## Why teach mathematics?

*“Mathematics is a creative and highly interconnected discipline that has been developed over centuries, providing the solution to some of history’s most intriguing problems. It is essential to everyday life, critical to science, technology and engineering, and necessary for financial literacy and all forms of employment. A high-quality education in maths therefore provides a foundation for understanding the world, the ability to reason mathematically, an appreciation of the beauty and power of mathematics, and a sense of enjoyment and curiosity about the subject.”*

– National Curriculum in England, 2014

## Mastery

St. Michael’s adopts a Teaching for Mastery approach across the school and is continually engaging with the Matrix Maths Hub to develop and embed this approach.

“Mastering means pupils acquiring a deep, long-term, secure and adaptable understanding of the subject. The phrase ‘teaching for mastery’ describes the elements of classroom practice and school organisation that combine to give pupils the best chances of mastering maths. Achieving mastery means acquiring a solid enough understanding of the maths that’s been taught to enable pupils to move on to more advanced material.” NCETM

Mastery at St. Michael’s looks like:

- We assume that everyone can learn and enjoy maths. We reject the idea that some ‘just can’t do maths’. All pupils are encouraged by the belief that by working hard at maths they can succeed.
- Learning is taught in small steps which build up learning slowly. This ensures all can access new learning and all secure deep understanding.
- Focus is on understanding the concepts behind the maths (conceptual understanding), not just being able to ‘do it’.
- A variety of examples, representations and models are carefully selected by teachers to show the maths in lots of different ways and to emphasise links and connections.
- A CPA approach is used across the school. This means that pupils will first use a strategy or method with concrete/practical resources and manipulatives (such as dienes, counters or cubes), then pictorially (such as diagrams, bar model or pictures of manipulatives), and finally abstractly (in their head or on paper).
- Procedural fluency (how to use a procedure, method or strategy) and conceptual fluency (how the procedure, method or strategy works) are developed in tandem because each supports the development of the other.
- Practice is a vital part of learning and opportunities for whole class, group, paired and independent practice are provided throughout the lesson.
- Teachers lead back and forth interaction (often called ping-pong teaching) which allows children to have a go throughout the lesson. Teachers will provide opportunities to question, discuss, consolidate and practise and will teach through instruction, explanation, illustrating and demonstrating.
- Pupils are taught through whole-class interactive teaching. The focus is on all pupils working together on the same lesson content at the same time to ensure all can master concepts before moving on. For those pupils who need further support to understand concepts, support and scaffolding is provided (e.g. through the use of adult support, the use of manipulatives or pictorial representations, the use of sentence stems or scaffolded calculations or representations).
- Precise mathematical language is used by teachers and pupils which enables all pupils to communicate their reasoning and thinking effectively. Sentence stems or sentence starters are often used to aid pupils in constructing their verbal explanations.
- Significant time is spent developing a deep understanding of the key ideas before moving on the next learning steps.
- Key mathematical facts are learnt to automaticity to avoid cognitive overload in working memory and enable pupils to focus on new learning. In Key Stage One and Key Stage Two, in addition to the daily maths lesson, a daily fluency session is provided to focus on the retrieval and practise of these key facts. In Key Stage One, this session follows the NCETM’s Mastering Number Programme. In Key Stage Two, this session follows the school’s fluency plan.

The Essence of Maths Teaching for Mastery

## Equal Opportunities

One of the key elements to mastery teaching and learning is a mindset that all are able to achieve, and that by working hard all can succeed. Positive attitudes towards mathematics are encouraged so that all pupils, regardless of race, gender, cultural background, ability, SEN or disability, including those for whom English is an additional language, develop an enjoyment and confidence within mathematics. All pupils receive high quality inclusive teaching, and high expectations are set for all. This is enhanced through the use of:

- mixed ability flexible pairings and groupings to encourage collaboration.
- careful planning of tasks and activities based on systematic, accurate assessment of pupils’ prior skills, knowledge and understanding.
- representations, structures and manipulatives for all pupils and abilities.
- timely support and intervention; systematically and effectively checking pupils’ understanding throughout lessons.

*“The expectation is that the majority of pupils will move through the programmes of study at broadly the same pace. However, decisions about when to progress should always be based on the security of pupils’ understanding and their readiness to progress to the next stage. **Pupils who grasp concepts rapidly should be challenged through being offered rich and sophisticated problems before any acceleration through new content.** Those who are not sufficiently fluent with earlier material should consolidate their understanding, including through additional practice, before moving on.” *– National Curriculum in England, 2014

## Fluency, Reasoning and Problem Solving

We follow the National Curriculum aims for mathematics, which are to ensure that 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;
- reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language;
- 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.

Within mathematics lessons, a wide variety of teaching and learning styles are used. We aim to develop pupil’s knowledge, skills and understanding, as well as ensure problem solving and investigative skills are at the heart of mathematics teaching and learning. Fluency, reasoning and problem solving are embedded throughout mathematical teaching and learning and opportunities are provided for all learners.

Fluency:

Fluency is developed across the school in order to avoid cognitive over-load. The working memory has limited capacity and knowing core facts to automaticity will free working memory and build confidence. Having a secure knowledge of number facts will also support pupils to think mathematically as they can use them to reason, see structures and patterns, and make connections.

Fluency development is developing pupils’ understanding of mathematical concepts and skills, how they link and how they ‘work’, in order for pupils to be competent and confident in using a variety of mathematical strategies and approaches independently. It is developed through regular discussion about strategies and methods and through a recognition and understanding that there is not one right way of solving a calculation or problem.

Fluency is developed and taught within the main mathematics lesson across KS1 and KS2, and is taught in conjunction with reasoning and problem solving. Separate practice and retrieval time is then provided during daily fluency sessions within KS1 and KS2. These are approximately 15 minutes in length and do not teach any new concepts or skills. Instead, they embed concepts and skills and provide opportunities for clarification, practice, application and revisiting. Within KS1, these sessions follow the NCETMs Mastering Number Programme and within KS2, progression of fluency development is outlined in the school’s fluency medium term plan.

Reasoning:

Reasoning development is developing pupil’s language and ability to discuss and explain their strategies, solutions and approaches to mathematical activities. It is developed through regular opportunities to reason verbally and in writing, the use and modelling of high quality mathematical language, the use of diagrams, manipulatives and calculations to support and structure pupil’s reasoning, and the use of a variety of questions, such as Why? Why not? What if? How?

You may see pupils across the school use APE as a strategy to structure their verbal and written reasoning. This strategy is adapted to suit the age of the pupils, and the detail and complexity of the reasoning increases through the year groups.

A= Answer. What is the answer to the question, problem or discussion prompt?

P= Prove. How do you know that the answer is correct? Explain with pictures, diagrams, calculations, manipulatives, sentences or in another way.

E= Explain. Explain their approach, strategies and thinking.

Problem Solving:

Problem Solving within mathematics refers to activities and experiences where maths is used and applied, where there is more than a simple, straightforward calculation, and where mathematical thinking is required. A variety of skills need to be developed in order for pupils to be successful in solving problems, such as:

- adopting, developing and evaluating approaches and strategies; understanding and using facts and procedures,
- selecting appropriate resources; posing and answering questions,
- representing problems using symbols, words, diagrams or pictures; developing and choosing ways to record,
- organising work, working systematically and checking results,
- using mathematical language and constructing reasoned arguments,
- predicting,
- making connections,
- concluding and explaining,
- identifying patterns,
- forming generalisations in words, pictures or with resources,
- justifying answers, solutions, methods and conclusions and supporting with examples,
- organising work, working systematically and checking results.

Problem solving skills are developed through the use of a range of contexts, activities and problems with different questions and approaches; integration throughout the mathematics curriculum; opportunities for all to encounter mathematical problems; and opportunities to discuss and reason problems.

A four step plan is used as a problem solving strategy across the school. This can be summarised as What? How? Do! Ok?

## Early Years Foundation Stage

All pupils are given ample opportunity to develop their understanding of mathematics through varied activities that allow pupils to use, enjoy, explore, practice and talk confidently about mathematics. Mathematics teaching and learning in Foundation Stage involves providing pupils with opportunities to develop and improve their skills in counting, understanding and using numbers, calculating simple addition and subtraction problems; and describing shape, spaces and measures.

*“Developing a strong grounding in number is essential so that all children develop the necessary building blocks to excel mathematically. Children should be able to count confidently, develop a deep understanding of the numbers to 10, the relationships between them and the patterns within those numbers. By providing frequent and varied opportunities to build and apply this understanding – such as using manipulatives, including small pebbles and tens frames for organising counting – children will develop a secure base of knowledge and vocabulary from which mastery of mathematics is built. In addition, it is important that the curriculum includes rich opportunities for children to develop their spatial reasoning skills across all areas of mathematics including shape, space and measures. It is important that children develop positive attitudes and interests in mathematics, look for patterns and relationships, spot connections, ‘have a go’, talk adults and peers about what they notice and not be afraid to make mistakes.”*

– EYFS Statutory Framework, 2023

Within Nursery, the Master the Curriculum scheme is used which has been chosen as it links closely to the curriculum through the rest of the school (particular links to White Rose Maths).

Within Reception, the NCETM Mastering Number curriculum is followed for 4 days of the week and White Rose Maths curriculum is used on the 5^{th} day to provide opportunities for development of geometry, measurement and data skills and concepts.

## Key Stage One

*“The principal focus of mathematics teaching in Key Stage 1 is to ensure that pupils develop confidence and mental fluency with whole numbers, counting and place value. This should involve working with numerals, words and the four operations, including with practical resources (e.g. concrete objects and measuring tools). At this stage, pupils should develop their ability to recognise, describe, draw, compare and sort different shapes and use the related vocabulary. Teaching should also involve using a range of measures to describe and compare different quantities such as length, mass, capacity/volume, time and money. By the end of Year 2, pupils should know the number bonds to 20 and be precise in using and understanding place value. An emphasis on practice at this early stage will aid fluency. Pupils should read and spell mathematical vocabulary, at a level consistent with their increasing word reading and spelling knowledge at Key Stage 1.”*

*National Curriculum in England, 2014*

Within Years 1 and 2, the main mathematics teaching follows the school’s medium term planning. Additional fluency sessions are provided on a daily basis which follow the NCETM Mastering Number Programme. Continuous provision is provided in both year groups and mathematics activities are regularly planned and provided within this to further enhance learning opportunities.

## Lower Key Stage Two

*“The principal focus of mathematics teaching in lower Key Stage 2 is to ensure that pupils become increasingly fluent with whole numbers and the four operations, including number facts and the concept of place value. This should ensure that pupils develop efficient written and mental methods and perform calculations accurately with increasingly large whole numbers. At this stage, pupils should develop their ability to solve a range of problems, including with simple fractions and decimal place value. Teaching should also ensure that pupils draw with increasing accuracy and develop mathematical reasoning so they can analyse shapes and their properties, and confidently describe the relationships between them. It should ensure that they can use measuring instruments with accuracy and make connections between measure and number. By the end of Year 4, pupils should have memorised their multiplication tables up to and including the 12 multiplication table and show precision and fluency in their work. Pupils should read and spell mathematical vocabulary correctly and confidently, using their growing word reading knowledge and their knowledge of spelling.”*

*National Curriculum in England, 2014*

Within Key Stage Two, the main mathematics teaching follows the school’s medium term planning. Additional fluency sessions are provided on a daily basis which follow the school’s fluency medium term plan.

## Upper Key Stage Two

*“The principal focus of mathematics teaching in upper Key Stage 2 is to ensure that pupils extend their understanding of the number system and place value to include larger integers. This should develop the connections that pupils make between multiplication and division with fractions, decimals, percentages and ratio. At this stage, pupils should develop their ability to solve a wider range of problems, including increasingly complex properties of numbers and arithmetic, and problems demanding efficient written and mental methods of calculation. With this foundation in arithmetic, pupils are introduced to the language of algebra as a means for solving a variety of problems. Teaching in geometry and measures should consolidate and extend knowledge developed in number. Teaching should also ensure that pupils classify shapes with increasingly complex geometric properties and that they learn the vocabulary they need to describe them. By the end of Year 6, pupils should be fluent in written methods for all four operations, including long multiplication and division, and in working with fractions, decimals and percentages. Pupils should read, spell and pronounce mathematical vocabulary correctly.”*

*National Curriculum in England, 2014*

Within Key Stage Two, the main mathematics teaching follows the school’s medium term planning. Additional fluency sessions are provided on a daily basis which follow the school’s fluency medium term plan.

## Calculation Strategies

Pupils are encouraged to develop and use a variety of formal and informal methods for recording their mathematical learning, appropriate to their age. Pupils are encouraged to compare and discuss different methods, and are supported in choosing an appropriate and effective strategy for the activity. Mental mathematics and arithmetic are incorporated throughout all lessons and mental calculation strategies are discussed and developed continuously. A secure foundation in mental calculation, mental strategies and recall of number facts is established, before written strategies and methods are introduced.

In order for pupils to understand and reason why concepts, approaches and strategies are true and effective, pupil’s conceptual understanding is developed alongside their understanding of how strategies and concepts work. Pupils are taught new strategies using CPA approach (concrete, pictorial, abstract). This means that pupils will first use a strategy or method with concrete resources and manipulatives (such as dienes, counters or cubes), then pictorially (such as diagrams, bar model or pictures of manipulatives), and finally in their head or on paper.

- Parent Calculation Booklet- Whole School
- Parent Calculation Booklet- EYFS and KS1
- Parent Calculation Booklet- Lower KS2
- Parent Calculation Booklet- Upper KS2

Representations and Manipulatives

A wide range of resources and manipulatives are used throughout the school to develop understanding. All pupils are encouraged to use these to support their learning, and each class has similar resources to ensure progression, consistency and continuity. These could be physical (manipulative or practical resource), visual (picture), written symbols, verbal (spoken) or contextual (real situations). It is important that all forms of variation are used to promote deep thinking and deeper level of understanding.

**Manipulatives**

- Multilink Cubes or Unifix Cubes
- Place Value Counters
- Numicon
- Dienes or Base Ten
- Bead Strings to 20 or 100
- Tens frame
- Hundred Square or Thousand Square
- Multiplication Square
- Number Line
- Cuisenaire Rods
- Number Mat
- Number Track
- Place Value Chart
- Sorting objects such as dinosaurs, bears, fruit, vehicles.
- Rekenrek
- Counters

**Representations**

Pupils are encouraged to develop their understanding of number and calculation by representing them in different ways. The representations that we use in school are:

**Part – Whole Model or Cherry Diagram:**

These can have as many parts as you like and show ways in which a number can be regrouped. The parts added together make the whole.

**Bar Model:**

This can be one bar, two bars on top of each other (as shown) or more. Bar models can be used in lots of different ways. In this example, the parts add together to make the whole (7 + 7 + 7 + 7 = 28), but it also could show a multiplication (4 x 7 = 28) or division (28 ÷ 4 = 7).

Bar models can also be used to compare fractions, add or subtract fractions or solve fraction problems.

**Tens Frame:**

A 5 x 2 rectangle that is used with counters or dots to represent numbers up to 10. Two frames are then used to represent numbers to 20 which is useful to show how to count up to and beyond 10.

**Array:**

These are used to represent multiplications and divisions. This array represents 6 x 3 or 3 x 6 and 18 ÷ 6 and 18 ÷ 3

## Vocabulary

The use of high quality mathematical language is important in the teaching and learning of mathematics.

Calculation or operation: add, subtract, multiply or divide.

Equivalent: equal or the same.

Number fact: a fact that pupils should be able to recall rapidly.

Representation: a way of drawing, writing or showing a calculation. This could be a number line, with practical resources (dienes, numicon etc.), a diagram or written calculations.

Resource or manipulative: equipment used to support their calculations.

Inverse: calculations that are opposite (addition and subtraction, multiplication and division).

Exchanging or regrouping: when we change one place value to another (e.g. ones to tens, hundreds to tens etc.) This is used when we complete written methods, and is used instead of the terms borrowing and carrying.

Place Value: the value of digits in a number (note that in the new curriculum, we use the terminology ones, rather than units).

Partition or regroup: a number is split into its place values (256 = 200 + 50 + 6).

Regroup: represent the number in a different way. There are many ways in which the pupils could do this which they will explore as they move through the school. e.g. 46 could be 40 + 6 20 + 20 + 6 10 + 10 + 10 + 10 + 5 + 1 etc.

Addition: add, more, and, sum, total, altogether, double, one more, ten more, how many more to make…?, how many more is … than…?, plus, near double, how much more is…?, addition, increase, plus.

Subtraction: take away, less, how many are left?, how many have gone?, one less, ten less, how many fewer is… than…?, difference, how much more is…?, subtract, minus, how much less is…?, subtraction, decrease.

Multiplication: double, near double, lots of, groups of, times, multiply, multiplied by, multiple of, once, twice, times as, repeated addition, array, row, column, equal groups of, multiplication, product, square, square root.

Division: halve, half, share, share equally, one each, equal groups of, divide, divided by, divided into, left, left over, division, remainder, factor, quotient, divisible by… factorise, prime number, prime factor.

## Mathematics at Home

For Key Stage One and Lower Key Stage Two pupils, we highly recommend the White Rose Maths 1 minute app. This covers lots of different maths topics and the idea is for pupils to have a go for 1 minute everyday to build fluency of key mathematical facts.

https://whiteroseeducation.com/1-minute-maths#download

Useful websites for children to access at home:

- MathsFrame Games
- Topmarks Maths Games
- Math Playground Maths Games
- Family Maths Toolkit
- Oxford Owl Maths at Home
- Mathematics Shed
- Times Tables Games
- Maths Games
- EYFS and KS1- NumberBlocks You Tube Channel
- Useful documents for ways in which you can help your child at home:
- Helping your Child with Maths
- Dice and Card Games to Practice Maths Facts
- Useful websites for reference when you or a pupil are unsure of a concept or procedure:
- KS1 BBC Bitesize Maths
- KS1 Oak Academy Maths
- KS2 BBC Bitesize Maths
- KS2 Oak Academy Maths

There are a variety of number facts and concepts that pupils need to learn. The aim is for pupils to be able to recall these facts quickly and accurately, and have a secure understanding of these concepts and terms. Regular and continuous practice of these at home will be invaluable.

**EYFS**

See documents under the Early Years Foundation Stage section above for ideas.

** Year 1**

- Addition facts to 10 and subtraction facts from 10.
- Addition facts to 20 and subtraction facts from 20.
- Practice of simple addition, subtraction, multiplication (as groups of) and division (as sharing). Practical practice of these is ideal.
- Counting in steps of 2, 5 and 10 from 0.
- Find one more and one less for numbers up to 100.
- Counts to and across 100, forwards and backwards from any number.
- Regrouping one-digit in different ways (two-digit numbers if confident)

**Year 2**

- Addition facts to 20 and subtraction facts from 20.
- Addition facts to 100 and subtraction facts to 100 (multiples of ten).
- Multiplication and division facts for the 2, 5 and 10 times tables (and 3, 4, 6 and 8 if confident)
- Odd and even numbers.
- Counting in steps of 2, 3 and 5 forwards and backwards, from and to 0.
- Count in steps of ten from any number.
- Regrouping one-digit and two-digit numbers in different ways.

**Year 3**

- Doubling and halving one-digit and two-digit numbers.
- Regrouping two-digit and three-digit numbers in different ways.
- Multiplication and division facts for the 2, 5, 10, 3, 4 and 8 times tables (and the other times tables if confident)
- Counts in steps of 4, 8, 50 and 100 forwards and backwards, from and to 0.
- Finds 1,10 or 100 more or less than one, two and three-digit numbers.

**Year 4**

- Multiplication and division facts for all the times tables up to 12 x 12.
- Multiples, factors, factor pairs.
- Understanding what a decimal number is.
- Count in steps of 6, 7, 9, 25 and 1000 forwards and backwards, to and from 0.
- Count beyond zero to include negative numbers.
- Count in intervals of 10, 100 or 1000 from one, two, three and four-digit numbers.
- Regrouping three-digit and four-digit numbers in different ways.

**Year 5**

- Multiples, factors, factor pairs, common factors, prime numbers, prime factors, composite numbers, square numbers, cube numbers.
- Multiply and divide whole numbers and decimal numbers by 10, 100 and 1000.
- Count forwards and backwards in powers of 10 from any number up to 1,000,000.

**Year 6**

Revision and practice of previous years.

Click here for the Year 6 page of our school website for lots of resources and ideas