Building confident and capable mathematicians

Mathematics helps children understand patterns, relationships, quantity, space and change. It gives them the knowledge and tools to make sense of everyday life and provides foundations for learning across science, technology and many other areas of the curriculum.

At Waterloo, we believe that every child can learn and succeed in mathematics. Mathematical attainment is not a fixed ability, and no child is simply “not a maths person”. Children become successful mathematicians through carefully sequenced teaching, regular practice, meaningful discussion and opportunities to apply what they know.

Our mathematics curriculum is designed to ensure that children develop:

• secure number sense;

• fluent recall of important facts;

• efficient mental and written calculation;

• a deep understanding of mathematical concepts;

• precise mathematical language;

• the ability to represent ideas in different ways;

• confidence in explaining and justifying their thinking;

• resilience when an answer is not immediately apparent;

• the ability to reason and solve problems independently.

Secure basic skills are essential, but they are not the final destination. They reduce the demands placed on children’s working memory and give them the capacity to notice relationships, make connections, reason and solve increasingly complex problems.

Our curriculum approach

In Nursery, mathematics is taught through the United Learning Early Years Curriculum. This establishes the knowledge, language and experiences children need to access Mathematics Mastery successfully in Reception.

From Reception to Year 6, we use Ark Mathematics Mastery as the toolkit through which our mathematics curriculum is planned and taught.

Mathematics Mastery provides:

• a cumulative curriculum from Reception to Year 6;

• a clear sequence of mathematical knowledge;

• systematic progression in calculation;

• time to explore important concepts in depth;

• regular rehearsal of core facts and methods;

• consistent mathematical models and representations;

• structured opportunities for mathematical talk;

• varied reasoning and problem-solving;

• diagnostic assessment and responsive teaching.

The materials support teaching but do not replace teachers’ professional judgement. Teachers remain responsible for understanding their pupils, assessing what they know and adapting teaching so that all children can participate and succeed.

The approach reflects Ark’s belief in high expectations for every child, conceptual understanding, problem-solving and depth before breadth. Ark Mathematics Mastery

What does mastery mean?

Mastery does not mean that every child works at exactly the same speed or completes the same task without support.

A child has mastered a mathematical concept when they can:

• recall the relevant facts and knowledge;

• represent the concept in different ways;

• explain it using appropriate mathematical language;

• recognise it when presented in an unfamiliar form;

• select an appropriate method or strategy;

• make connections with other mathematical ideas;

• apply it when reasoning and solving problems;

• retain and use it over time.

Children who understand a concept securely do more than repeat a procedure. They understand why the procedure works, when it is useful and how it connects to other learning.

Our aim is therefore not to move children rapidly through content. It is to build secure, connected understanding that can be used independently.

Mathematics from Nursery to Year 6

Phase	What children learn	How children learn
Nursery	Early number sense; counting; cardinality; comparison; subitising; pattern; shape; position; measure; time and mathematical vocabulary.	Through the United Learning EYFS Curriculum, explicit adult interaction, practical exploration, stories, songs, routines, games and mathematically purposeful continuous provision.
Reception	Comparison and matching; pattern; numbers within and beyond 10; counting; subitising; number bonds; addition and subtraction; grouping, sharing, doubling and halving; shape, space, pattern and measure.	Through Ark Mathematics Mastery lessons, Maths Meetings, practical exploration, talk, stories, games and mathematical opportunities within continuous and enhanced provision.
Years 1 and 2	Secure place value; addition and subtraction; number bonds; multiplication and division; fractions; shape; position and direction; measurement, time, money and early statistics.	Through explicit Mathematics Mastery teaching, concrete and pictorial representations, structured talk, deliberate practice, Maths Meetings, reasoning and problem-solving.
Years 3 and 4	Increasingly large numbers; mental and written calculation; multiplication tables; multiplication and division; fractions and decimals; measures; geometry, position and statistics.	Through cumulative units that connect new learning to secure foundations, with regular retrieval, mathematical discussion, variation, reasoning and application.
Years 5 and 6	Large numbers and decimals; increasingly efficient calculation; fractions, decimals and percentages; ratio and proportion; algebra; geometry; measures; statistics and multi-step problems.	Through deeper conceptual study, precise mathematical language, fluent calculation, comparison of strategies, increasingly complex reasoning and non-standard problems.
Specialist provision	Mathematics selected from the developmental stage most appropriate to each child, including early mathematical development, functional mathematics and elements of the Mathematics Mastery curriculum.	Through explicit, heavily adapted teaching using meaningful contexts, practical resources, visual representations, repetition and opportunities to generalise learning.

Mathematics in Nursery

Young children begin developing mathematical ideas long before they encounter written calculations.

In Nursery, we use the United Learning Early Years Curriculum to establish strong foundations in number, pattern, shape, space and measure. The curriculum is carefully sequenced but is brought to life through children’s play, interests, daily routines and the themes being explored within the setting.

Children develop early understanding of:

• number names and the stable order of counting;

• one-to-one correspondence;

• cardinality—understanding that the final number counted tells us how many there are;

• subitising—recognising a small quantity without counting each object;

• matching and comparing quantities;

• more, fewer and the same;

• composing and separating small numbers;

• repeating patterns;

• shape and its properties;

• position, direction and spatial relationships;

• size, length, height, weight and capacity;

• sequence, routine and the language of time.

Children experience mathematics through:

• counting songs, rhymes and stories;

• construction and block play;

• sorting and matching;

• loose parts and natural materials;

• water, sand and malleable materials;

• role-play and everyday routines;

• puzzles and games;

• movement and outdoor learning;

• purposeful adult-led teaching;

• mathematical opportunities within continuous provision.

Learning through play does not mean leaving mathematical development to chance. Adults notice the mathematics within children’s activity and deliberately extend it. They model precise vocabulary, ask carefully chosen questions, demonstrate strategies and help children represent and explain what they notice.

The environment is planned so that children meet important concepts repeatedly and in different contexts. This helps them begin Reception with the language, representations and mathematical habits on which Mathematics Mastery can build.

The United Curriculum is based on entitlement, coherence, mastery, adaptability, representation and education with character, with knowledge carefully sequenced over time. United Learning Curriculum

Mathematics in Reception

Reception provides the bridge between early mathematical exploration and increasingly structured mathematical learning.

Children begin Ark Mathematics Mastery while continuing to learn through purposeful play, exploration and continuous provision.

Teaching develops children’s understanding of:

• matching and comparing sets;

• counting accurately;

• recognising and representing numbers;

• subitising;

• the composition of number;

• one more and one less;

• number bonds;

• zero;

• addition and subtraction;

• grouping and sharing;

• doubling and halving;

• numerical patterns;

• shape and spatial relationships;

• pattern;

• time and sequence;

• length, height, weight and capacity.

Explicit lessons introduce and develop important concepts, while the wider environment gives children opportunities to practise and apply them. A child may encounter comparison during a taught lesson, apply it while building towers and then use the same language when comparing quantities at snack time.

Adults make these connections visible. Children are encouraged to show, say, make, draw and explain their thinking.

Key Stage 1

In Years 1 and 2, children build secure foundations in the number system and the relationships between numbers.

Teaching gives particular attention to:

• counting forwards and backwards;

• place value;

• comparison and ordering;

• number bonds;

• addition and subtraction;

• equal groups;

• multiplication and division;

• fractions;

• time, money and measures;

• properties of shapes;

• position and direction;

• representing and interpreting information.

Children move between concrete objects, pictorial representations and abstract symbols. For example, an addition may first be constructed using counters, shown using a bar model and then recorded as an equation.

Representations are not used as decoration or simply because children are young. They are selected to expose the mathematical structure and help children understand the relationships involved.

By the end of Key Stage 1, we want children to possess a secure and flexible understanding of number. This provides the platform for the increasingly complex calculation, fractions, geometry and problem-solving encountered in Key Stage 2.

Lower Key Stage 2

In Years 3 and 4, children extend their understanding to larger numbers and increasingly efficient methods.

They develop knowledge of:

• place value within larger numbers;

• mental and written addition and subtraction;

• multiplication and division;

• multiplication tables and related division facts;

• fractions and decimals;

• length, mass, capacity, money and time;

• perimeter and area;

• properties of shapes, angles and position;

• statistics.

Important number facts and calculation methods are deliberately practised so that they become fluent. However, fluency is always connected to understanding.

Children compare methods, identify patterns, explain relationships and consider which strategy is most efficient. They learn that successful mathematicians do not simply calculate: they make choices, check whether answers are reasonable and explain why a method works.

By the end of Year 4, children should recall multiplication and related division facts fluently, providing an essential foundation for fractions, ratio, proportion and algebra.

Upper Key Stage 2

In Years 5 and 6, children work with increasingly complex numbers, relationships and problems.

They develop knowledge of:

• place value in large numbers and decimals;

• mental and formal written calculation;

• factors, multiples and prime numbers;

• fractions, decimals and percentages;

• ratio and proportion;

• algebra;

• converting measures;

• perimeter, area and volume;

• angles and properties of shapes;

• coordinates, position and direction;

• statistics;

• multi-step and non-standard problems.

Children are expected to select methods rather than wait to be told which operation to use. They justify choices, evaluate strategies, identify errors and apply knowledge across unfamiliar contexts.

By the end of Year 6, children should possess the fluency, understanding and mathematical resilience needed to access the secondary curriculum successfully.

Basic skills: the foundations of mathematical thinking

Basic skills are explicitly taught, practised and revisited throughout the school.

They include:

• counting and recognising quantity;

• subitising;

• comparing and ordering;

• number bonds;

• place-value knowledge;

• addition and subtraction facts;

• multiplication tables and related division facts;

• equivalence between fractions, decimals and percentages;

• mental calculation strategies;

• accurate and efficient written methods;

• mathematical vocabulary.

These skills develop cumulatively.

Foundation	Early development	Later development
Counting	Saying number names, one-to-one correspondence and cardinality.	Counting from different starting points and in different steps, including through zero and with decimals and fractions.
Number relationships	Comparing quantities and recognising one more or one less.	Understanding place value, magnitude, equivalence and proportional relationships.
Composition	Recognising that a number can be made from smaller parts.	Using number bonds, partitioning, distributive reasoning and algebraic relationships.
Calculation	Combining, separating, grouping and sharing practical objects.	Selecting efficient mental and written methods across whole numbers, fractions and decimals.
Pattern	Copying, continuing and describing repeating patterns.	Identifying sequences, generalising relationships and expressing them algebraically.
Fact fluency	Subitising and recalling small number facts.	Automatic recall of number bonds, multiplication facts and key fraction, decimal and percentage equivalences.
Mathematical language	Using words such as more, fewer, same, under, behind and next.	Explaining relationships, justifying conclusions and constructing precise mathematical arguments.

Fluency does not mean completing large numbers of identical calculations as quickly as possible. It means possessing accurate, efficient and flexible knowledge that can be drawn upon when reasoning and solving problems.

The Dimensions of Depth

Mathematics Mastery is organised around three Dimensions of Depth. Together, these enable children to become confident problem-solvers.

Conceptual understanding

Children understand the mathematical ideas beneath methods and procedures.

They:

• use objects, images, symbols and language to represent a concept;

• connect new learning with what they already know;

• compare different examples and strategies;

• explain why a method works;

• recognise the same structure in a different context.

A pupil who understands multiplication, for example, recognises equal groups, arrays, repeated addition, scaling and the relationship with division.

Language and communication

Children learn to express mathematical thinking clearly and precisely.

Teachers explicitly introduce and model mathematical vocabulary. Children use full sentences, sentence stems and structured partner talk to explain methods and justify conclusions.

Mathematical language develops from informal descriptions towards precise terminology, signs and symbols. Children learn to distinguish between everyday meanings and mathematical meanings of words such as difference, product, face, mean and volume.

Talk allows teachers to hear children’s thinking. It also enables pupils to clarify ideas, rehearse explanations and learn from one another.

Mathematical thinking

Children learn to behave as mathematicians.

They are taught to:

• notice patterns;

• make connections;

• classify and compare;

• work systematically;

• generate examples;

• make and test conjectures;

• generalise;

• prove or disprove statements;

• identify what is the same and what is different;

• select and evaluate strategies.

Mathematical thinking is not reserved for children who finish routine work quickly. It is an entitlement for every child and an integral part of learning mathematics.

Reasoning and problem-solving

Problem-solving sits at the heart of our mathematics curriculum.

Children are not expected to solve complex problems without the necessary knowledge. Teachers first establish relevant concepts, vocabulary, facts and methods. Children then learn to combine and apply these foundations in increasingly unfamiliar situations.

Our progression moves from:

1. experiencing a mathematical relationship;

2. representing and describing it;

3. practising the associated facts or procedure;

4. explaining why it works;

5. comparing it with other relationships or methods;

6. applying it within familiar problems;

7. selecting and applying it within unfamiliar problems;

8. generalising what has been learned.

Children encounter a range of problem types so that they do not associate a particular operation only with one familiar form of question.

Reasoning may involve explaining an answer, finding an error, completing a statement, comparing methods, proving whether something is always true or creating an example that satisfies particular conditions.

Problem-solving is therefore not an occasional Friday activity or an extension for a small number of pupils. It is the purpose towards which fluency, conceptual understanding and mathematical language are developed.

How we teach mathematics

Mathematics teaching across Waterloo is explicit, interactive and responsive.

A Mathematics Mastery lesson may include:

Do Now

A short, accessible task enables children to retrieve previous learning and settle quickly into mathematical thinking.

New Learning

The teacher introduces the main concept using clear explanation, precise modelling and carefully selected representations.

Talk Task or Let’s Explore

Children explore the concept and rehearse mathematical language with a partner. Sentence stems help all pupils participate and express complete mathematical ideas.

Develop Learning

Examples, questions and representations are used to deepen understanding, expose structure and address misconceptions.

Independent Task

Children practise and apply the intended learning. Tasks may include fluency, reasoning and problem-solving, with appropriate support and challenge.

Plenary

The lesson concludes by revisiting the key mathematical idea, checking understanding and identifying important connections.

This is a flexible structure rather than a rigid script. Teachers may return to modelling, discussion or guided practice in response to what assessment reveals.

Mathematical representations

Children use concrete, pictorial and abstract representations throughout the school.

These may include:

• real objects;

• counters and cubes;

• bead strings and rekenreks;

• ten frames;

• part-whole models;

• number tracks and number lines;

• arrays;

• bar models;

• place-value charts;

• fraction models;

• diagrams, tables and graphs;

• equations and mathematical symbols.

Older pupils are not automatically moved away from practical or visual representations. A representation remains valuable whenever it makes a structure visible or supports reasoning.

Children are also taught to move between representations. They explain how an object, picture, diagram and equation represent the same mathematical relationship.

Maths Meetings

Maths Meetings provide regular opportunities to rehearse, revisit and connect important knowledge outside the main sequence of a mathematics lesson.

They may include:

• counting;

• number facts;

• multiplication tables;

• time and calendars;

• money;

• measures;

• shape;

• mathematical vocabulary;

• previously taught concepts.

This frequent, spaced practice helps important knowledge move into long-term memory. It also ensures that learning requiring regular rehearsal remains secure when the main lesson has moved to another area of mathematics.

Maths Meetings are purposeful teaching sessions, not simply rapid-fire recall. They may include discussion, representations, songs, movement, games and explanation.

Adaptation and inclusion

All pupils should encounter an ambitious mathematics curriculum. We seek to provide one curriculum with appropriate pathways into it, rather than separate, permanently lowered curricula for different groups.

Teachers identify the prerequisite knowledge children need and adapt teaching through:

• pre-teaching and revisiting vocabulary;

• additional concrete and pictorial representation;

• smaller instructional steps;

• worked examples;

• guided practice;

• manipulatives;

• sentence stems;

• visual prompts;

• additional processing time;

• reduced recording demands;

• repeated practice;

• carefully selected questions;

• targeted adult support;

• opportunities to revisit earlier concepts.

Children who need support may work with smaller numbers or more accessible representations while exploring the same mathematical structure.

Children who grasp a concept quickly deepen their understanding. They explain, prove, generalise, compare methods and apply learning in unfamiliar contexts rather than simply moving to content from a later year group.

Mathematics in specialist provision

Children in our specialist provisions share the same entitlement to become confident and increasingly independent mathematicians.

Their curriculum is determined by developmental stage, existing understanding, communication profile and individual outcomes—not simply by chronological age.

Depending on their needs, children may access:

• the United Learning EYFS progression;

• Ark Mathematics Mastery from an appropriate developmental stage;

• heavily adapted learning linked to their mainstream year group;

• mathematical learning connected with Cornerstones projects;

• a personalised pathway in early or functional mathematics;

• a combination of these approaches.

Mathematics Mastery materials are used when they provide an appropriate pathway for the child. Staff select the relevant concepts, representations, language and tasks rather than expecting every child to complete unadapted age-related materials.

Teaching may focus on:

• attention and engagement with quantity;

• cause and effect;

• anticipation and sequence;

• object permanence;

• matching and sorting;

• one-to-one correspondence;

• counting and cardinality;

• subitising;

• comparison;

• number composition;

• pattern;

• position and spatial awareness;

• time, money and measure;

• calculation;

• functional problem-solving.

Teaching remains explicit and purposeful. Practical and sensory experiences are used when they help children understand, communicate or generalise a mathematical concept—not as a substitute for clearly identified learning.

Adaptation may include:

• highly familiar and meaningful contexts;

• objects of reference;

• visual schedules and communication systems;

• reduced language with precise mathematical vocabulary;

• additional modelling and repetition;

• smaller learning steps;

• opportunities to practise in different settings;

• alternative methods of responding or recording;

• assistive technology;

• increased time to process and respond.

Links with mainstream classes are planned where they are educationally and socially meaningful. Children may join mainstream mathematical learning, work on related concepts within specialist provision or apply mathematics through shared projects and experiences.

Progress may be shown through increased attention, anticipation, matching, selecting, representing, communicating, calculating, explaining or independently applying learning. Every child is expected to make meaningful progress from their starting point.

Assessment and responsive teaching

Assessment is woven through mathematics teaching.

Teachers use:

• observation;

• questioning;

• mathematical talk;

• children’s representations;

• written and practical work;

• Do Nows;

• diagnostic questions and quizzes;

• independent tasks;

• Maths Meetings;

• periodic assessments.

Assessment identifies the precise knowledge or misconception affecting a child’s understanding. It should tell us more than whether an answer is right or wrong.

Teachers consider:

• Does the child understand the mathematical concept?

• Can they recall the necessary facts?

• Can they represent the idea?

• Can they explain their method?

• Can they select an efficient strategy?

• Can they apply learning in a different context?

• Can they retain and retrieve it later?

• What prerequisite knowledge may be missing?

Where a gap is identified, teachers respond through additional modelling, guided practice, intervention or opportunities to revisit the relevant foundation. Intervention should help children reconnect with the main curriculum rather than permanently remove them from it.

What success looks like

Our mathematics curriculum is successful when children:

• see themselves as capable mathematicians;

• approach mathematics with curiosity and confidence;

• possess secure number sense;

• recall important facts fluently;

• select efficient mental and written methods;

• use mathematical language accurately;

• represent ideas in different ways;

• explain and justify their thinking;

• recognise patterns and make connections;

• persevere when a solution is not immediately apparent;

• identify and learn from errors;

• reason and solve familiar and unfamiliar problems;

• apply mathematics independently across school and everyday life;

• make strong progress from their individual starting points;

• leave Waterloo ready for the mathematical demands of secondary education and later life.

Above all, we want children to understand that mathematics is something they can make sense of. Through secure foundations, careful teaching and worthwhile challenge, every child can learn to think mathematically.