the Gattegno Tens Chart, in which Alf describes the way a Year 3/4 class (7-9 year olds) used a Gattegno chart to gain an awareness of place value and the inverse relationship between multiplication and division. Here Alf outlines six activities using the Gattegno chart which begin by developing learners' awareness of reading, writing and ordering numbers, then progress to gaining awareness of multiplication facts, multiplication and division by 10, onto fractions as operations and, finally, to finding percentages of any number. These activities were created in the context of a project run through the charity5x5x5=creativity, in collaboration with Penny Hay, Sara James, Kate Parsons, George Samios, Hannah Scott, Helen Shoubridge, Sam Thacker, Ellie Walsham, Carol Watts, Grace Webster.

An example of the Gattegno Chart

Activity 1: Naming Numbers
Aim
When introducing the Gattegno chart to a group for the first time, students needs to see how numbers are named on the chart. Rather than concern about the meaning or place value of numbers, the focus of this activity is on how to say and write numbers and to gain awareness of how they are ordered.

A starting point
Tap on a number in the units row and get the class to chant back in unison the number name. Continue for other numbers of the units row and extend to numbers in the tens row.

Tap on "4" (class chants FOUR) and then "40" (class chants FOUR-TY); tap on "6" and then "60"; tap on "8" and then "80". Focus attention on how the number name changes. All we do is add "-ty" to the end of the digit number. Practise this (you may want to allow students to say "five-ty" and "three-ty" for 50 and 30).

Having established these names, tap on "40" followed by "2" - students need to chant back "FOUR-TY-TWO".

Keep tapping out two-digit numbers. When students seem confident in calling back these number names, invite a student to the board to write "forty-two" - or, if this is new to students, show them how to write it and invite someone to write 43, 44, etc.

Students could count up in 1s from different starting points, focusing on how the digit names keep repeating in sequence.

A task
Students, in pairs or small groups, could be challenged to create their own "100 square", i.e., order the numbers 1 - 100 (or as an extension, 100-200). The Gattegno chart can be kept in view as support.

Activity 2: Multiples
Aim
Support students in exploring patterns in number, leading to awareness of multiplication table facts.

A starting point
With no introduction, say to the class: "One hundred". Invite a response - someone will usually say "Two hundred", you can then reply "Three hundred" and get the class to continue chanting multiples of 100. As they do this, begin tapping out the numbers on the Gattegno chart. When you have gone slightly higher than feels comfortable, invite the class to give you another starting number, or choose one for the class (e.g. 10 or 5 or 2).

Work on multiples of the new starting number, get the class to chant in unison - again, tapping out the numbers they say on the chart. Someone can write down the numbers as well. Invite the class to talk about any patterns they notice.

A task
Students pick their own starting number and write out multiples, going as high as they can. Students can be invited to pay attention to any patterns they notice. At some point you might want to gather the class on the carpet and invite individuals to describe the patterns they have seen. Other students can be asked if they can continue these patterns.

Activity 3: Addition and Subtraction Journeys
Aim
Students to develop an awareness of what happens at the 'transition' between tens and/or hundreds

A starting point
Tap on a number and get students to chant back the number one higher. Choose different numbers, particularly those ending in a "9".

A task
Students choose a starting number and add one. They then keep on adding one, looking for patterns in their answers, writing, for example:
17+1=18
18+1=19
19+1=20
20+1=21
When they have gone as far as they want, they choose a new starting number.

The same process could be gone through with subtracting one, or going up/down in numbers other than one.

Activity 4: Multiplication and Division by Powers of 10
Aim
Students to gain an awareness of how to multiply and divide by powers of ten and awareness of the relations between these operations.

A starting point
Tap on a number, get the class to chant back the number that is 10 times bigger. Initially chose single-digit numbers, then progress to others, returning to single digit if the class lose confidence. At some point, invite someone to say how, on the chart, they get their answer. Focus attention on how you can get the answer simply by moving down one row on the chart. Try to steer the class away from mention of changing numbers of zeros as such methods do not extend to decimals. Return to chanting with the awareness of movement. Then point on a number and invite the class to chant back that number divided by 10. Repeat for multiplication and division by 100 (and possibly 1000).

A task
Students have to choose a number on the chart, then go on a 'journey' multiplying or dividing by 10, 100, etc. Their challenge is to return to the number they started with. Do one 'journey' with the class and then let them try out their own.

You may want to show some of the decimal rows of the chart in order to help students extend their journeys. Be prepared for some big numbers!

After some time, gather students to discuss what they have done. They might have questions they want to explore further. This activity has continued with energy over four hours and more with some groups. Questions might emerge about how long they can make their journeys, or whether they can go on a journey and get back in one go, or whether they can get journeys that shift between columns (see Fraction Journey activity below).

Activity 5: Fraction Journeys
Aim
Let students develop an awareness of fractions as operations.

A starting point
This activity is best done following on from: 'Multiplication and Division by Powers of 10' (see above). In that activity, students may have noticed that their 'journeys' only move vertically (up or down). It may be that a student asks how they could do different journeys. This activity is best set up if this question has arisen in such a way.

The key to the starting point is to establish, through chanting on the chart, that to get from the first (1s) column to, say, the fifth (5s) column, we multiply by 5. To go in the reverse direction, we multiply by one fifth. To go from the 1s column to the 7s column, we multiply by 7. To go in the reverse direction we multiply by one seventh.

A task
Students have to pick two numbers from the chart: a starting number and a finishing number. Their task is to find a journey from one to the other.

e.g., From 200 to 70
200 x ½ = 100
100 Ã·10 = 10
10 x 7 = 70

You may need to do a few such journeys all together. We have seen year 3 classes (7-8 year olds) confidently working with fractions in this way, having spent time beforehand working on multiplication and division by 10 on the Gattegno chart.

Activity 6: Percentages
Aim
Develop a method for finding a percentage of any number.

A starting point
Tap on a number, get the class to chant back 10% of that number. Repeat many times. As with multiplication and division by powers of ten, at some point focus attention on what you are doing visually (going up one row). Then repeat with finding 1% of a number (going up two rows).

A task
Students can set challenges for each other of finding percentages of a number. They should find their percentages initially by splitting it up into 10s and 1s.

They may notice ways to get 20% or 5%, based on 10% and this can speed up their calculations.

A harder challenge is to choose two numbers from the chart and find, as close as possible, one as a percentage of the other.

e.g., choose numbers 16 and 60, what percentage of 60 gives us an answer as close as possible to 16?

10% of 60 = 6
10% of 60 = 6
5% of 60 = 3

So, 25% of 60 is 15
Can you get closer?

# data[].attributes.attachment_link.description
In this article, Alf outlines six activities using the Gattegno chart, which help to develop understanding of place value, multiplication and division.

# data[].attributes.attachment_link.description
Mathematics is the study of numbers, calculations, shapes and equations, written using special notations and symbols.

# data[].attributes.attachment_link.readable_text
These PowerPoints include links to relevant resources and pupil-facing activities. They are ideal to be used with small groups of pupils to review, practise, and consolidate learning. The five-minute video at the foot of this page demonstrates this in more detail.

The pupil activities are intended to stimulate interaction and discussion. We suggest you use them flexibly and are guided by pupil response, repeating activities where pupils lack confidence. The ready-to-progress criteria are intended as goals for the end of the year. When used at the start of a year, you might want to use the materials from the previous year group. If you are teaching Year 4 for example, the Year 3 materials are used to review, practice and consolidate learning from the previous year.

# data[].attributes.attachment_link.description
79 PowerPoints, each one focusing on one of the ready-to-progress criteria in the new DfE maths guidance for KS1 and KS2

# data[].attributes.attachment_link.title
Session2   Number and Number Processes, Primary

# data[].attributes.headline
National Improvement Hub:           The Moderation Cycle 

# data[].attributes.subject
National Improvement Hub:           The Moderation Cycle 

# data[].attributes.attachment_link.description
The Moderation Cycle is designed for use by practitioners to support their understanding of moderation. Engaging in the moderation process with colleagues will assist practitioners in arriving at valid and reliable decisions on learners' progress towards, and achievement of, a level. Consequently, this resource will support the consistency of teachers' professional judgement, both locally and nationally.

# data[].attributes.body
What are the counting principles?

# data[].attributes.attachment_link.title
The Counting Principles - Number and Number Processes

# data[].attributes.headline
Session 3 Slides - Fractions, Decimal Fractions & Percentages

# data[].attributes.subject
Session 3 Slides - Fractions, Decimal Fractions & Percentages

# data[].attributes.headline
Session 3 Recording - Fractions, Decimal Fractions and Percentages

# data[].attributes.subject
Session 3 Recording - Fractions, Decimal Fractions and Percentages

# data[].attributes.attachment_link.title
Introduction to teaching Fractions, Decimal Fractions and Percentages Primary

# data[].attributes.headline
Session 4 - Gender Balance and Equality in Maths

# data[].attributes.subject
Session 4 - Gender Balance and Equality in Maths

# data[].attributes.attachment_link.title
Gender Balance and Equality within Maths Prim NQTs 6 Dec

# data[].attributes.headline
Gender Balance and Equality in Maths Links

# data[].attributes.subject
Gender Balance and Equality in Maths Links

# data[].attributes.headline
Session 5 - Developing a Problem Solving Approach

# data[].attributes.subject
Session 5 - Developing a Problem Solving Approach

# data[].attributes.attachment_link.description
Same But Different is a powerful routine for use in math classrooms. The images can be used for teaching a way of thinking: grayscale thinking, categorical thinking; helping students to build a network of ideas and an approach for learning and thinking about mathematics.

# data[].attributes.headline
Developing a Problem Solving Approach Links

# data[].attributes.subject
Developing a Problem Solving Approach Links

# data[].attributes.attachment_link.title
Links Developing a Problem Solving Approach

# data[].attributes.headline
Recording of Session 5 - Developing a Problem Solving Approach

# data[].attributes.subject
Recording of Session 5 - Developing a Problem Solving Approach

# data[].attributes.attachment_link.title
Developing a Problem solving approach Primary

# data[].attributes.headline
Session 7 Slides - Teaching Data & Analysis

# data[].attributes.subject
Session 7 Slides - Teaching Data & Analysis

# data[].attributes.attachment_link.title
Effective questioning to support assessment in numeracy and mathematics

# data[].attributes.attachment_link.description
Education Scotland is a Scottish Government executive agency responsible for supporting quality and improvement in Scottish education.

# data[].attributes.body
3 x First Level
1 x Second Level
These will give you a starting point to build your own ideas on.

# data[].attributes.headline
Session 7 Recording - Teaching Data & Analysis

# data[].attributes.subject
Session 7 Recording - Teaching Data & Analysis

# data[].attributes.attachment_link.title
Introduction to Teaching Data and Analysis

# data[].attributes.headline
Session 8 Slides - Taking Maths Outdoors & Effective Questioning

# data[].attributes.subject
Session 8 Slides - Taking Maths Outdoors & Effective Questioning

# data[].attributes.attachment_link.title
Session 8 - Maths Outdoors and Effective Questioning

# data[].attributes.headline
Session 8 - Maths Outdoors & Effective Questioning in Maths

# data[].attributes.subject
Session 8 - Maths Outdoors & Effective Questioning in Maths

# data[].attributes.attachment_link.readable_text
0:00 / 1:09:46•Watch full video

# data[].attributes.headline
Session 9 Slides - Ideas of Chance & Uncertainty

# data[].attributes.subject
Session 9 Slides - Ideas of Chance & Uncertainty

# data[].attributes.attachment_link.title
Session 9 - Ideas of Chance and Uncertainty

# data[].attributes.headline
Session 9 Recording - Ideas of Chance & Uncertainty

# data[].attributes.subject
Session 9 Recording - Ideas of Chance & Uncertainty

# data[].attributes.attachment_link.readable_text
Chance and Uncertainty

0:00

54:20

Back

# data[].attributes.body
Please complete the evaluation if you attended any of the sessions or accessed any of the recordings.

# data[].attributes.attachment_link.readable_text
Loading…

# data[].attributes.attachment_link.description
Easily create surveys and polls to collect customer feedback, measure employee satisfaction, and organize team events.