ST MICHAEL'S C OF E PRIMARY ACADEMY

Mathematics 

At St. Michael's, mathematics will provide children with practical, hands on learning opportunities to support their development and understanding of facts, methods and strategies. 

At St. Michael's, we follow the ‘White Rose Hub’ Schemes of Learning across school. Children experience opportunities for regular fluency, reasoning and problem solving tasks. These tasks are completed when exploring the entire mathematics curriculum, organised into a progressive long term plans covered during a child's time at St. Michael's. 

The Fundamentals of Mathematics at St. Michael's 

• Practical opportunities to develop mastery in mathematics through fluency, reasoning and problem solving tasks.
• Use of the CPA (concrete, pictoral and abstract) approach to embed and deepen understanding. 
• Daily key skills teaching, focused on building, revising and securing the foundation skills in mathematics.
• Times tables practise, through TT Rockstars, Numbots and CGP Daily Practice books, focused on improving speed and accuracy of instant recall facts 

St. Michael's Maths Handbook - Download here

Mathematics Intent 

The children at St. Michael's are provided with a mathematics curriculum designed to:

• build on and develop core mathematical concepts in a well sequenced and structured progression across their time at St. Michael's.
• support children’s acquisition and progressive understanding of declarative, conceptual and procedural mathematical knowledge.
• enable children to become fluent in the fundamentals of mathematics.
• equip children to develop conceptual understanding recall and apply.
• provide opportunities for frequent and varied practise of reasoning and problem solving.
• allow children to gain and use a range of transferable key skills both in mathematics and across the full curriculum.
• offer opportunities for explicit teaching of problem solving activities.

Mathematics Implementation: Knowledge  

Progression of skills and understanding is vital in the successful implementation of our mathematics curriculum. In order for children to progress, they need to have a firm understanding of facts (declarative knowledge), before developing methods (procedural knowledge) and implementing strategies (conditional knowledge). Teachers ensure that children’s understanding of facts is secure, before developing and building on understanding of methods and strategies.

As a school, we recognise the importance of problem solving in mathematics and, therefore, children are provided with explicit opportunities for gaining these skills and knowledge.

Declarative Knowledge

The statutory programmes of study from the National Curriculum are taught through the school’s adopted ‘White Rose Hub’ Scheme of Learning. Clear progression of skills and concepts is identified with the scheme across Early Years, Key Stage One and Key Stage Two.  Using White Rose Hub materials, including ‘Small Steps’ (based on Ready to Progress Criteria), mathematics lessons at St. Michael’s focus on learning facts, methods and strategies, which have been broken down into small steps. This ensures children deepen their knowledge and understanding of core concepts. The exploration of the types of mathematical knowledge (declarative/facts, procedural/methods and conditional/strategies) is made explicit to the children in teaching and learning.

TT Rockstars, Numbots and NCETM Mastering Numer (EYFS and KS1) are used to teach and embed core number and multiplication facts.

Each day starts with 15-30 minutes of mathematical knowledge revision via the use of TT Rockstars and/or CGP revision guides. Each lesson begins with daily revision, focused on building, revising and securing the foundation facts in mathematics.

Procedural Knowledge

Mathematics at St. Michael’s involves practical opportunities to develop mastery in mathematics through fluency, reasoning and problem solving tasks, underpinned by the use of a CPA approach (concrete, pictorial and abstract) when teaching procedural knowledge.  

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. 

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.

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.

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.

Teachers will go back and forth between each stage to reinforce concepts. This supports children to craft powerful mental connections between the concrete, pictorial, and abstract phases.

By ensuring that concrete representations aren’t removed too early, it allows children to build a conceptual mathematical understanding that can propel them through their education.

The CPA model is a progression. By the end of KS1, children need to be able to go beyond the use of concrete equipment to access learning using either pictorial representations or abstract understanding.

Conditional Knowledge

Conditional knowledge is the knowledge is the understanding of strategies which can be used to reason and solve problems.  This extends to combinations of conceptual / declarative and procedural knowledge which then become strategies for particular types of problems.  This type of knowledge can typically be described as ‘I know when…’

At St. Michael’s, our mathematics curriculum aims to ensure all pupils become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately. Once secure in this area, children can reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language. Finally, children can solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions.

 Mathematics Implementation: Scheme of Learning

The statutory programmes of study from the National Curriculum are taught through the school’s adopted ‘White Rose Hub’ Scheme of Learning. Clear progression of skills and concepts is identified with the scheme across Early Years, Key Stage One and Key Stage Two. 

Using White Rose Hub materials, including ‘Small Steps’ (based on Ready to Progress Criteria), mathematics lessons at Sleights focus on learning facts, methods and strategies, which have been broken down into small steps. This ensures children deepen their knowledge and understanding of core concepts. The exploration of the types of mathematical knowledge (declarative/facts, procedural/methods and conditional/strategies) is made explicit to the children in teaching and learning.

Mathematic Implementation: Summary

The statutory programmes of study from the National Curriculum are implemented at St. Michael's through:

• White Rose Hub Schemes of Learning in Early Years, Key Stage One and Key Stage Two.
• In addition to White Rose Maths, the EYFS and Key Stage One curriculums are enhanced by the use of NCETM's Mastering Number Programme. This programme aims to develop solid number sense, including fluency and flexibility with number facts, which will have a lasting impact on future learning for all children. It also involves high quality professional development for teachers.
• Daily mathematics lessons, consisting of key skills and knowledge development, focused on acquiring and building on mathematical facts, methods and strategies.
• Quality first teaching, designed to meet the needs of all children.
• Effective and purposeful use of a wide range of manipulative resources to support conceptual, pictorial and abstract understanding - CPA approach. 
• Ongoing development of the learning environment, including working walls, manipulatives and challenges.
• A robust and progressive approach to calculation and times tables.
• Support for children with additional needs, through booster or intervention programmes.

Memory: The Retention of Learning 

The fundamental idea behind our curriculum design is to support pupils to be able to perform simpler tasks so they can then move on to perform more complex tasks. For example, we cannot expect pupils to add two numbers together before they understand what each individual number represents.

This thinking gives rise to a typical sequence of ‘blocks’ of mathematics that you will see in most of our year groups.

Within each of these blocks we then have ‘small steps’ which are again sequenced in order of difficulty and dependency. Here are the first seven steps (of 18) in our Year 3 Addition and Subtraction block:

1. Add and subtract multiples of 100 
2. Add and subtract 3-digit and 1-digit numbers - not crossing 10 
3. Add and subtract 3-digit and 1-digit numbers - crossing 10 
4. Subtract a 1-digit number from a 3-digit number - crossing 10 
5. Add and subtract 3-digit and 2-digit numbers - not crossing 10 
6. Add and subtract 3-digit and 1-digit numbers - crossing 10 
7. Subtract a 2-digit number from a 3-digit number - crossing 10

 As you can see, nothing is left to chance – each step builds carefully from the previous step, building on pupils’ prior knowledge to develop new skills, with nothing left out. Pupils are ready for this having covered addition with 2-digit numbers in Year 2 and Place Value up to 1,000 in the first block of Year 3.

Our curriculum is designed to use skills that have already been learnt in different contexts (sometimes called ‘interleaving’) whenever we can. This helps pupils to remember and to make connections between different parts of the curriculum.

Taking the Year 3 example, after the Addition and Subtraction block, pupils will revisit and practice these skills again in these blocks later in the year:

• Multiplication and Division
• Money
• Length and Perimeter
• Mass and Capacity

…and then they are built on and extended in Year 4 and beyond.

 White Rose Maths includes this revisiting in their example questions, and also in the worksheets that accompany the small steps available via our Premium Resources subscription. The subscription also includes other useful resources to help pupils remember:

• Flashback 4 – a daily starter activity consisting of one question each from a topic covered last lesson, last week, two or three weeks ago and last term or last year
• True of False – a question for each step that can be used whenever the teacher wants to bring that topic back to the front of pupils’ minds.

Our maths curriculum combines the best of both ‘mastery’ and ‘spiral’ approaches in its design. It certainly follows many of the mastery principles – spending longer on topics to help gain deeper understanding, making connections, keeping the class working together on the same topic and a fundamental belief that, through effort, all pupils are capable of understanding, doing and improving at mathematics. But we also recognise that just spending a good chunk of time on a topic doesn’t mean that all pupils will ‘master’ it the first time they see it, and that they need to see it again and again in different contexts and in different years to help them truly develop their understanding on their journey to mastery, so this has been built in the revisiting and reinforcing features of spiral curricula too.

Fluency, reasoning and problem solving are key components of learning mathematics and they are included in all the White Rose small steps. We do not simply complete all of the fluency in a block first, then the reasoning and then the problem solving. These three key components should be integrated into classroom practice as much as possible in the order that is appropriate for the step, e.g. the process of division may be introduced by a problem about sharing or grouping for which we need to become fluent at the procedure. 

Mathematics in EYFS 

Our maths curriculum supports the ethos of the EYFS whilst at the same time enabling teachers to create a mathematically rich curriculum. Additionally, it allows for key mathematical concepts to be revisited and developed further across the year.  

The White Rose Maths EYFS scheme of work is divided into ten phases and provides a variety of opportunities to develop the understanding of number, shape, measure and spatial thinking. 

Staff in Reception also use the Mastering Number Programme alongside the EYFS White Rose Curriculum. This programme focuses on the key knowledge and understanding needed in Reception classes, and progression through KS1. Staff were fully trained in this programme during the 2022/23 academic year.

•  White Rose Maths: EYFS Talk and Learn - Phase 1 
• White Rose Maths: EYFS Talk and Learn - Phase 2
• White Rose Maths: EYFS Talk and Learn - Phase 3 
• White Rose Maths: EYFS Talk and Learn - Phase 4
• White Rose Maths: EYFS Talk and Learn - Phase 5 
• White Rose Maths: EYFS Talk and Learn - Phase 6
• White Rose Maths: EYFS Talk and Learn - Phase 7 
• White Rose Maths: EYFS Talk and Learn - Phase 8
• White Rose Maths: EYFS Talk and Learn - Phase 9
• White Rose Maths: EYFS Talk and Learn - Phase 10

Mathematics Pedagogy 

At St. Michael’s, when teaching maths, the whole class moves through topics and lessons at broadly the same pace, starting from the same point. All children are given the same opportunities and the support or intervention given, will vary from lesson to lesson. Each topic is studied in depth and the teacher does not move to the next stage until children demonstrate that they have a secure understanding of the mathematical concepts. 

Children are given time to think deeply about the maths and really understand concepts at a relational level, rather than as a set of rules or procedures. This slower pace leads to greater progress because it ensures that students are secure in their understanding and teachers don’t need to revisit topics once they have been covered in depth.

Though the whole class goes through the same content at roughly the same pace, there is still plenty of opportunity for differentiation. Those pupils who grasp concepts quickly are challenged with rich and sophisticated problems within the topic. Children who have grasped the fluency elements (pitched at the age related expectations) are then given the opportunity to challenge themselves by completing reasoning and problem solving challenges, some of which will begin their independent activities during the lesson at this point. Those children who are not sufficiently fluent are provided with additional support to consolidate their understanding before moving on. This element is in the form of same day intervention, both in class during the lesson or as a follow up intervention if needed.

Concrete, pictorial, abstract (CPA) is a highly effective approach to teaching that develops a deep and sustainable understanding of maths in pupils. Often referred to as the concrete, representational, abstract framework, CPA was developed by American psychologist Jerome Bruner. It is an essential technique within the Singapore method of teaching maths for mastery.

The basics of the CPA Approach:

• An essential technique of maths mastery that builds on a child’s existing understanding.
• A highly effective framework for progressing pupils to abstract concepts like fractions.
• Involves concrete materials and pictorial/representational diagrams.
• Based on research by psychologist Jerome Bruner.
• Along with bar modelling and number bonds, it is an essential maths mastery strategy.

Concrete

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.

Pictorial

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

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.