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Analysis of school census data on the English proficiency of pupils with English as an additional language (EAL).

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In spring 2018, an additional question was included in the school census to collect data on the English proficiency of pupils with English as an additional language (EAL).

Schools were required to assess the position of their EAL pupils against a five-point scale of reading, writing and spoken language proficiency and make a 'best fit' judgement as to the proficiency stage for each pupil.

This release presents an analysis of this data.

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English proficiency: pupils with English as additional language

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Analysis of school census data on the English proficiency of pupils with English as an additional language (EAL).

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Context

Mathematics is a universal language that helps us to understand the world, and it is a core part of the curriculum. As well as teaching about numbers, shapes, statistics and patterns, it provides important tools for work in areas such as physics, architecture, medicine and business. It helps learners to develop logical and methodical thinking, to focus and to solve a wide range of mathematical problems. Success can be 'coordinated' when leaders ensure all the elements of pupils' mathematics education are supporting each pupil's progress in the subject. Success in mathematics leads to many opportunities for further study and employment. Mathematics is a core subject of the national curriculum. It is an important entitlement for all pupils in England's schools: academies, free schools and maintained schools.

At GCSE, with the exception of results in 2020 and 2021, the proportion of pupils attaining a grade 4 or above has steadily increased over time. In 2019, 71.5% of pupils obtained a grade 4 or above, and 20.4% of pupils obtained a grade 7 or higher. These are increases of 9.1 and 3.3 percentage points since 2013 The results of national reference tests, introduced by Ofqual in 2017, also reflect a steady improvement in pupil performance between 2017 and 2020. Data from 2021 does not follow this trend, which is likely due to the impact of pandemic restrictions. However, national reference test results in 2022 suggest that attainment is improving again, although it has not yet reached the levels recorded in the 2020 test.

Mathematics is increasingly the most popular choice at A level. Within the top 15 subjects chosen by students, mathematics and further mathematics are the subjects that have the highest proportion of A* and A grades awarded. In addition, level 3 core mathematics is an increasingly popular post-16 course taken by pupils on top of 3 A levels, or equivalents.

Although pupils in England, on average, perform better in mathematics than pupils in many other countries, there is a large gap between the lowest and highest achievers, and between disadvantaged and advantaged pupils. Studies and media reports continue to show that there is a shortage of specialist mathematics teachers and that this is a longstanding issue found in other countries.

This report evaluates the common strengths and weaknesses of mathematics in the schools inspected and considers the challenges that mathematics education faces. The evidence was gathered by His Majesty's Inspectors as part of routine inspections. The report follows our mathematics research review published in 2021, which sets out our idea of a high-quality mathematics education. The report is split into findings in primary schools and those in secondary schools, and includes evidence from Reception classes and sixth forms. Within each of these sections, we talk about:

It is important to note that we evaluate schools against the criteria in the school inspection handbooks. Findings from this report will not be used as a 'tick list' by inspectors when they are inspecting schools: we know that there are many different ways that schools can put together and teach a high-quality mathematics curriculum.

During the period of evidence-gathering, schools were facing many challenges because of COVID-19. As other studies have shown, pupils have been affected by remote learning, lockdowns and national restrictions. Therefore, some of the evidence gathered for this report may not represent 'business as usual'. However, by focusing on the curriculum and its implementation over time, we hope that this risk has been reduced.

Key terms used in this report

Knowledge in mathematics

Throughout this report, we use the same terminology for mathematics knowledge as we used in our mathematics research review. These are not necessarily terms that Ofsted would expect pupils or teachers to use:

declarative knowledge: facts, concepts, formulae

procedural knowledge: methods, procedures, algorithms

conditional knowledge: strategies formed from the combinations of facts and methods to reason and problem-solve

Main findings

Primary schools

In the last few years, a resounding, positive shift in mathematics education has taken place in primary schools. Curriculum is now at the heart of leaders' decisions and actions. Generic approaches, such as the expectation that all teaching should always be differentiated, have dissipated. We now see high quality curriculums, collaborative support for teachers and a focus on mathematics teaching. Leaders intend that pupils 'keep up, not catch up'. These approaches set out a better path to proficiency for pupils.

Teachers help pupils to understand new concepts. Networks of support, such as the Maths Hubs, provide regular and highly useful training. This helps teachers to adopt new and improved ways of explaining and modelling concepts. Often, teachers use physical resources and pictorial representations to help pupils see underlying mathematical structures. They also teach and model new vocabulary, regularly check pupils' understanding and swiftly pick up misconceptions.

There are some deficiencies in the quality and quantity of practice that pupils undertake. Even when teachers teach with clarity and precision, it is likely that these deficiencies undermine pupils' ability to remember important knowledge. For older pupils, these deficiencies affect their ability to attain procedural fluency (speed and accuracy).

Pupils' gaps in knowledge tend to be centred around, but not limited to, addition facts in younger year groups. This was for some, but not all pupils. These early gaps in knowledge may not become apparent until a significant amount of time has elapsed. This is because it is possible, in the medium term, for pupils to understand what is being taught and then keep up with extra classroom support and slower calculation. However, this is at the expense of later ability to access the curriculum.

Accountability measures and wide spreads of attainment tend to influence leaders' decision making and resource allocation for Year 6 cohorts. Allocating additional resources to year 6 leaves leaders with fewer resources to invest in pupils' earlier education. Further, a goal of true proficiency is superseded by 'age related expectations' which roughly equates to 50% accuracy in end of key stage tests. As a result, many pupils aren't as prepared for the rigours of secondary education as they could be.

Secondary schools

Notable improvements have taken place in mathematics education in recent years. Widespread weaknesses identified at the time of Ofsted's last mathematics subject report, around deficiencies in curriculum guidance and weaknesses in ongoing professional development for staff, are now much less likely to be evident in schools. However, some weaknesses identified in that report persist and continue to limit pupils' learning of mathematics. The teaching of disparate skills to enable pupils to pass examinations but not equip them for the next stage of education, work and life, and weaknesses in the teaching of mathematical problem solving, remain areas of weakness across many schools. These weaknesses are disproportionately likely to be evident in schools that struggle to recruit and retain specialist mathematics teachers.

Recruiting and retaining high-quality, specialist, maths teachers is a challenge for many schools. Leaders' curriculum decisions are increasingly influenced by the need to cope with these difficulties. Some schools that identify this problem do not take steps to develop the subject knowledge and subject specific pedagogical knowledge of less experienced and non-specialist teachers, usually provide a weaker mathematics education to their pupils.

A high quality of education leads to strong pupil outcomes, but this is not necessarily true in reverse. Strong exam outcomes do not, necessarily, indicate a high-quality mathematics education because, in some schools, pupils are taught a narrowed curriculum that allows them to be successful in exams without securing the mathematical knowledge they need to be successful later. These decisions are made because leaders and teachers are acutely aware of the impact of pupils achieving certain threshold grades in terms of post-16 opportunities, and implications for school accountability.

Curriculum planning around the teaching of mathematical facts and methods is usually strong. Where it is weaker, however, it tends to take less account of what pupils have learned previously and what they will study later. In these schools, pupils often are taught a series of disconnected mathematical methods and 'tricks' that apply only in specific circumstances.

Long term curriculum planning to develop pupils' ability to use the facts and methods they have been taught to solve familiar, and unfamiliar, problems is uncommon. Curriculum decisions about problem solving are often left to individual class teachers. The quality of these decisions is variable. As a result, some pupils, particularly those who find learning mathematics more challenging and those taught by non-specialist teachers, are not effectively taught how to solve problems mathematically.

Maths leaders, and teachers, consistently emphasise the importance of clarity and technical accuracy in written, and spoken, mathematics. Where teachers model this accurately and explicitly, it is more likely to be present in pupils' communication.

In most schools, exercises and activities are used by teachers, but in some schools, pupils are asked to undertake exercises and activities that are not carefully designed, or some pupils are moved on without having had sufficient practice to consolidate new learning.

Pupils who are learning mathematics more slowly than their peers frequently receive a mathematics education that does not meet their needs. They are often rushed through the study of new content, in order to 'complete the course', without securely learning what they are studying. This frequently results in pupils repeating content, in key stage 4 that they have already studied, but not learned, in key stage 3 (and 2). Often the curriculum for these pupils is narrowed with little teaching of how the facts and methods learned can be used to solve problems mathematically. Many of these pupils develop a negative view of mathematics.

Leaders' application of GCSE grade thresholds to internal assessments gives false assurance about pupils' learning. As a result of teachers and leaders accepting this level of achievement in internal assessments, some pupils are progressing through the mathematics curriculum with significant, and growing, gaps of knowledge.

In the vast majority of secondary schools, department meeting time is allocated to improving the quality of provision in mathematics as opposed to undertaking administrative tasks. Many leaders and teachers note that this had been a significant and positive change over the recent years.

Discussion of the findings

The overall picture of mathematics education in England is broadly healthy. This positive picture did not arise through chance but through the commitment of school leaders, teachers and members of the mathematics subject community.

Leaders prioritise creating or adopting a high-quality mathematics curriculum. They give careful and ongoing consideration to the effective teaching of that curriculum. Leaders make good use of support and resources from Maths Hubs, the National Centre for Excellence in the Teaching of Mathematics ( NCTEM) and from commercial providers. As a result, many teachers receive high quality, subject-specific continual professional development ( CPD) and there are flourishing formal and informal networks of teaching professionals.

This is a significant shift compared to when Ofsted's last mathematics subject report, Made to Measure, noted that 'very few schools provided curricular guidance for staff, underpinned by professional development that focused on enhancing subject knowledge and expertise in the teaching of mathematics, to ensure consistent implementation of approaches and policies'.

Given this very positive picture, this report explores the features of this widespread effective practice to support its replication. This report also considers the factors which might explain the continued weaknesses in outcomes for some children.

The phrase 'coordinating mathematical success' describes how effective schools make sure that curriculum plans, teaching approaches, pupil tasks, assessments and mechanisms for evolving these align well. When successful, each individual element is of high quality, and the elements work in harmony, together supporting pupils to learn effectively. It means setting out a path to proficiency in the subject, checking pupils are on that path and helping them to stay on that path.

Many of the features of the conception of quality, as outlined in our research review, are prevalent in the schools visited. For example, curriculum sequencing that includes 'small steps' approaches towards increasing mathematical proficiency, teaching that helps pupils to understand, and carefully curated opportunities to practise. In most schools, teachers routinely assess whether pupils have the necessary prior knowledge to undertake new learning. Primary school teachers quickly identify misconceptions, including through computerised tests.

However, it was when each element of mathematics education was of high quality, and those elements worked together, that pupils learnt most effectively. A centralised approach, with a carefully sequenced curriculum at its core, also supports schools faced with higher teacher turnover and/or teachers with less experience and subject knowledge. Even in schools without these challenges, teachers' shared understanding of curriculum progression and high-quality teaching contributes to high quality mathematics education.

It is now common for teachers, in both primary and secondary schools, to receive regular subject-specific professional development. In primary schools, this support is often provided through Maths Hubs and is put in place by leaders who have a clear focus on developing teachers' subject-specific teaching knowledge. Leaders' strong understanding of high-quality mathematics education tends to be reflected in their monitoring foci. In secondary schools, it is much more common for professional development to take the form of departmental meetings that focus on curriculum design and effective curriculum practice.

In schools with less experienced or non-specialist teachers, there is a need to develop a shared understanding of curriculum progression and features of effective practice. Schools with more experienced subject-specialist staff engage in professional debates about how to teach aspects of the mathematics curriculum most effectively. Schools where mathematics provision is strongest ensure that other adults working with pupils, including teaching assistants and tutors, understand the curriculum and its implementation. Where these other adults do not have this shared understanding, the effectiveness of their support is limited.

There are examples in some schools of less successful practices. At primary level inspectors encountered curriculums that lack specified detail in the Reception Year, that allocate geometry to the summer term only or do not provide for enough learning of conditional knowledge. Sometimes questioning causes pupils to guess rather than recall. In other cases, multiple representations cause confusion rather than clarity.

In some schools, towards the end of primary and secondary phases, the focus of assessment shifts away from identifying pupils' needs and moved towards exam or test preparation. This phenomenon is often coupled with an increase in resources to provide for interventions and, in primary schools, reduced class sizes. The need for significant 'last-minute' intervention in some schools suggests deficiencies in the curriculum, teaching or rehearsal earlier on in pupils' mathematical education.

In most secondary schools visited, decisions about GCSE entry tiers are, in practice, taken at the end of Year 9, based on levels of attainment at that time rather than, more appropriately in Year 11. These early decisions about tiers of entry determine the curriculum pathway for these pupils in key stage 4 and this limits the mathematics that some pupils learn.

In some schools, the choice of mathematical methods taught is left to individual teachers' preferences rather than taken as a broader curriculum decision. This means that the approaches that are taught may only be useful to answer questions of the precise type identified in the curriculum for a particular year or key stage. The opportunity to engineer success over time is lost and pupils in these schools typically develop what can be characterised as 'disconnected pieces of knowledge'.

Pupil practice is sometimes limited in quality and quantity in both primary and secondary schools. This happens when leaders see practice as an activity, rather than focusing on its outcomes - whether pupils have practised until they have learned, to automaticity, the intended mathematical knowledge. There is often no consensus among leaders about benchmarks for optimal quality and quantity of practice that gives assurance that pupils have learned what is intended.

In some secondary schools, lack of revisiting conditional knowledge and, in primary schools, less emphasis on procedural automaticity (which develops through practice) is likely to compound this situation. Leaders are increasingly identifying the need for more practice, but not the reasons for this need: deficiencies that might be occurring in curriculum quality, teaching, or in the quality and quantity of practice in pupils' main lessons, for example.

An ambitious curriculum is one that maximises the mathematics that pupils learn. In some schools, teachers move on before ensuring pupils have learned important knowledge and committed that knowledge to long term memory. In schools where this is common, leaders focus on what pupils study, rather than on what pupils learn. Moving on when pupils are not mathematically ready gives the illusion of progress but creates ever greater gaps that will take more time to address in the future. Some leaders gain false assurance about the effectiveness of their curriculum design and practice through internal assessments that closely align with 'expected performance thresholds' of external assessments. This approach often leads to an acceptance of pupils moving through the mathematics curriculum with significant gaps in their knowledge and leaders failing to make necessary adjustments to their curriculums. In these schools, some pupils would be better served by studying less, but securely learning more.

Curriculum thinking about problem-solving and reasoning differs between primary and secondary schools. Pupils need to learn strategies and the most useful combinations of facts and methods to solve types of problem. Since it is not possible for pupils to encounter every possible problem, a suitable curriculum can identify strategies to solve an identified range of problem types. In some primary schools, pupils self-selected the problems they will solve and in many secondary schools there is a lack of curriculum planning to ensure all pupils encounter a range of problem types and have practice solving these problems. In some cases, pupils have little or no opportunity to use the mathematical knowledge they have to reason mathematically or solve problems. This is especially an issue for pupils who find learning mathematics more challenging.

Recommendations

Curriculum

All schools should make sure that:

curriculums emphasise secure learning of, rather than encountering, mathematical knowledge.

curriculum sequencing prepares pupils for transitions between key stages and phases

Primary schools should make sure that:

they identify and sequence small steps in the Reception Year curriculum

all pupils learn to apply facts and methods to wider problem-solving

geometry knowledge is sequenced throughout, rather than at the end of, each year's curriculum

Secondary schools should make sure that:

the curriculum specifies the mathematical methods that leaders want all pupils to learn, and that these form a coherent, 'forward-facing', base of mathematical knowledge rather than a collection of disconnected algorithms and tricks

key stage 4 curriculums focus on maximising pupils' learning, and that decisions on GCSE tier of entry are based on what pupils know and can do towards the end of key stage 4 rather than what pupils knew and could do at the end of key stage 3, or earlier

the curriculum plans how pupils will learn conditional (problem-solving) knowledge over time, for example by:

clearly identifying the range of problems to which pupils should be able to apply their new declarative (facts) and procedural (methods) knowledge

teaching pupils how their new facts and methods can be used to solve a range of problems

providing all pupils with sufficient practice in solving problems using newly learned facts and methods

ensuring that all pupils have enough opportunities to practise solving problems, after they have first being taught, and that these opportunities require pupils to make decisions about how best to solve these unfamiliar problems

Pedagogy and assessment

All schools should:

make certain that teachers routinely check whether pupils have secure knowledge and understanding of prerequisite mathematics and address any gaps identified, before moving on to the next stage of learning

make sure that teachers regularly connect new learning to what pupils have learned before, including showing pupils how it connects with learning in other subjects

make sure that all pupils practise and consolidate new learning through well-designed exercises and activities, including sequences of problem-solving

check that pupils are developing 'procedural fluency' (speed and accuracy of recall of methods) and address gaps in pupils' procedural knowledge at the earliest possible opportunity

Primary schools should:

consider using routines, keeping noise levels low and making sure that pupils are facing the teacher when they are explaining new content and giving instructions, to help them focus on what is being taught

help younger pupils to learn their addition facts by heart and regularly check their recall of this knowledge

reflect on the extent to which additional afternoon practice is due to deficiencies in the early curriculum and its implementation

aim for pupils to become proficient and ready for Year 7, rather than just meet age related expectations for end of key stage tests

make sure that questioning helps all pupils to recall and make connections, rather than allowing pupils to guess

provide pre-teaching, additional teaching and extra practice for most pupils with special educational needs and/or disabilities ( SEND)

Secondary schools should:

make sure that pupils have sufficient opportunities to practise reasoning, explaining and problem-solving using the facts and methods they have been taught

make sure that assessment information is used to reflect on the effectiveness of the curriculum and the way it is implemented, to guide future improvements

Systems at subject and school level

All schools should:

provide continuing professional development for teaching assistants, and other adults working with pupils, to help them to understand the intended school mathematics curriculum and the way it is put into practice

Primary schools should:

make sure that discussions with leaders about progress specifically address the needs of the lowest attaining younger pupils

aim to prioritise resourcing for younger year groups, to better engineer success from the start of a pupil's mathematics journey

when leaders observe lessons, focus on pupils' thinking and the quality and quantity of practice they undertake

Secondary schools should:

make sure that non-specialist teachers receive the necessary professional development, including subject knowledge and subject specific pedagogical knowledge, to teach mathematics effectively

Other organisations

Other organisations should:

provide sufficient quality and quantity of practice within scheme resources

develop and offer computerised tests of facts and methods that share information about progress and attainment with pupils, teachers and leaders. Benchmarks for attainment should be based on proficiency

extend the remit of Maths Hubs to make sure that most schools benefit from them

offer the Mastering Number programme to all schools as an example of good practice in early mathematics

Those responsible for recruiting teachers should:

make sure that all pupils are taught by teachers with appropriate levels of subject knowledge and subject teaching knowledge

The Department for Education, Ofqual and Awarding bodies should:

explore whether the current design of the mathematics GCSE, including the tiers of entry offered and current typical grade thresholds, contribute to practices in schools that are not in pupils' best interests

Primary

Curriculum intent: identifying what pupils need to know and do

Summary of the research review

The curriculum should identify and sequence, in small steps, declarative, procedural and conditional knowledge, and plan for pupils to learn this in small steps. This will make sure pupils' so that pupils' knowledge builds steadily over time. Linked facts and methods should be sequenced to take advantage of the ways that knowing facts helps pupils to learn methods, and knowing methods helps them to learn facts. Declarative and procedural knowledge can be combined and taught as strategies for problem-solving. A well-sequenced curriculum, and systematic teaching and opportunities for practice help pupils to become proficient in mathematics. This leads to success and motivation in the subject.

1. Whether adopted or 'home-grown', most primary schools' mathematics curriculums covered the national curriculum and were carefully sequenced. Leaders and teachers were aware of the importance of setting out a coherent path of progression for pupils. Often, planning included 'small steps' for each lesson, revisiting of previous learning and scheduled 'buffer zones' to allow teachers to respond to pupils' needs.

2. Occasionally, subject areas such as geometry were moved to later in the academic year. This was so that pupils could focus on foundational number and arithmetic at the start of the academic year. A small proportion of pupils were relatively insecure in their knowledge of geometry facts. This was possibly because geometry topics had been allocated to the summer term, leading to long gaps before pupils revisited them. In contrast, some schools had organised their curriculums so that a day a week was devoted to geometry, to make sure that pupils were regularly revisiting this topic.

3. Leaders' reasons for adopting commercial schemes included a desire to improve sequencing and resources. These schemes often included planned sequences of recapping and revisiting previously learned content. Leaders realised that this reduced the planning workload for teachers. This is a positive development.

4. 'Home-grown' curriculum plans were sometimes the result of collaboration across a multi-academy trust. In many schools, the NCTEM 's 'ready-to-progress' criteria provided a useful framework for content. This helped leaders and teachers to see how concepts were interconnected and built over time. Additional resources from commercial schemes, to provide much-needed extra practice for pupils, were common.

5. Leaders wanted staff to follow the curriculum plans closely. This made sure that pupils experienced a smoother path to proficiency, supported with consistent language and routines, rather than a series of disjointed lessons. However, leaders also encouraged teachers to adapt lessons in response to the needs of their classes. The NCETM's 'ready-to-progress' criteria helped many teachers to prioritise key content.

6. In some schools, curriculum sequencing in the Reception Year was less detailed. Occasionally, leaders used early learning goals for curriculum planning. Potentially, this approach is problematic because the early learning goals are not frameworks to be used in this way. In contrast, the most effective Reception Year curriculum planning was as detailed as the planning that teachers of older pupils had access to. In many schools, staff had worked together to make sure that the curriculum prepared children for Year 1.

7. In some schools, leaders made sure that parents and pupils could find out what pupils would learn. One way of doing this was through sharing 'knowledge organisers'. These were used as well as, rather than instead of, curriculum planning.

Declarative knowledge

Summary of the research review relevant to declarative knowledge

The curriculum should identify and sequence key facts, formulae, concepts and vocabulary. This helps pupils to avoid relying on derivation, guesswork or looking for clues.

8. In many schools, staff wanted pupils to learn key mathematics facts by heart. Pupils knew that this knowledge was important, too. The introduction of the times tables check in Year 4 has undoubtedly raised the profile of this type of knowledge. Curriculums generally emphasised mathematics facts, such as times tables, alongside helping pupils to understand the connections within 'families' of numbers. It is likely that these factors are the reason why it was rare to see pupils at key stage 2 having to rely on times tables grids.

9. Pupils in key stage 1 were often expected to develop flexibility and 'deep understanding' when thinking about number. However, there appeared to be less emphasis on learning addition and subtraction tables (number bonds) by heart. This is potentially problematic, as pupils need to be able to recall this type of knowledge quickly in order to access more complex mathematics in key stage 2.

10. In some schools, the curriculum in the Reception Year and key stage 1 emphasised both understanding and quick recall of addition facts. The NCETM's Mastering Number programme was particularly helpful. In these schools, pupils were successful and received lots of praise. They were learning how to subitise (recognise a number of objects without having to count), understand numerical concepts and recall addition facts. Key features of this programme, in addition to a carefully sequenced curriculum, include:

whole-class teaching

use of a rekenrek (a type of counting frame)

low-distraction dice patterns for subitising

clear diagrams and representations

regular questioning

videos for staff training

11. Many schools' curriculums identified and sequenced mathematical vocabulary, sentence stems and speaking frames. It was rare to see pupils relying on a 'working wall' for such vocabulary. Teachers explained and modelled important, age appropriate, mathematical vocabulary, such as 'part - whole', 'lowest common multiples', 'array' and 'integer'. In Reception Year, songs and rhymes to helped with early language learning. Older pupils used technical vocabulary in their discussions and their writing. Occasionally, pupils had gaps in their vocabulary that affected their ability to reason and problem-solve.

12. In most schools, teachers quickly identified rare misconceptions in declarative knowledge. Older pupils were often accurate in their recall of declarative knowledge. However, in younger year groups, some pupils were unable to subitise or easily recall addition facts. Staff noted the impact of the pandemic restrictions on pupils' ability to recall this knowledge. In response, leaders adapted planning to give them additional opportunities to revisit it. Occasionally, pupils' books showed that they inverted numbers. This phenomenon may be linked to a lack of coherent sequencing, or insufficient modelling and practice in the Reception Year.

13. Younger pupils' inability to subitise or easily recall addition facts hampers their progress. They may be able to understand a teachers' instruction in, for example, Year 2 or 3, but they struggle to complete tasks with the speed and accuracy of their peers. They eventually obtain the correct answers ('getting by'), thus demonstrating their understanding, but are less likely to remember this new knowledge. This cycle continues, but with pupils increasingly unable to understand, let alone apply, new knowledge. This is likely to be one of the reasons why interventions are so ubiquitous in Year 6: pupils' internal struggles manifest after a significant amount of time has elapsed.

An example of stronger practice, in which pupils' success with fractions was underpinned by strong knowledge of mathematics facts:

In one school, a joined-up approach to curriculum, teaching and pupil practice had helped older pupils to gain knowledge and confidence in working with fractions. Among many other positive features, the mathematics lead was helping to steer teachers towards consistency in carrying out the school's chosen scheme of learning. Teachers were encouraged to slow the pace of learning, if necessary, to make sure that pupils mastered this knowledge before moving on. Learning key mathematics facts in class, for homework and through competitions between classes was part of a recently revised strategy to help pupils learn mathematics facts to automaticity (the point at which they could use them automatically). The mathematics lead ran a mathematics club for pupils who were less likely to be able to do homework at home. A Year 6 book scrutiny that focused on fractions showed that the sequence of lessons on fractions built in complexity in a logical order. Pupils successfully used different pictorial and concrete representations to help them to understand fractions. They were getting enough practice in calculating fractions. Pupils' highly accurate work with equivalent fractions showed the impact of leaders' work to make sure that pupils learned their mathematics facts to automaticity.

Procedural knowledge

Summary of the research review relevant to procedural knowledge

There is a difference between methods that help pupils to understand concepts and perform mental calculations and methods that are efficient and useful now and in the next stage of learning. The curriculum needs to carefully sequence the teaching of mathematical methods. It should allow for some early methods, such as one-to-one counting, parsing, derivation and complex diagrams, to fade over time ('designed obsolescence'). Pupils should learn the most efficient, systematic and accurate mathematical methods, so that they can use them for more complex calculations and in their next stage of learning

14. Many primary schools' policies for calculation set out how pupils will learn procedural knowledge in a logical way. This made sure that language and models were consistent. Sequences tended to move from expanded and informal methods to compact and formal methods. This approach prioritises pupils developing an understanding of mathematical ideas. For example, using the 'grid method' of multiplication helps pupils to understand place value and the concepts that underpin multiplication. However, this can be at the expense of developing automaticity in using efficient and formal methods. The expectation that pupils should develop significant automaticity in these procedures appeared to be less of a priority. This is problematic because efficient and formal methods, once learned, enable pupils to engage with problems using larger numbers, for example.

15. In some schools, leaders encouraged pupils to learn and then choose from a wide range of methods. The goal was to ensure that pupils had different methods at their disposal to solve a range of problems, and to practise selecting from them. However, such an approach could lead to pupils choosing 'easier' methods and not getting enough practice in using the methods they will need most in the future. This would make it more difficult for them to progress through the curriculum step by step

16. In some schools, younger pupils could use calculation tools quickly and accurately. For example, they used number lines or a rekenrek to help with addition and subtraction. This is likely to be the result of careful teaching of how to use these tools and of pupils' secure knowledge of 'one-to-one correspondence' (the ability to match an object to the corresponding number and recognise that numbers represent a quantity), subitisation and 'addition facts'.

17. Knowing how to set out written work is another form of procedural knowledge. In the best examples, leaders saw presentation as a part of the mathematics curriculum. Teachers carefully modelled and taught this form of knowledge. Textbooks and worksheets also helped to guide and support pupils' presentation, which gave them a sense of pride. Careful presentation is also likely to help pupils spot patterns and identify their own mistakes.

18. When leaders talked to us about gaps in pupils' knowledge, they often mentioned fractions. This was interesting, as this topic was generally sequenced well. Pupils grasped the conceptual knowledge underpinning this topic. Some leaders referred to the difficulties in teaching this topic online. However, working with fractions involves knowing and using procedural knowledge. Pupils can encounter difficulties when teachers have not prioritised procedural automaticity enough.

19. The bar model had been integrated into mathematics curriculums, which was a positive development. In addition to helping pupils understand proportion, for example, it also helped older pupils to solve word problems and algebraic equations.

20. In some schools, some older pupils could not remember methods of multiplication and division. Pupils need procedural fluency to be able to solve a range of problems and to then learn which types of problems a method is useful for. Both of these develop with practice. Lack of procedural fluency is likely to be one of the reasons why pupils eventually need interventions in Year 6. Their lack of procedural fluency may not be apparent until they encounter a sample test paper that requires them to choose which method to use.

Conditional knowledge

Summary of the research review relevant to conditional knowledge

Pupils should be able to recall facts and methods to some level of automaticity before using them for wider problem-solving. The curriculum should reflect this optimal ordering.

'Problem-solving' is not a generic skill, and pupils cannot become problem-solvers by imitating the activities of experts. Pupils need to learn strategies and the most useful combinations of facts and methods to solve types of problem. Since it is not possible for pupils to encounter every possible problem, a suitable curriculum identifies strategies to solve a range of problem types (topic-specific). For younger pupils, these include how to interpret word problems that are more common at their stage of learning. This approach helps pupils to know what to do, without relying on guesswork.

21. Leaders in all schools wanted pupils to be able to reason and solve a wide range of problems. However, curriculum approaches differed. In many schools, curriculum plans specified the models, explanations and sentence stems that would teach pupils strategies for wider problem-solving. Often, teachers dedicated a section of the lesson to this form of knowledge. This was a positive approach. One pupil said to us, 'They do it in a structured way by looking at a problem and then we do a similar one.'

22. In some schools, however, reasoning and problem-solving were an activity or task, and something that pupils could choose. This approach may result in some pupils skipping ahead of vital practice of facts and methods, or sticking with repeated practice of already-secure knowledge. This is problematic because pupils are entitled to learn all types of knowledge.

23. Some leaders had identified lack of fluency in procedures and lack of language and comprehension as barriers to reasoning and wider problem-solving. This shows that leaders were increasingly aware of the forms of knowledge that pupils need, to be able to reason and problem-solve.

24. A lack of conditional knowledge ultimately leaves pupils unable to choose the best method when completing a mixed set of questions, for example during a test. Multiple factors contribute to this. For example, pupils may lack automaticity in using declarative knowledge and procedural fluency. If pupils' declarative and procedural knowledge is secure, it is likely that they do not know the ranges and boundaries of the problems to which the methods apply. This comes from teaching and practising when to use methods, based on identifying underlying mathematical structures. In schools where pupils choose problem-solving challenges, some pupils will discover this form of knowledge, but many will not. This contributes to differences in pupils' attainment and a need for more intervention in Year 6.

Meeting the needs of pupils

Summary of the research review relevant to meeting pupils' needs

A well-sequenced path to proficiency, with the small steps identified, is important for all pupils and crucial for pupils with SEND. This helps pupils to keep up and reduces the need for catch-up support. Many pupils with SEND benefit from explicit, systematic instruction and from practice in using declarative and procedural knowledge. They may also need more time to complete tasks and opportunities to practise, rather than different tasks or curriculums. The value of knowing crucial facts and methods to automaticity is even more important for some pupils, such as those with autism spectrum disorder. This is because it frees up working memory for listening, learning and thinking about new knowledge.

25. The 'keep up, not catch up' approach, often directly referred to by leaders, made sure that pupils really understood and remembered what was being taught before moving on. This is an inclusive approach, particularly for pupils with SEND, provided there is enough support in place to help them keep up. Many pupils with SEND were following the same curriculum, with support and adaptations in class. For example, they received the same teaching as the main class, and then the teaching assistant supported them during their independent practice. However, this support may circumvent, rather than close gaps in knowledge.

26. Some of the more effective examples of additional help included pre-teaching and same-day interventions. This support gave pupils, including those with SEND, vital additional opportunities to review and practise core knowledge. Some SEND coordinators had introduced an additional 'precision teaching' programme for pupils who were working well below age-related expectations.

27. In some schools, leaders tried to maintain the ideal of all pupils moving on together when this was not a successful approach for pupils who were working at significantly below age-related expectations. Pupils in some schools, particularly those with SEND, were less likely to be 'keeping up and catching up' in lower year groups. Further, some pupils were less secure in the basic facts than their peers. Teachers' instructions and explanations of relatively advanced mathematical concepts were beyond their comprehension, even if the teacher explained them well. In these situations, the 'appearance' of inclusivity may be taking the place of real inclusivity. These pupils' needs might be better met if they were to learn different content practised using different tasks. This could be in groups of pupils with a similar level of attainment. This approach would be similar to how schools manage the teaching of early reading for older pupils who have yet to master basic reading skills.

28. Interventions were common. They were more likely to happen in Year 6 and, to a lesser extent, Year 2. In some schools, leaders had reduced class sizes and introduced setting in older year groups. As one school leader reported, the 'gaps just got bigger and bigger'. These year groups are associated with external accountability measures, in the form of end of key stage tests. The focus on these year groups shows that leaders were having to balance the needs of individual pupils with the pressures of accountability. But leaders knew that some pupils needed to spend more time developing fluency and confidence in 'the basics' in earlier years. We observed differences in the quality and quantity of practice and handwriting proficiency in pupils' books. These are likely to be contributing to holding back pupils with and without SEND.

Pedagogy: teaching the curriculum

Summary of the research review relevant to teaching

The novice, whether they are starting school or starting a new topic, needs more instruction rather than less. Teaching should help them on the journey to expertise. Ideally, teaching should be systematic, follow the curriculum sequence and help pupils to understand. A systematic approach works well for pupils of all ages and stages.

29. In many schools, a consistent approach to designing and implementing the curriculum, with an emphasis on content and 'small steps' sequencing, involved a shift of responsibility for the curriculum from the individual teacher to the school's leadership. This approach assured leaders that pupils' progression through the curriculum was joined up and balanced and that teachers were using mathematical language and representations consistently. Additional whiteboard resources, often associated with commercial schemes, included representations that were clear and consistent. This helped pupils to understand underlying mathematical structures better.

30. In the Reception Year, mathematics teaching sessions happened daily and tended to be quite short. This is to be expected, given the shorter attention spans of children in this age group. Many practitioners planned to use fun, interesting books, songs and rhymes to help children learn the language and concepts of early mathematics. This is positive, considering leaders' observations that speech and language delays were increasingly common.

31. Teachers often showed strong subject teaching knowledge in the classroom. They used careful explanations and demonstrations. They were able to break learning down into small steps for pupils. This shows the effect of strong networks of support, such as from the Maths Hubs. Where available, accompanying progression documents were also useful. By setting out what pupils had learned and what they would learn in the future, teachers could understand how each lesson fitted into the bigger picture of mathematics progression. Teachers adapted plans to make sure that pupils were confident before moving on to new learning.

32. Where the curriculum was chosen well and sequenced, and adapted to meet pupils' needs, this was reflected in the activities in pupils' books. These showed how learning was built from small component tasks to more complex mathematical processes.

33. Leaders often viewed teaching and use of mathematical vocabulary as 'non-negotiable' in lessons. Teachers introduced new vocabulary at the start of lessons and used it throughout. They frequently gave pupils opportunities to revisit this form of knowledge. This approach helped pupils to understand and remember more.

34. Teachers often used the concrete-pictorial-abstract approach to teach new ideas and methods. Pupils were often able to use the objects used in demonstrations themselves. This is helpful for pupils, and leaders were keen for this to happen. However, occasionally, there was a tendency for teachers to use too many concrete and pictorial demonstrations. When this happened, pupils ended up confused from multiple representations.

35. In many schools, leaders expected teachers to use frequent questioning in lessons. They understood that a lack of back-and-forth interactions had been one of the limitations of online learning during the pandemic. Questioning tended to be used well. Familiar sets of questions were almost routine: 'What do you notice?' 'What's the same and what's different?' and 'Convince me'. Many teachers used questioning deftly throughout lessons to check whether pupils were ready to learn the material, to check their understanding and to encourage their reasoning. In some schools, teachers targeted questioning at pupils who they knew needed extra practice. This made sure that opportunities to take part in the lesson were distributed fairly. However, in a minority of schools, questioning was less effective. Pupils were expected to second-guess what was going to be taught. More confident pupils would be able to volunteer, while others would stop concentrating. It is likely that, in these situations, pupils who need systematic, explicit instruction were missing out on opportunities to understand and learn new knowledge.

36. In many schools, lessons were efficient because of the use of routines. For example, in one school, pupils recited times tables during the transition time between instruction and going to their tables. Many teachers used routines and hand signals that younger pupils were familiar with from their phonics lessons. Pupils knew what was expected of them and what was going to happen next. This meant they could focus on new learning. The use of routines was associated with pupils being on task.

37. In some schools, pupils did not always listen or take part. This included, but was not limited to, pupils with SEND. Some pupils with challenging behaviour had gaps in their workbooks where there should have been evidence of practice. These pupils were practising much less than their peers. This may lead to a vicious circle where pupils with poor behaviour increasingly do not understand what is happening in the classroom, fall behind and are ultimately labelled as having SEND.

38. In some schools, the classroom layout affected pupils' learning opportunities. In classrooms where pupils faced the teacher, pupils engaged more and were better able to listen and pay attention. This makes sense, as teachers can better gauge pupils' reactions and know whether they need another explanation or worked example. In contrast, pupils found it difficult to concentrate in classrooms where they had been split into multiple groups for teaching and practice, mainly because of the noise.

Pedagogy: pupils' practice

Summary of the research review relevant to pupils' practice

Practice helps pupils to understand and remember mathematical knowledge. There are broadly 2 types of practice. Type 1 involves retrieving and rehearsing facts, methods and strategies to the point of familiarity, speed and accuracy. Type 2 is more exploratory. It requires pupils to explain relationships, prove that they understand them and describe their reasoning. Both types are important. Pupils need quantity and quality of practice to help them understand and commit knowledge to long-term memory. This practice does not always involve textbooks and worksheets. It can include songs, games and rhymes.

Tasks should help pupils to focus on the mathematics to be learned. They should provide for overlearning and, ideally, include variation. Support for learning and understanding should be gradually withdrawn over time. Tasks should give pupils opportunities to be successful, rather than having to rely on guesswork or unstructured trial and error.

39. In most schools, staff and pupils knew that practice, including the need for overlearning, was important. This is very positive. However, practice was not always accompanied by checks to ensure that all pupils were learning the intended declarative, procedural and conditional knowledge, to automaticity, before moving on.

40. Leaders had moved away from an assumption that correct answers showed that work was 'too easy'. When choosing schemes of work, some leaders looked for opportunities for retrieval, including quizzes. Worksheets accompanying schemes of learning were generally well designed. They included worked examples to help pupils understand, and did not contain distracting pictures. This was very positive for pupils.

41. Teachers consistently built in opportunities for pupils to rehearse knowledge. For example, they would start lessons with a 'fluent in 5' approach. Schools were increasingly adding a short, discrete session of extra practice. This usually took place in the afternoon and included mental arithmetic. In one notable example, leaders had built 'fast fractions' into this extra session, and pupils had opportunities to practise finding fractions of amounts and counting in fractions. Some schools used the early morning registration period to provide extra opportunities for pupils to receive focused support or practise content they had been taught. These additional sessions helped pupils. However, the need for them may indicate deficiencies in the design and implementation of the curriculum, including opportunities for practice, in pupils' main lessons.

42. Opportunities for practice sometimes skipped plainer, 'type 1' practice and moved too quickly to wider problem-solving. Teachers provided additional workbooks when they thought pupils needed more practice of the basics. However, if this happens frequently it may indicate that, despite being carefully presented, worksheets and tasks do not always provide enough practice for pupils.

43. Linked to the issue of limited practice on worksheets, there was often no consensus among leaders on the amount of quality and quantity of practice that gives assurance that pupils have learned what was intended. Leaders and teachers often needed a better understanding of what an adequate amount of practice is.

44. Staff in the Reception Year frequently made sure that children had repeated exposure to mathematical language and concepts. They planned mathematics activities carefully. Simple, rather than complex, activities helped children to think about mathematical concepts. However, when they were not working in a group with an adult, children tended to choose opportunities for practice from what was laid out in their classroom. Schools need to have a system for monitoring children's access to mathematics-related tasks from these resources. Otherwise, some children will get more practice than others, and some children will make less progress.

45. Most pupils' workbooks showed some quality and quantity of practice. However, there was some variation in pupils' completion rates and access to wider problem-solving. Older pupils' confidence with recall was associated with high-quality book work. In addition to pupils' relative proficiency, quality and quantity of practice is likely to be linked to how well they focus on the work, and the amount of effort they make. Pupils who were confident working with fractions had also experienced plenty of practice in calculating with fractions and knew important mathematics facts to automaticity. However, where pupils' work in key stage 1 was less thorough, they struggled to recall number bonds, for example.

46. In many schools, songs and rhymes gave pupils low-stakes opportunities (where pupils can make mistakes without penalty) to practise counting, shapes and vocabulary. Times table starters (such as 'rolling your numbers') and songs in lessons helped pupils to remember important knowledge. This is positive, as it helps all pupils and creates a culture in which mathematics is celebrated. Pupils, too, when asked about what helped them remember, talked about the usefulness of songs.

47. Frequent use of choral response (responding in unison) for low-stakes practice of concepts, vocabulary and mathematical sentences was a positive theme. In one example, leaders and teachers informed us that this was similar to their phonics approach. They noted that the approach was more inclusive, helping pupils with SEND and reducing anxiety about mathematics. This 'choral response' is a key feature of the NCETM's Mastering Number programme for younger pupils. A 'my turn your turn' approach to modelling and reciting stem sentences helps pupils to learn new mathematical language and understand important concepts.

Example of stronger practice, incorporating choral response:

In one school, leaders and teachers informed us how the use of choral response was similar to their phonics approach. The approach was more inclusive, helping pupils with SEND and reducing mathematics anxiety:

'Lessons [include] lots and lots of practice, a "ping-pong" approach between teacher and children, partner work, small steps of progress with teachers circulating... There is a lot of whole-class choral response with the teachers... You can hear who has got it slightly wrong without having to draw attention to the child.'

'It does feel "repetitive", but this is necessary.'

48. In many schools, homework consisted of arithmetic practice and access to online platforms to rehearse key mathematics facts. The latter was very popular with pupils. Homework tended to require pupils to practise something they had recently learned. This was because leaders wanted homework to require minimal parental support. In some schools, Year 6 pupils were set questions similar to those in national curriculum tests as homework. This is potentially problematic, as some questions require pupils to know relatively complex strategies and have high levels of understanding. Pupils who are working below age-related expectations become confused or resort to guessing. Leaders were often careful to make sure that test questions were related to content that pupils had already been taught. Some leaders laid on an extra homework club for pupils who were unable to complete homework at home.

49. There has been a cultural shift away from the previously ubiquitous '3 levels of differentiation'. This was where pupils were assigned or chose tasks set at different levels. In practice, this reinforced different declarative, procedural and conditional knowledge. In schools where pupils chose their own level of task, some pupils found this choice motivational, as it was a source of pride to take on the difficult 'challenge'. However, only a subset of pupils would access the reasoning or problem-solving version of whatever was being learned. Further, some pupils would proceed to the 'extra challenge' that teachers had prepared for early finishers, while others might not finish basic tasks. This approach may cater for a wider range of proficiency in the moment, but it does not allow each pupil to reinforce the knowledge they need to learn. This reinforces diverging rates of progress and attainment. In some schools, leaders and teachers overcame this by setting a minimum expectation that all pupils should at least do the basic tasks to consolidate their knowledge of facts and methods. Pupils experienced independent practice that was largely similar, at least at the beginning of sequences of learning. Leaders were aware that doing the same thing can, in some circumstances, be 'OK', and that every pupil needed to be familiar with the basics.

Assessment

Summary of the research review relevant to assessment

Frequent low-stakes testing (that is, without risk of failure), with an element of timing, is useful for checking pupils' knowledge of key facts and methods. This helps pupils to remember and gives leaders an insight into gaps in pupils' knowledge. Assessment at the end of a year or phase should assess pupils on what they have learned and rehearsed, rather than on what they do not know and cannot do.

50. Many teachers used live marking in lessons, as well as whole-class feedback. They swiftly noticed pupils' successes, misconceptions and errors. They could direct pupils to revisit knowledge at the start of the lesson, adjust the next lesson for all or focus on pupils who needed additional support. In some schools, same-day marking fed into short interventions in the afternoon. The speed and responsiveness of this approach minimised workload for teachers, as it was less likely that pupils would carry forward their errors and misconceptions.

51. Most leaders checked children's knowledge on entry to Reception Year. Practitioners were responsive to children's needs, but there was little evidence that they were systematically addressing gaps in the children's mathematical knowledge. It is possible that, at this stage of learning, gaps in learning slip through the net. More positively, in some schools, leaders were assessing pupils' 'addition facts'. This reflected leaders' awareness of the importance of declarative knowledge.

52. Most schools used regular end-of-unit tests aligned with the school's curriculum. Pupils would be tested on what they had learned and practised. This highlighted to leaders which forms of knowledge pupils needed to revisit. Intervention plans and resources were available for teachers to use with some commercial test schemes. Occasionally, assessments did not include geometry, data and work with coordinates. Some leaders had developed their own assessments to test and identify gaps in knowledge. Some used multiple-choice questions to expose misconceptions. Both approaches are positive, because they increase the level of detail of diagnostics. In some cases, benchmarks for proficiency had been set at an expectation of 80% accuracy. This is a more appropriate benchmark than the benchmark for 'meeting age-related expectations' at the end-of-key-stage tests (currently only around 50% accuracy). However, there was less testing for, or understanding of, benchmarks for 'procedural fluency'. This indicated a lack of knowledge in this area.

53. Some leaders had used technology for testing. For example, pupils took end-of-unit quizzes on laptops, then received their overall score and information about which questions were correct. Pupils were keen to take a test again, motivated to do even better. Although these quizzes cannot check pupils' reasoning or wider problem-solving, they provide low-stakes practice and precise information about pupils' component knowledge and readiness to move to the next topic. This kind of system gives pupils and teachers instant and accurate feedback, while almost eliminating teachers' workload of marking and analysis.

54. Pupils liked frequent, low-stakes (and timed) testing because they could achieve personal bests. In some schools, pupils talked about testing being fun, with friendly class-against-class competition, the use of music and even a big voiceover. Given the frequency of low-stakes testing and pupils' positive view, it is unsurprising that no pupils talked about fear of national curriculum tests. These pupils were well prepared, due to high-quality curriculums, teaching and practice. There was no notion of teachers putting pupils under undue pressure or expecting them to compare themselves with others. Pupils knew that speed and accuracy of recall (of component knowledge) were important to aim for. They were proud of the amount of work they could complete, the proportion of answers that were correct and being seen to access the difficult task. This was set against a backdrop of a positive culture where mistakes were 'OK'.

55. Leaders often used summative tests to compare pupils with the national average and to carry out question-level analysis. Where this was due to a focus on national curriculum tests in Year 6, for example, leaders wanted to make sure pupils were going to be successful. However, there was little consensus about whether leaders had identified mechanisms for understanding the cause of pupils' lack of progress. Most schools used end of key stage tests, or similar papers. The most positive approaches kept things 'low key' and avoided repeated summative testing to 'show progress'. However, there are still limitations to this approach. Question-level analysis of summative tests does not accurately pinpoint gaps in pupils' component knowledge. Further, meeting age-related expectations (around 50% accuracy) does not assure leaders that pupils are ready for key stage 3. This is because the subject is hierarchical: new and increasingly abstract concepts integrate, and therefore depend on, mastery of foundational concepts. It is also possible that, because resources are limited, prioritising 'cusp' groups comes at the expense of pupils who need support the most.

Systems at the school level

Summary of the research review relevant to systems

School-level systems strengthen the consistency of a pupil's journey to proficiency. They include monitoring approaches, staff training, resource allocation, teaching and learning expectations, ways of raising the subject's status and ways of sharing information between stakeholders. Professional development should be a planned and purposeful pathway to expertise in teaching and subject leadership.

56. A well-established culture of collaboration has strengthened and extended networks of support. Leaders across different schools and phases often collaborated on curriculum progression, moderation, supporting professional development and sharing good practice. Larger organisational structures, such as the Maths Hubs and academy trusts, helped to 'cross-pollinate' information about high-quality mathematics education. Collaboration and information-sharing extended into staff meetings, leaders' work to support children's transition into Year 1, team planning and team teaching. Teachers appreciated opportunities to observe each other, and gained confidence in teaching outside their 'comfort zone'. This collaboration is positive for pupils' education and, to use a common safeguarding phrase, results in a large and interconnected 'team around the child'.

57. Professional development provided by the Maths Hubs and information produced by the NCETM have informed leaders about high-quality mathematics teaching. This knowledge could reduce their vulnerability to quick-fix approaches or unevidenced 'fads'. This, in addition to choosing or creating high-quality curriculums, appeared to be the key driver of improvement in mathematics education. This is very different from the situation we identified a decade ago, when the 'Mathematics: made to measure' report stated, 'Very few schools provided curriculum guidance for staff, underpinned by professional development that focused on enhancing subject knowledge and expertise in the teaching of mathematics, to ensure consistent implementation of approaches and policies.'

58. These combined aspects were highly positive, signalling a shift away from a culture of high-stakes lesson observations (used, in isolation, for accountability purposes) and planning scrutiny. Senior leaders made sure that subject leads had time to look at books, visit lessons and speak with pupils. Lesson observations were typically frequent, collaborative and low stakes. As teachers at one school told us, 'Nothing is sprung on us'. Leaders used their knowledge of high-quality mathematics education, rather than the teachers' standards, to inform observations. They would cross-check lessons with curriculum plans and check for consistency. Rather than generic approaches, they generally looked for features such as modelling vocabulary, using resources, recapping knowledge and supporting pupils with SEND. Feedback to individual staff and at staff meetings gave small, sequential development points to gradually improve teaching and teachers' understanding of progression.

59. The balance of observations, however, appeared to be weighted more towards what teachers were doing, than pupils' understanding, focus and practice. This may be because leaders relied on a checklist of observable 'features' of teaching. It is possible that a stronger focus on pupils would give leaders more information about pupils' understanding, and the variation in type and amount of pupils' practice.

60. Most primary schools are small, which means that classes are typically of mixed attainment. Occasionally, some year groups appeared to have higher-than-average proportions of pupils with SEND. This presented resourcing and teaching challenges to leaders and teachers. In some schools, leaders allocated experienced and knowledgeable staff to the pupils and cohorts with the greatest needs. In some schools, children with significant SEND would receive their mathematics instruction as a separate group.

61. In Year 6 and, to a lesser extent, Year 2, setting and smaller class sizes were relatively common. This indicates that, despite strong curriculums and teaching, not all pupils are making good progress. It was rare to hear of this approach being used with other year groups. It is likely that allocating more resources to the final year groups leaves little or no capacity to take the most proactive approach possible: to significantly intervene with the younger pupils, closing gaps right from the start.

62. In many schools, pupils spoke highly of the way that their school celebrated and encouraged mathematics. Pupils were immersed in a supportive culture that reduced anxiety and increased their confidence. Hard work, progress and achievement in mathematics were often celebrated in assemblies. Common approaches to encouraging friendly competition and celebrating pupils' successes were set in a context in which teachers helped pupils to understand why they made mistakes and that it was 'OK' to make mistakes. Wherever competition was mentioned, it was framed positively. For example, schools offered pupils opportunities to compete on online platforms or against other schools within a trust, similar to a sports tournament.

63. In many schools, leaders often shared information with parents. This included facilitating workshops, games sessions and interactive lesson observations. Parents would learn about age-related expectations, end-of-key-stage-2 tests, mathematics teaching, and how they could help their children at home. In some schools, leaders invited parents to share the ways mathematics helps them in their jobs. Leaders often signposted parents to useful websites and apps that would help their children.

64. When school leaders were held to account by governors and senior trust leaders, challenge was often based on important, reportable data. This is understandable, given the high-stakes nature of accountability. Governors and trust leaders also visited schools regularly to learn more about pupils' experiences. However, summative assessment data does not necessarily give senior leaders enough assurance about the level of younger pupils' proficiency. For example, it is possible for younger pupils to obtain correct answers, but this might mask a reliance on using fingers to count. Governors and trust leaders could easily identify gaps in foundational learning through simple checks, similar to checks of pupils' phonics knowledge, and then discuss them with senior leaders.

Secondary

Curriculum intent: identifying what pupils need to know and do

Summary from the research review

The curriculum should identify and sequence declarative, procedural and conditional knowledge so that pupils' knowledge builds steadily over time. Linked facts and methods should ideally be sequenced to take advantage of the way that knowing facts helps pupils to learn methods and knowing methods helps them to learn facts. Declarative and procedural knowledge can be combined and taught as strategies for problem-solving. A well-sequenced curriculum, systematic teaching and opportunities for practice help pupils to become proficient in mathematics. This leads to success and motivation in the subject.
Curriculum design
65. In just over half the schools visited, leaders considered their mathematics curriculum as a 5-year programme. Their curriculum did not distinguish between key stage 3 and key stage 4. This approach ensured that what pupils studied in key stage 4 built on what they had learned in key stage 3.

66. In a small number of schools that had distinct key stages 3 and 4 curriculums, the key stage 4 curriculum for many pupils who were expected to sit the foundation tier GCSE papers repeated all, or most, of what they had learned at key stage 3. This was often because leaders treated examination specifications as the curriculum and 'started at page 1 of the specification'. They did not design a key stage 4 curriculum that built on what pupils already knew and could do at the end of key stage 3.

67. A minority of schools had a 2-year key stage 3. This often led to pupils being rushed to complete the content of the key stage 3 national curriculum by the end of Year 8. As a result, some pupils' learning was insecure. This meant that topics had to be retaught in key stage 4.

68. In a very small number of schools, the key stage 3 curriculum was designed around projects or themes. Typically, the mathematics that pupils learned through each project was not clearly defined. Decisions about what mathematics to focus on in each unit were left to individual teachers. At the end of Year 9, pupils in these schools had very varied mathematical knowledge, depending on the combination of teachers they had been taught by over key stage 3. As a result, key stage 4 was mainly spent 'identifying and filling gaps' in pupils' knowledge.

69. In the majority of schools visited, decisions about GCSE tiers of entry were, in practice, taken at the end of Year 9, based on levels of attainment at that time. These early decisions determined the curriculum that pupils would study in key stage 4, rather than what pupils knew and could do at the appropriate stage of Year 11. This limited the mathematics that some pupils learned.

70. One of 2 scenarios was likely for pupils judged to be on the border between foundation and higher-tier classes. If they were placed in a foundation-tier class, they tended to progress through the foundation-tier content at a slower rate than they were capable of, or they would complete their mathematics learning up to a year before their GCSE examinations. If they were placed in a higher-tier class, they were expected to rush through the higher-tier curriculum in the hope that enough content would 'stick' for them to be able to pick up lots of 'method marks' in their GCSE exams. Both of these approaches denied pupils the opportunity to securely learn mathematics that would be useful in their future study and beyond. Mathematics leaders identified that pressure from pupils, parents and senior leaders to 'finish the course' was a significant factor in their decision to design their curriculums in this way.

71. A minority of schools, typically those with historically poor outcomes in mathematics, designed their mathematics curriculum to finish at the end of Year 10. In these schools, pupils spent most of Year 11 in a cycle of completing past GCSE papers, often under examination conditions. Teachers marked the papers and identified 'gaps in knowledge'. Teachers then re-taught the topics in which pupils had performed less well, focusing on question formats likely to be used in examinations. This approach reduces mathematics education to an exercise in preparing pupils to pass an external examination and does not provide pupils with a rich mathematical education. Leaders in these schools rarely considered why pupils had these gaps in knowledge or whether GCSE assessments were the best way of identifying them. They did not consider whether fewer gaps would exist if pupils were not rushed through the curriculum to finish new learning by the end of Year 10.

72. A minority of mathematics leaders designed their key stage 4 curriculum to maximise the amount of mathematics that each pupil would learn securely. They made a clear distinction between curriculum design and examination specifications. Their curriculums focused on ensuring that pupils finished Year 11 with as much secure mathematical knowledge and understanding as possible. Decisions about which tier of examination each pupil would sit were taken as late as possible. In these schools, some pupils sat the foundation-tier papers despite having securely learned some mathematics beyond the foundation specification. Others sat the higher-tier papers despite not having studied the entire higher-tier specification. This approach better prepared pupils for further study and work, as they had a greater breadth of secure mathematical knowledge.

Example of stronger practice:

Leaders in one school had worked with local post-16 colleges to identify the important mathematical knowledge that pupils might need to be successful in a range of post-16 courses. Leaders used this information to inform their curriculum design for 'intermediate' pupils whose learning would extend beyond the foundation-tier specification but not complete the higher-tier specification. This supported pupils to be successful in their post-16 study.

73. Almost all schools visited had curriculums that developed learning sequentially. Leaders had ensured that pupils learned the necessary prerequisite knowledge before beginning to study new mathematical content.

74. All schools had some classes that were split between 2 or more teachers. In some, this was due to timetabling constraints. In others, it was to make sure no classes were taught entirely by non-specialist teachers. Year 7 classes and lower sets in other year groups were disproportionately likely to be taught by more than 1 teacher.

75. Most schools visited had thought carefully about pupils' progression through the curriculum when they were taught mathematics by more than one teacher. In some schools, different units or curriculum strands were allocated to different teachers. In others, the teachers taught sequentially, with one teacher 'picking up where the other left off'.

76. Where teachers taught different units or curriculum strands, learning was most successful when leaders had made sure that pupils were not taught new mathematics that required knowledge that their other teacher had not yet taught. When this was not the case, and when teachers did not identify gaps in pupils' knowledge, pupils struggled to learn new mathematics.

77. In schools where teachers taught sequentially, new learning was less successful when teachers and leaders did not make sure accurate information was shared between teachers after each lesson.

78. A minority of mathematics curriculums explicitly took account of what pupils were learning in other subjects. In these schools, mathematics and other subject leaders had worked together to agree, where appropriate, common approaches to teaching and applying mathematics. For example, in one school, the mathematics leader and science leader had worked together to make sure that the approach to rearranging formulae was consistent across both subjects. They made sure that the examples and exercises used in mathematics drew on formulae that pupils would study in science.

Declarative knowledge (facts) and procedural knowledge (methods)

Summary from the research review

The curriculum should identify and sequence key facts, formulae, concepts and vocabulary. This helps pupils to avoid relying on derivation, guesswork or looking for clues.

There is a difference between methods that help pupils to understand concepts and perform mental calculations and methods that are efficient and useful now and in the next stage of learning.

The curriculum needs to sequence the teaching of mathematical methods carefully. It should allow for some early methods, such as parsing, derivation and complex diagrams, to fade over time (designed obsolescence). Pupils should learn the most efficient, systematic and accurate mathematical methods, so that they can use them for more complex calculations and in their next stage of learning.

79. In the majority of schools visited, leaders explicitly considered the important mathematical vocabulary that pupils were expected to learn at each stage, when designing the curriculum. Making this explicit ensured that teachers developed and checked that pupils were using mathematical vocabulary accurately.

80. In most lessons visited, teachers carefully and consistently modelled the use of correct mathematical terminology and expected pupils to use it when talking or writing mathematically. On the few occasions when teachers did not use vocabulary accurately, for example referring to the 'bottom of a fraction' when talking about the denominator, many pupils also failed to use accurate mathematical terminology when talking about mathematics.

81. Teachers consistently emphasised the importance of pupils presenting work carefully so that others could follow lines of reasoning. In some cases, they presented this as important 'so that you can get method marks in the exam even if you make a mistake'. However, more often, teachers emphasised the fact that mathematics is 'a communication subject'.

82. When new mathematical procedures were being taught, modelling of high-quality mathematical presentation for pupils was common. Pupils were then expected to present their mathematics similarly. This explicit teaching of accurate mathematical presentation increased the likelihood of pupils communicating their mathematical understanding clearly.

83. In schools where teachers modelled inaccurate mathematics (for example 'stringing equals signs': 3/5 of 60 = 60 ÷ 5 = 12 x 3 = 36), similar errors were seen in pupils' work. Leaders ascribed such errors to gaps in teachers' subject knowledge.

84. The majority of schools had identified core mathematical methods and approaches that they wanted pupils to be taught at the various stages of their learning journey. Most focused on selecting methods and approaches that were:

'forward facing', in that future learning would build on them

'backward facing', in that new learning was deepened because it built on what pupils already understood

85. In these schools, the teaching of mathematical methods and techniques was carefully sequenced so that all pupils learned more mathematical knowledge, irrespective of the combination of teachers they were taught by. Leaders in these schools saw the range of mathematical methods to be taught, and the order in which they are taught, as a curriculum decision.

Example of stronger practice:

One school had identified that pupils would initially be taught to expand and simplify pairs of brackets using the 'grid' approach. This had been selected following extensive discussion in departmental meetings. It was chosen because it:

built on pupils' existing knowledge of multiplication of numbers using the grid approach

was extendable to more complex situations, and would, ultimately, support pupils to expand brackets without the need for drawing out grids

86. In a minority of schools, the choice of mathematical methods taught was seen as a teaching decision. The goal was that pupils would be able to answer questions of a specific type. This sometimes led to 'cul-de-sac' approaches being taught, which could be used to solve questions of the precise type identified in the curriculum for this year, but were not extendable as pupils developed mathematically. This risks pupils seeing mathematics as a collection of unconnected algorithms to be memorised and applied in specific situations. In these schools, pupils were often taught a range of methods and approaches, but no links were drawn between them. The range of methods and approaches taught, and the order in which they were taught, were not picked carefully to deepen understanding. Instead, a range of methods were taught in the hope that one or more would 'stick'. In these schools, leaders and teachers were more likely to talk about choosing approaches that enabled pupils to answer examination questions or gain method marks, rather than to develop their understanding.

Example of weaker practice:

In one school, the curriculum aims were identified as 'I can do' statements, for example: 'I can expand and simplify a pair of brackets of the form (ax + b)(cx + d)'. Decisions about the mathematical method(s) to be taught were left to individual teachers. Some teachers taught pupils to complete questions of this type using the 'FOIL' mnemonic ('First. Outside. Inside. Last'). This standalone approach to multiplying a pair of brackets did not build on pupils' existing knowledge of multiplication. It did not support their future learning, for example being able to expand and simplify expressions of the form (ax + by + c)(dx + e).

Example of stronger practice:

Mathematics leaders in a middle school had worked closely with first schools and high schools in their area to ensure that the methods and approaches they used with children were consistent and coherent as pupils made progress in mathematics. In particular, the schools worked together to explore how bar models could be used effectively to teach new concepts throughout pupils' mathematics education. The curriculum made explicit how bar models should be used. This 'joined-up thinking' ensured that pupils' mathematical progress did not slow at points of transition between schools.

87. When the choice of method was left to individual teachers, inappropriate methods were sometimes taught. This happened when teachers were not aware of 'what comes next' (including education beyond the age range of the school) or 'what came before' (including education before the age range of the school). This was because they lacked knowledge of the curriculum beyond the current year group or key stage.

88. A few leaders and teachers identified that high-stakes accountability of internal and external assessments led to them teaching 'tricks', such as 'keep, flip, change (KFC)' for dividing fractions, that taught pupils to answer questions of the specific type likely to come up in assessments but did not prepare them for future learning. These approaches often led to pupils viewing mathematics as a list of unconnected methods to remember and apply.

Conditional knowledge (strategies)

Summary from the research review

Pupils should be able to recall facts and methods to some level of automaticity before using them for wider problem-solving. The curriculum should reflect this optimal sequencing.

'Problem-solving' is not a generic skill, and pupils cannot become problem-solvers by imitating the activities of experts. Pupils need to learn strategies and the most useful combinations of facts and methods to solve types of problem. Since it is not possible for pupils to encounter every possible problem, a suitable curriculum identifies strategies to solve a range of problem types (topic-specific).

89. All leaders said that their curriculums were designed to match the national curriculum aim that '[pupils] 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'.

90. In the majority of schools, leaders said that problem-solving was integrated within their curriculum and pupils were taught to apply new facts and methods to solve problems within each unit of work. In most of these schools, however, teachers were given little guidance on the types of problems pupils needed to be taught to solve in the unit of study. There was no common understanding of how to teach problem-solving effectively.

91. In most schools, decisions about teaching problem-solving were left to individual teachers. This often led to lack of fairness. Some classes benefited from considerable teaching of how to solve structurally similar problems using recently taught facts and methods. Other classes had very little. This lack of teaching was disproportionately an issue in lower sets, and in classes taught by less experienced or non-specialist staff.

92. In some schools, pupils were not explicitly taught how to apply the mathematics they had recently learned to mathematical problems. Their only exposure to solving mathematical problems was through answering the final few questions of a predominantly procedure-focused exercise. Often, many pupils did not reach this stage of the exercise. These pupils, therefore, had very little experience of applying mathematical methods beyond routine and established applications. Pupils in these schools were notably less confident when solving mathematical problems.

93. In a minority of schools, problem-solving was explicitly planned into the curriculum. Teachers understood the importance of demonstrating how to apply mathematical methods to problems and giving pupils multiple opportunities to practise applying these methods to structurally similar problems. In the most successful lessons, teachers clearly 'drew out' the similarities between problems to help pupils identify the mathematical techniques that might be useful for different types of problem.

Example of stronger practice:

One department had clearly identified the range of problems that they wanted pupils to be able to solve at various stages of the curriculum. In lessons, teachers modelled how pupils could use new learning to solve mathematical problems. They drew out the mathematical similarities in a range of problems that, on the surface, looked unlinked. They gave pupils opportunities to practise solving problems of a mathematically similar nature.

Meeting the needs of pupils

Summary from the research review

A well-sequenced path to proficiency, with the small steps identified, is important for all pupils and crucial for pupils with SEND. It helps pupils to keep up, minimising the need for catch-up support. Many pupils with SEND benefit from explicit, systematic instruction and from practice in rehearsal of declarative and procedural knowledge. They may also need more time to complete tasks and opportunities to practise, rather than different tasks or curriculums.

94. In all schools visited, teaching assistants supported classes in which some pupils had SEND. It most cases, they supported individual pupils or small groups of pupils in the classroom. A small number of schools used teaching assistants to support the work of the wider class, potentially following teacher-led whole-class instruction, while the teacher provided additional support to the pupils with SEND.

95. Teaching assistants who were working directly with individual pupils with SEND were most successful in supporting mathematical development when they had a secure knowledge of the mathematics curriculum and of the range of pedagogical approaches being used by the teacher. In these cases, it was common for the teaching assistant to be assigned to work mainly within the mathematics department, effectively as 'a mathematics-specialist TA', or for the teaching assistant to have received mathematics-specific CPD that focused on the curriculum and appropriate choices of teaching method.

96. Teaching assistants who did not have this specialist knowledge were sometimes restricted to supporting pupils in more generic ways. For example, they offered encouragement or repeated the teacher's explanations. In a very small number of cases, the teaching assistant's lack of mathematical knowledge actively hindered pupils' progress. For example, one teaching assistant advised pupils of 'tricks' to follow rather than explaining the mathematical methods the teacher intended.

97. Almost all schools visited structured their classes in sets based on current attainment, at some stage of the pupils' journey through school. Most had school timetables that allowed pupils to move between sets during the school year. A small minority of schools had explicit systems in place to support pupils who moved from one teaching group to another. For example, they provided interventions to close any gaps in knowledge between the pupil who was moving and their new class. This enabled pupils to be more successful in their new group. Most other schools took ad-hoc approaches to supporting pupils who had moved set. These had varying degrees of success.

Example of stronger practice:

One school employed a subject specialist whose role included providing extra support for pupils who needed it. When a pupil moved teaching group, they received additional small-group or one-to-one teaching from the intervention tutor to make sure that they did not have any gaps in necessary prerequisite knowledge. This made it more likely that the pupil would integrate successfully into the new class.

98. In some schools visited, lower-attaining pupils completed a key stage 4 entry-level qualification. This worked most effectively when leaders designed the mathematics curriculum so that pupils could also be entered for GCSE mathematics, where appropriate. Using the entry-level qualification in this way made sure that the curriculum for these pupils was not narrowed and did not limit the qualification that they could achieve.

99. In a small number of schools, high-attaining pupils were given the opportunity to study for additional mathematics qualifications beyond GCSE, including level 2 additional mathematics and level 3 free-standing mathematics qualification ( FSMQ) additional mathematics. Typically, pupils took the examinations for these qualifications at the end of Year 11, alongside their GCSE mathematics, rather than taking their GCSE mathematics early. This approach benefited pupils, as it allowed the curriculum to be structured in a coherent manner. It made the links between GCSE mathematics and the additional qualification clear, rather than artificially dividing the mathematical knowledge between GCSE and the additional qualification.

Pedagogy: new learning

Summary from the research review

The novice, whether they are starting school or starting a new topic, needs more instruction rather than less. Teaching should help them on the journey to expertise. Ideally, teaching should be systematic, follow the curriculum sequence and help pupils to understand. A systematic approach works well for pupils of all ages and stages.

100. In many schools where mathematics provision had historically not been strong, leaders discussed the challenges of staff recruitment and having less-experienced mathematics teams, often with a number of non-specialist teachers. In these schools, it was common to see teachers using plans and resources developed by others - either from commercial schemes, from multi-academy trust subject teams or developed 'in house' by more experienced staff. Textbooks were used in very few schools. When they were, they were usually associated with the specific commercial mathematics scheme that the school was using. Requiring teachers to use provided lesson plans and resources gave leaders a level of quality assurance of the mathematics provision that pupils would receive from teachers who lacked some subject knowledge or subject specific pedagogical knowledge.

101. In almost all schools, leaders said that teachers were expected to adapt and adjust provided resources to meet the specific needs of pupils in their classes. It was, however, relatively rare to see adaptations take place to resources usually being used without any adjustments. On occasions, this led to pupils studying new content without being secure with necessary pre-requisite learning or pupils moving onto new content before they were ready. Some leaders identified that pre-prepared lesson plans and resources were a 'necessary but insufficient' step on the department's journey of improvement. They had correctly identified that some teachers' subject knowledge and subject teaching knowledge needed to improve to allow them to make informed decisions about appropriate necessary adjustments to meet the needs of pupils in their classes.

102. In schools where mathematics provision had historically been strong, teachers were often given more flexibility to write or source the resources to use with their classes. This was a successful approach in this context. Often, these departments would have physical and electronic stores of shared resources to reduce teachers' workload. The departments typically had settled staffing, and new staff were inducted relatively informally into the department's agreed approaches.

103. In some schools, we saw misconceptions being introduced by teachers. These errors usually occurred in lessons taught by less experienced or non-specialist teachers, who lacked the subject knowledge or subject teaching knowledge to go 'off script' when responding to pupils' questions or when they observed misconceptions.

Example of stronger practice:

The mathematics leader in one school had identified that the use of pre-prepared resources and lesson plans was necessary, in their context, to support the number of non-specialist staff in their team but was not sufficient to ensure high-quality provision. They had identified that 'off-script' provision (when teachers needed to respond to pupil questions, misconceptions or errors) was sometimes poor. Senior leaders had dedicated significant subject meeting time to providing CPD to teachers in the department. This aimed to improve subject knowledge and subject teaching knowledge.

Example of stronger practice:

One small trust had developed a central 'resource hub' of high-quality resources that teachers could use with their classes. These resources, why they were viewed as being high quality and how they aligned with the curriculum were discussed in department meetings. This approach had been introduced because leaders had noticed that the resources being selected by teachers did not always align with the planned curriculum and this was limiting pupils' learning.

The 'resource hub' was constantly evolving as teachers from across the trust developed their understanding of the curriculum and effective resource design, and added new resources that were quality assured by mathematics leaders. Leaders and teachers identified that this approach:

provided teachers with the professional autonomy to select resources that met their pupils' needs

ensured that resources selected aligned with the curriculum

developed teachers' expertise

helped teachers to manage their workload while meeting the needs of pupils in their classes

Pedagogy: consolidation of learning

Summary from the research review

Practice helps pupils to understand and remember mathematical knowledge. There are broadly 2 types of practice. Type 1 involves retrieving and rehearsing facts, methods and strategies to the point of familiarity, speed and accuracy. Type 2 practice is more exploratory. It requires pupils to explain relationships, prove that they understand them and describe their reasoning. Both types are important. Pupils need quantity and quality of practice to help them understand and commit knowledge to long-term memory. This practice does not always involve textbooks and worksheets. It can include songs, games and rhymes.

Tasks should help pupils to focus on the mathematics to be learned. They should provide for overlearning and, ideally, include variation. Support for learning and understanding should be gradually withdrawn over time. Tasks should give pupils opportunities to be successful, rather than having to rely on guesswork or unstructured trial and error.

104. In all schools, pupils were given the opportunity to practise new content in lessons. In most schools, teachers and leaders carefully selected exercises that became more complex over time. This ensured that pupils' knowledge deepened as they worked through the exercises set over a sequence of lessons. In a minority of schools, however, teachers had not planned for exercises to become more complex over time. As a result, pupils remained unable to apply their knowledge to more complex problems.

105. Most schools gave pupils opportunities to practise mathematics after they had first learned it. For example, homework was routinely used to practise prior learning, rather than focusing on current learning. Teachers also provided frequent low-stakes quizzes on prior learning and lessons in which pupils routinely spent time answering questions on topics they had studied previously. These approaches helped pupils to develop accuracy and speed in their mathematics.

106. There was much planned practice in the schools visited. However, in only a few schools had teachers agreed on how much practice was sufficient. Often, the amount of practice was determined by how long was left in the lesson once the teacher had modelled new content. It was not based on judgements about whether pupils had successfully engaged with sufficient, well-designed practice to learn the intended mathematics to automaticity. Some pupils were moved on to new content before they had had a chance to get to grips with what they had learned. This was most typically an issue for pupils who were grasping new concepts more slowly. These pupils' workbooks showed that, in some cases, they would complete just a fraction of the practice of their classmates. They were not benefiting from the carefully developed exercises, and their practice was restricted to a few routine applications that were not practised enough for security. This left many pupils unable to build successfully on these concepts later.

107. In most schools visited, pupils did not practise problems enough that required them to explain, prove, justify or describe relationships. This runs the risk that pupils have a shallower understanding of mathematics, with their knowledge restricted to routine application of learned algorithms.

Assessment

Summary from the research review

Frequent, well-timed, low-stakes testing is useful for checking pupils' knowledge of key facts and methods. This helps pupils to remember and gives leaders an insight into the gaps in pupils' knowledge. Summative assessment should assess what pupils have learned and rehearsed, rather than what they do not know and cannot do.
Assessment for learning
108. In all schools, leaders said that teachers would routinely make sure that pupils had securely learned conceptual knowledge before starting new learning that built on it. However, this assertion was consistently accurate in very few of the schools visited.

109. Teachers used a range of approaches to assess pupils' pre-existing knowledge/understanding. These included quick low-stakes quizzes, with pupils' answers displayed on mini-whiteboards or electronic devices. When used with well-considered questions, these approaches gave teachers appropriate assurance that pupils had the knowledge and understanding necessary to move on to the next stage of the curriculum.

110. However, there were weaknesses in how effectively teachers struck a balance between assumption and assessment when checking pupils' pre-existing knowledge. In a minority of lessons, teachers assumed that the majority of pupils had a firm grasp of pre-existing knowledge based on the answers of only 1 or 2 pupils. In others, teachers knew that pupils had been taught prerequisite knowledge and assumed that they had learned it. In these lessons, teachers moved on to new content while significant numbers of pupils had gaps in their knowledge or a lack of automaticity that would limit their chances of successfully learning new mathematics.

111. The mathematics national curriculum states, 'Decisions about progression should be based on the security of pupils' understanding and their readiness to progress to the next stage. Those who are not sufficiently fluent should consolidate their understanding, including through additional practice, before moving on'. Despite this, in just under half the schools visited, teachers' implicit, and occasionally explicit, focus was on 'covering the curriculum' rather than on securing learning. In these schools, teachers often said that they felt pressure to 'stick to the timings in the scheme of work' for fear of not completing the scheme by the end of the academic year.

112. In the other schools, teachers' focus was more clearly on securing learning. In these schools, teachers moved on to the next conceptual step only when pupils were ready. Some schools achieved this by making sure that there was sufficient slack in the curriculum's suggested timings. Others did so by ensuring that teachers worked through the curriculum at the pace suitable for pupils' needs, and had secure methods for passing information on to the next teacher at the end of each academic year.
Assessment as learning
113. The majority of schools visited provided frequent opportunities for pupils to recall and apply mathematics they had learned previously. These were usually brief activities carried out in most, or sometimes all, mathematics lessons.

114. In most of these schools, teachers monitored pupils' success in these recall activities. In the most successful lessons, when teachers identified a lack of secure knowledge, they quickly decided whether the best course of action was to spend a short time addressing the difficulties as a class or to revisit the topic in more detail in a future lesson. In less successful lessons, teachers sometimes tried to address the difficulties in the same lesson, but without careful planning or appropriate resources. This tended to be more time-consuming and was usually less successful.

Example of weaker practice:

In one lesson visited, pupils were adding fractions with different denominators as part of a 'retrieval practice' activity. A significant number of pupils in the class simply added the numerators and the denominators together. The teacher tried to address this common misunderstanding by re-teaching the concept to the class. The teacher selected 5/12 + 3/8 as the example to use to explain it. The teacher changed both fractions into 96ths, rather than 24ths, while talking about 'lowest common multiple'. The lack of planning led to a poor choice of example to re-explain the method and limited the impact of the intervention.

115. In the majority of schools, teachers had a common understanding about the purpose of these opportunities: to practise the most crucial knowledge and skills that pupils had learned previously, but that was not yet learned to automaticity. When pupils practised retrieval for this purpose, they generally had a high success rate, as they were embedding content they already knew. In some schools, teachers did not have a common understanding of the purpose. Some teachers used 'retrieval practice' for practising exam-style questions or for assessing whether pupils were ready to move on to new learning. This lack of common purpose meant that retrieval practice was less successful in achieving its primary aim: to embed previously learned knowledge more deeply in long-term memory.

Example of stronger practice:

In one school, leaders had identified what important mathematics content needed to be learned to automaticity from each stage of their curriculum. They had collectively developed resources to be used for these brief retrieval activities that took place in lessons, and pupils recorded their answers on mini-whiteboards. Teachers had a common understanding of how to follow up these activities. They expected a high success rate. Any minor errors dealt with 'in the moment'. More often, the teacher noted the errors, which fed into their planning for future lessons.

116. In some schools, pupils were practising mathematics that they had not fully understood at the time of first learning. Pupils were repeating this failure including misconceptions. These misconceptions were becoming embedded.

Example of weaker practice:

One department had a policy that every mathematics lesson began with pupils being asked to complete 'fast 5' questions drawn from prior learning. A significant number of pupils in the class incorrectly found 5% by dividing by 5. Following the department policy, the teacher displayed the correct answers to the 'fast 5' questions on the board, and pupils self-corrected any wrong answers. Pupils' workbooks showed that many of them were regularly unable to recall prior learning. They had made this error repeatedly, and dutifully written the correct answer, during spaced practice activities over a number of months. Further investigation of pupils' workbooks showed that many had consistently made this error when they were first taught how to find percentages without a calculator earlier in the year. At no stage had the teacher been concerned that pupils were regularly unable to recall the intended curriculum content or identified that pupils had this particular misconception and taken steps to address it.

117. These approaches were, usually, limited to the recall of a restricted repertoire of mathematical facts and procedures. Where used consistently well, these approaches were highly effective in supporting pupils to develop automaticity in recalling and accurately applying whatever mathematics needed to be learned.

118. In very few schools were pupils required to practise solving problems using the facts and methods they had previously learned. The exception was when they completed exercises when first applying new mathematical techniques to problems. This lack of practice, except at the point of first learning, prevented pupils from becoming fluent at identifying likely appropriate mathematical approaches to use when faced with unfamiliar problems.

Example of stronger practice:

In one school, leaders had identified that, as well as initially completing exercises with structurally similar problems in class, pupils needed to decide on the appropriate approaches to use when faced with an unfamiliar problem. Leaders had designed their curriculum to ensure, through spaced practice activities, that pupils were solving problems after the point of first teaching. These spaced practice activities required pupils to recall prior learning and choose the most appropriate approach to solve familiar and superficially unfamiliar problems. The problems that pupils had to solve varied from short word problems to longer problems, such as the 'Frogs' investigation. These were aligned with pupils' prior learning.
Assessment of learning - have curriculum goals been achieved?
119. Most schools used termly or half-yearly tests to find out whether pupils had successfully learned the taught curriculum. In most of these schools, the tests were carefully designed to assess what pupils had been taught. In a minority of schools visited, pupils were asked to take tests that included topics that they had not studied and questions that were therefore impossible for them to answer. This was particularly the case when the school was using commercial tests or past GCSE examination papers. This is an inefficient use of pupils' time, which could be better spent learning new mathematics. It could also harm pupils' perceptions of their mathematical capability.

120. Teachers routinely followed summative assessments with activities designed to review pupils' performance in the tests. While summative assessments do not lend themselves to formative purposes, they were most helpful when teachers used them to:

121. Going over the test by modelling correct answers was not so successful. It did not identify the specific gaps in pupils' knowledge or provide pupils with additional teaching or opportunities to practise the areas of mathematics they had struggled with.

122. In the majority of schools, test outcomes were converted into some form of grade or attainment descriptor. The thresholds for expected performance were most often set in line with those for GCSE exams. In many schools, little attention was paid to identifying gaps in pupil knowledge and effectively addressing them. Accepting this low level of pupil success as 'good enough' and moving on results in many pupils moving to the next stage of their learning with significant gaps in their mathematical knowledge.

123. In some schools, information from internal and external summative assessments was used to systematically provide feedback on the effectiveness of the way the curriculum was designed or implemented. In these schools, leaders used the assessment information to inform the ongoing development of the curriculum. As a result, the design and implementation were increasingly effective in supporting pupils to learn.

124. In a small number of schools, assessments served little purpose other than determining a current or forecast grade to report to parents and deciding whether a pupil should change teaching class.

125. Many schools visited use the outcomes of tests to track pupils' progress against GCSE, or similar, targets. Teachers were sometimes expected to make intervention plans for pupils who were 'behind target'. This resulted in some pupils with identical mathematical needs not receiving the support given to others. Allocating intervention resources in this way inappropriately limits pupil access to support based on the mathematics they knew potentially 5 years previously.

Systems at the school level

Summary from the research review

School-level systems strengthen the consistency of a pupil's journey to proficiency. They include monitoring approaches, staff training, resource allocation, teaching and learning expectations, and ways of raising the subject's status, and ways of sharing information between stakeholders. Professional development should be a planned and purposeful pathway to expertise in teaching and subject leadership.

126. Schools visited had taken differing approaches to timetabling mathematics lessons. Averaging over 5 years, the majority of schools ensured that pupils had between 3.5 and 4 hours of mathematics lessons per week. In a small number of schools, pupils had only 3 hours.

127. Schools varied in how they split mathematics curriculum time across the week. In most schools, pupils had mathematics lessons on 3 or 4 days per week. However, in some schools, pupils had 2 much longer lessons per week instead. In a small minority of schools, pupils had a mathematics lesson every day of the week.

128. In schools that varied the number of mathematics lessons per week between year groups, it was usual for pupils in Year 11 to have slightly more mathematics lessons per week than pupils in key stage 3. Often, this additional teaching time in Year 11 was at the expense of non-examined subjects such as personal, social, health and citizenship education ( PSHCE), religious education or physical education. In a small number of schools, Year 11 pupils received an extra lesson of mathematics teaching per week in compulsory 'after-school' lessons.

129. At some stage, all schools visited arranged teaching groups based on pupils' current levels of attainment. Usually, this was based on current levels of mathematics attainment (sets), although occasionally grouping was based on attainment across a wider range of subjects (streams). Almost all schools arranged the school timetable so that pupils could change teaching group midway through the school year if this was judged to be necessary to support their mathematical development.

130. When pupils were placed into classes based on current attainment, it was often on the basis of key stage 2 national curriculum test scores, or baseline assessments taken as pupils started Year 7. It was rare for leaders to include information from Year 6 teachers when deciding classes. This sometimes led to pupils being placed in a teaching class that did not meet their needs, when their key stage 2 assessment did not accurately reflect their level of mathematical knowledge.

131. Where mathematics groupings were not based on current attainment, this was usually in Year 7. A number of leaders said that they had introduced mixed-attainment teaching in Year 7 as a temporary response to the disrupted education and lack of key stage 2 national curriculum test results due to COVID-19. Many of these schools were intending to return to 'sets' once they were confident that they had sufficient and accurate assessment information to do so.

132. A small number of schools timetabled an additional mathematics lesson per fortnight for Year 7 pupils who had arrived with significant gaps in their mathematical knowledge. These pupils were typically identified through discussions with primary schools and through their key stage 2 scores. These groups tended to study a curriculum with an assumed earlier starting point rather than identifying and addressing specific gaps in knowledge.

133. Several schools identified that external tutors had had limited success with their pupils. This was because the tutors' work was not always aligned with the school's curriculum thinking. These schools had had more success through employing tutors directly or by spending time making sure that external tutors understood and applied the school's mathematics curriculum.

134. Most schools used departmental meeting time to improve the curriculum and the way it was put into practice. Many subject leaders said that, over the last few years, there had been a significant move from using subject meeting time for administration activities to using it as an opportunity to think about effective teaching.

135. There were significant differences in how often subject teams met. In some schools, subject teams met almost weekly. In other schools, meetings were limited to once or twice a half term. Providing limited opportunities for members of the mathematics department to work together increases the risk of incoherence in implementing the mathematics curriculum.

136. There was a noticeable difference in teachers' CPD between schools with historically stronger provision and those where it had been weaker. In schools that had historically stronger provision, CPD was most often department-led, and often focused on effective teaching of specific parts of the mathematics curriculum. In schools with historically weaker provision, CPD was often at whole-school level on more generic themes, such as 'retrieval'. In these schools, each department was sometimes given time to consider how this CPD could improve mathematics provision in the school.

Appendix

Methodological note

This thematic report draws on findings from 50 research visits to schools in England. These visits were carried out between September 2021 and November 2022.

Deep dives into mathematics took place as part of scheduled school inspections under the education inspection framework. They were carried out by inspectors with relevant expertise in mathematics education who had received training for this work. They carried out a deep dive as part of our methodology for evaluating the quality of education. Inspectors gathered a rich range of data from speaking to senior leaders, subject leaders and teachers, visiting mathematics lessons and speaking to pupils. They also reviewed pupils' work in mathematics.

Various criteria were monitored consistently to identify characteristics for the sample that risked being underrepresented. These criteria were: region, inspection outcome, disadvantage quintile, size of school, and a rural or urban location. We made sure that the sample was broadly representative of the national picture and there was some representation from schools with different characteristics. The deep dive evidence collected was split evenly between primary and secondary schools.

Inspectors gathered qualitative evidence about mathematics education in schools they visited. The range of evidence gathered across these visits enabled us to identify common themes in mathematics education which are likely to be relevant in a wide range of schools.

Inspectors focused on gathering evidence which related to the following areas:

When analysing this evidence, we drew on the conception of quality in mathematics education which we outlined in our mathematics research review. This enabled us to consider how mathematics education in English schools relates to our best evidence-based understanding of how schools can ensure a high-quality mathematics education for all pupils.

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Coordinating mathematical success: the mathematics subject report

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Mathematics is a universal language that helps us to understand the world, and it is a core part of the curriculum. As well as teaching about numbers, shapes, statistics and patterns, it provides important tools for work in areas such as physics, architecture, medicine and business.

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Guidance to provide support for early years practitioners when completing the early years foundation stage (EYFS) progress check at age 2.

Applies to England

Documents

Details

This guidance is for early years practitioners working with children aged between 2 and 3, including:

registered childminders

nursery schools and nursery classes

other early years settings

It will help early years practitioners to:

review a child's development and progress in the 3 prime areas of learning and development in the EYFS framework

identify any areas of concern or additional development needs

work with parents and other professionals to put in place appropriate support and intervention

The guidance informs, supports and offers suggestions. It does not replace the professional judgement of practitioners.

It should be read alongside Development Matters which sets out how children develop and learn.

What to expect in the early years foundation stage: a guide for parents can support practitioners to help parents find out more about their child's learning and development in the EYFS.

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Guidance to provide support for early years practitioners when completing the early years foundation stage (EYFS) progress check at age 2.

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What is an Education Health and Care Plan?

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What is an Education, Health and Care (EHC) plan?

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There are still too many people who are being held back by poor maths both in their careers and in their personal lives.

Prime Minister Rishi Sunak has said that he wants to see all students in England study some form of maths to 18 to ensure that they leave school better equipped for the jobs of the future.

Just half of all 16-19-year-olds study maths. Yet in a world where data is everywhere and statistics underpin every job, our children's jobs will require more analytical skills than ever before.

The Prime Minister set out the plans in a speech on 4 January 2023.

Here's what you need to know.

Why is maths so important?

So many jobs and careers require a good level of maths - particularly quantitative and analytical skills. But at the moment around 8 million adults in the UK have maths skills lower than those expected of a nine-year-old meaning lots of rewarding jobs might not be available to them. Part of the problem is that at the moment only around half of 16-18-year-olds study maths. But it goes beyond careers and employment, so many aspects of everyday life - from managing personal finances to understanding offers in the supermarket - require a decent level of maths.

Many other countries - including Australia, Canada, France, Germany, Finland, Japan, Norway and the USA - ensure students take some form of maths until they're 18 and the Prime Minister has made clear that England will do likewise.

How will 16-18-year-olds study maths under these plans?

Further detail will be set out in due course but we are exploring existing routes, such as the Core Maths qualifications and T-Levels, as well as more innovative options. This does not mean compulsory A level maths for everyone.

What level of maths do pupils study at the moment?

Maths is compulsory on the curriculum for all pupils in England up to age 16. We expect them to leave primary school having mastered the basics before going on to apply those fundamental principles to more complex problems in secondary school. Pupils that don't get a grade 4 or above in their maths GCSE have to resit the exam if they want to carry on studying in a school or college after they're 16 - this is already helping make sure fewer young people leave school without basic maths skills.

Since 2010 we have seen a number of key changes in the way maths is taught in school. These include the introduction of 'mastery teaching' - based on a Singaporean model of teaching that fosters a deeper understanding of the subject - and of Maths Hubs which help teachers around the country share and develop best practice in maths teaching.

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Studying maths to 18 - what you need to know

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There are still too many people who are being held back by poor maths both in their careers and in their personal lives. Prime Minister Rishi Sunak has said that he wants to see all students in England study some form of maths to 18 to ensure that they leave school better equipped for the jobs of the future.

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Mathematic programmes of study for ks 1 and ks2

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Mathematic programmes of study for ks 1 and ks2

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A review of studies of interventions that support and improve parental engagement in the education of children aged 5 to 19 years old.

Documents

Details

Parental engagement has a large and positive effect on children's learning. It is therefore a priority to identify interventions that are effective in supporting parental involvement, particularly those parents who are either not significantly involved in their children's education, or who are not involved at all.

The aim of the review is to highlight findings and conclusions from the evidence reviewed and to identify important themes and messages for practitioners and school leaders.

It is confined to studies of interventions looking to support and improve parental engagement in their children's education and which offer evidence on educational outcomes. Priority is given to studies undertaken in the period 2000 to 2010, although earlier, frequently cited studies are also included.

A broad interpretation of 'parental engagement' is adopted, which includes learning at home, school-home and home-school communication, in-school activities, decision-making (e.g. being a parent governor) and collaborating with the community.

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Review of best practice in parental engagement

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A review of studies of interventions that support and improve parental engagement in the education of children aged 5 to 19 years old.

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On Monday 17 April, Prime Minister Rishi Sunak set out the Government's plans to get young people in England to study maths up to the age of 18.

This will give young people better maths knowledge to make sure they're equipped with the skills they need for the future and to support them in their careers, including those in the creative industries.

Here's why we're expanding maths study for 16 to 18-year-olds and what it means for you.

Will all 16 to 18-year-olds be made to take maths A Level?

No. When we talk about all young people studying maths to 18, this doesn't mean everyone will have to take maths A Level.

Instead, it's about making sure that all young people, whatever path they take after school, have access to high-quality maths education that is suited to their needs. For example, we're collaborating with the Institute for Apprenticeships and Technical Education to work out how maths can be incorporated in a way that works for apprentices and employers.

Will there be a new maths qualification that pupils will have to take instead?

We're currently looking into the options available, including the best way to adapt existing qualifications and programmes to make sure young people get the maths skills they need.

This research will also look at whether there are any clear gaps which would make a brand-new qualification helpful.

We're developing plans to bring maths into a variety of post-16 routes from T Levels to apprenticeships and beyond. The aim is to make maths a core feature of everyone's education until the age of 18.

What level of maths do pupils currently study?

Maths is already a compulsory part of the curriculum for all pupils in England up to the age of 16.

When pupils start secondary school, they are expected to have mastered the basics of the subject, ready to move on to tackling more complex problems as they start to prepare for GCSEs.

To make sure young people have the core maths skills they need to succeed in life, pupils need to have a grade 4 or above in their maths GCSE by age 16, or continue to work towards that as part of their post-16 study.

Why is there such a focus on maths?

Many other countries - including Australia, Canada, France, Germany, Finland, Japan, Norway and the USA - already make sure that students take some form of maths until they're 18.

So many jobs require a good level of maths. As technology progresses, more jobs will require analytical and data skills than ever before.

However, maths is also a crucial skill in countless aspects of daily life, from managing personal finances, to finding the best mortgage deal, to taking measurements when decorating your home.

Currently, eight million adults in the UK have maths skills lower than those expected of a nine-year-old. By ensuring all young people study maths until the age of 18, we can support young people to start adulthood with the skills necessary to thrive in both work and home life.

Does this mean maths is more important than arts and humanities subjects?

Bringing maths into the education of 16 to 18-year-olds should not come at the expense of other subjects.

It's not just careers in science, technology, engineering and maths (STEM) which need good maths abilities. In fact, maths is an important skill across jobs, including the arts and creative industries.

By ensuring all young people study maths, we will strengthen young people's skillset and prepare them for the modern world of work, helping them thrive in a wide range of careers.

What are you doing to put these plans into action?

A new Expert Advisory Group is being organised to provide advice on our plans. We're also conducting research to better understand how other countries make sure that young people are studying maths until they're 18.

On top of this, we're committed to increasing maths teaching capacity and quality. This includes developing a new maths National Professional Qualification (NPQ) to support the professional development of maths teachers in primary schools from February 2024.

At the same time, we're increasing the support available from Maths Hubs across the country. These are partnerships of schools, colleges and other organisations who work together to provide support for maths teaching in their region.

Further details will be set out in due course, based on the findings of the Expert Advisory Group.

Will all teachers be required to take the new maths NPQ?

NPQs are voluntary qualifications to support the professional development of teachers and school leaders. As with all NPQs, the new maths NPQ will be optional.

Teachers work hard to deliver excellent maths teaching already - this NPQ is being designed to support them further.

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Maths to 18: Is maths A Level being made compulsory for 16 to 18-year-olds?

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On Monday 17 April, Prime Minister Rishi Sunak set out the Government's plans to get young people in England to study maths up to the age of 18. This will give young people better maths knowledge to make sure they're ...

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post 16 - Expectations on maths and english

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post 16 - Expectations on maths and english

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16 to 19 funding: maths and English condition of funding

Information for institutions on maths and English condition of funding.

Context

Achieving a level 2 qualification, and in particular a GCSE grade 9 to 4 or A* to C, in both maths and English helps students to progress to further study, training and skilled employment. The maths and English condition of funding ensures that all 16- to 19-year-olds have the best chance of achieving this standard, and get the necessary support to do so.

Condition of funding interactive tool

We recommend that you use our condition of funding interactive tool with this guidance. This interactive tool presents the maths and English condition of funding in a step-by-step, user-friendly way.

Summary

Students must study maths and/or English as part of their programme in each academic year. This applies to students:

aged 16 to 18 and 19 to 25 with an education, health and care (EHC) plan who do not hold a GCSE grade 9 to 4, A* to C (a standard pass grade) or equivalent qualification in these subjects

doing a programme of 150 hours or more, which started on or after 1 August 2014

This requirement is a condition of funding for all students on 16 to 19 study programmes, or who start on T Levels from academic year 2022 to 2023, and is part of the funding regulations. It also forms part of institutions' contracts with ESFA. We will remove funding from future allocations for students who do not meet the condition of funding. The other elements of the funding formula are not affected.

The qualifications that meet the condition of funding are GCSEs, functional skills level 2 and other 'stepping stone' qualifications.

Students with a grade 2 or below in maths or English can study towards a pass in functional skills level 2 (legacy or reformed) or they can still study towards a GCSE grade 9 to 4. Once they have achieved this, there is no requirement to undertake further maths or English qualifications to meet the condition of funding. A student does not need to continue studying these qualifications with their current institution, if these students have already achieved either with a previous institution.

This allows institutions and students with prior attainment of GCSE grade 2 and below to choose which level 2 qualification is most appropriate.

Part-time students with a grade 3 can study any qualification approved to meet the condition of funding as a stepping stone towards achieving a GCSE grade 9 to 4.

Students with a pass grade in maths or English functional skills level 2 (legacy or reformed) should be recorded in the individualised learner record (ILR) and school census as follows:

ILR

For English, record FAM Type ECF and code 5 (learner holds a pass grade for functional skills level 2 in English)

OR

For maths, record FAM Type MCF and code 5 (learner holds a pass grade for functional skills level 2 in mathematics)

School census

For English, record code F in the English GCSE funding exemption field

OR

For maths, record code F in the Maths GCSE funding exemption field

Progress measures

We include a headline progress measure in performance tables to recognise the progress that students make in maths and English. This holds schools and colleges to account for the progress of students in maths and English. We align the methodology of this progress measure as closely as possible with the condition of funding rules. We publish any revisions to the measure in the Technical Guide: 16 to 19 accountability headline measures.

Qualifications approved for teaching to 16 to 19-year-olds who meet the condition of funding

To meet the condition of funding, a student has to study an approved qualification. These are GCSEs, functional skills level 2 and certain other qualifications, known as 'stepping stone' qualifications. Full-time students who have a GCSE grade 3 or grade D must study a maths and/or English GCSE.

Stepping stone qualifications include:

functional skills at level 1 and below

English for speakers of other languages (ESOL) and

other qualifications nominated by institutions and approved by the Department for Education

Qualifications that meet the condition have a validity record in LARS. Institutions must check the status of qualifications using LARS before enrolling students to ensure the qualification is approved and valid at the time of enrolment.

Qualifying periods

Students have to pass a qualifying period for each maths and English qualification before it counts. The qualifying period is:

6 weeks for a qualification with a planned length of 24 weeks or more

2 weeks for a qualification with a planned length of less than 24 weeks

Students meet the condition of funding when they are studying for a qualification. It is up to institutions to decide when they are ready to take the exam.

Students withdrawing from their study programme

Where a student has enrolled on a maths and/or English qualification that meets the condition of funding but does not complete it, provided they have studied the qualification for the qualifying period, they will still meet the condition of funding. If a student's core aim was maths and/or English and the student drops out entirely then funding is reduced through the retention factor in the usual way. We have published further information on core aims in programmes in the funding rates and formula guidance.

Notes on individual qualification types

dual English language and literature qualifications count for prior attainment when determining whether students must study towards GCSE English to meet the condition of funding

dual English language and literature qualifications do not meet the condition of funding for students who must study towards GCSE English as part of their ESFA-funded study programme

international GCSEs no longer meet the condition of funding; however, they do continue to count as equivalent to GCSEs for the purposes of recognising prior attainment

holding the International Certificate in Christian Education (ICCE) exempts a student from further study in maths and English; when students have achieved an ICCE, institutions can record an exemption for holding an overseas equivalent qualification

What students can and must study

A summary of what qualifications students must be studying is below.

For the purposes of the condition, full-time and part-time are defined as:

full-time: 580 hours or more for 16 to 17-year-olds, or 485 hours or more for those aged 18 and older

part-time: between 150 and 579 hours for 16 to 17-year-olds, and between 150 and 484 hours for those 18 and older

Student

Programme

What a student must study to meet the condition of funding

Prior attainment with GCSE English literature and GCSE English language

English language GCSE grade held

English literature GCSE grade held

What student can and must study

Level 3 qualifications

A range of level 3 maths and English qualifications such as A and AS levels, core maths, Pre-U and the International Baccalaureate can be taught as an alternative to GCSE when it is appropriate for the student.

Students retaking maths and English qualifications

A retake is defined as a qualification retaken by a student, including learning and attending lessons other than revision lessons. Students are funded to retake maths and/or English until they achieve a GCSE grade 9 to 4. For students with a grade 2 or below, this can be either a functional skills level 2 or GCSE grade 9 to 4.

Prior attainment

Students meet the condition of funding when they have already achieved GCSE grade 9 to 4 or A* to C in maths, and in English language, English literature, or English language and literature.

We treat some qualifications in maths, English language, English literature, and English language and literature as equivalent to GCSE grade 9 to 4 or A* to C. This means that students who have studied in England or elsewhere who hold the following qualifications do not have to study maths and/or English to meet the condition of funding:

international GCSEs, regulated or unregulated, or equivalent level 1/level 2 certificates grade A* to C/9 to 4

GCSEs grade A* to C, obtained in Wales, including maths numeracy

GCSEs grade A* to C, obtained in Northern Ireland

Scottish Intermediate 2 certificates grade A to C

Scottish National 5 certificates grade A to C

Scottish Higher grade A to C

Scottish Advanced Higher grade A to C

Scottish National Certificate Unit (SCQF Level 6) only Communication at pass grade

international O levels grade A* to C

A and AS levels at any pass grade

full International Baccalaureate (IB) level 3 Diploma (LARS reference 50034157)

IB Middle Years Programme Certificates in maths and/or English at grade 3 or above

IB Certificate Level 3 Certificates in maths and/or English A at grade 3 and above

core maths qualifications at level 3

Students who start their 16 to 19 study programmes with a grade 2 or below and then go on to achieve a pass in functional skills level 2 do not need to undertake any further maths and English qualification to meet the condition of funding.

International GCSEs for the purpose of recognising prior attainment

International GCSEs, regulated or unregulated, or equivalent level 1/level 2 certificates in maths and English all count as equivalent to GCSEs for the purposes of recognising prior attainment in the 16 to 19 maths and English condition of funding.

Students who hold these qualifications in maths and English at GCSE grade 9 to 4 or A* to C will not have to continue their study of these subjects.

Overseas qualifications

Students who have achieved an overseas qualification that is equivalent to a GCSE in maths and/or English language are exempt from the condition of funding.

Guidance on particular student circumstances

19+ continuing students

19+ continuing students are now all funded through the standard ESFA 16 to 19 allocations process. The condition of funding in maths and English does not apply to 19+ continuing students.

Home educated students

A student who has previously been home educated and wishes to continue their education at an FE institution must comply with the condition of funding.

Students who speak limited English studying maths qualifications

A student who speaks limited English will still need to have maths in their programme, at a suitable level. This could be phased into the programme as their ability in English improves. For example, a student could study an ESOL qualification in the first term and enrol on a maths qualification in the second term.

Students on short study programmes

To ensure those that want to study short courses are not discouraged from doing so, there is flexibility for those on small part-time programmes of less than 150 planned hours. Students on these programmes do not have to study maths and English qualifications approved for the condition of funding. However, we expect institutions to include some maths and English content in programmes taken by such students to facilitate their learning.

Students achieving a GCSE or equivalent outside of their study programme

If a student achieves a GCSE grade 9 to 4, A* to C or a qualification deemed as equivalent by other means, then they will have met the condition of funding for the remainder of their programme. This also applies to students with prior attainment of GCSE grade 2 who pass functional skills level 2. Institutions must make sure that they record the achievement grade in the data return, to tell us that the student has met the condition of funding.

Students on traineeship programmes

Students on traineeship programmes are subject to the condition of funding in the same way as students on other study programmes.

The condition of funding grade 3 or grade D requirement applies to students on traineeship programmes according to whether the student is full-time or part-time.

Full-time students with prior attainment of GCSE grade 3 or grade D in English and/or maths must study GCSE to meet the condition; part-time students can study towards a GCSE or a valid stepping stone qualification, irrespective of prior attainment.

Students on apprenticeship programmes

Students on apprenticeship programmes are not subject to the maths and English condition of funding. However, similar requirements are in place to ensure all apprentices have achieved a good level of literacy and numeracy by the end of their apprenticeship to support their immediate and future career choices.

Students on T Levels

The condition of funding will apply to all students starting T Levels from academic year 2022 to 2023 in the same way as it does to students on other 16 to 19 study programmes. It does not apply to students who started T Levels in the academic years 2020 to 2021 or 2021 to 2022. However, we expect these students who have yet to achieve a GCSE grade 4 in English and/or maths, to continue to work towards a level 2 in these subjects.

Students on supported internships

Students on supported internships are not automatically exempt from the condition of funding. Some students on internships may meet the criteria to be exempt from the condition. Students on supported internships with an exemption should still be studying maths and English at an appropriate level, even when they are not able to work towards a stepping stone qualification or GCSE.

Exemptions to the condition of funding

There are 2 exemptions to the condition of funding:

students with special educational needs and/or disabilities (SEND), who are assessed as not able to study either GCSE or stepping stone qualifications

students with overseas qualifications that are established as equivalent to GCSE grade 4 or grade C

Students with SEND exemption

Where a student with SEND does not hold a GCSE grade 9 to 4, A* to C or equivalent qualification in maths and/or English, an institution may decide that it is not appropriate for them to study for a GCSE or a stepping stone qualification. In this case, in addition to the student's EHC plan, the institution must hold an evidenced assessment that the student is not able to study these subjects.

All exemptions must be considered on a case-by-case basis. There is no blanket exemption for whole institutions.

Assessments for exemptions must be authorised by an appropriate professional in the institution, such as the head of SEND or student support. The assessment should be structured and documented. We do not expect to give further detailed advice on the format or on who should conduct the assessment. This is for institutions to decide. Exemptions will be monitored from the data institutions submit rather than on an individual basis.

Institutions should keep evidence that justifies their decision that a student is exempt. This information must be available to auditors appointed to test the use of ESFA funding and to Ofsted inspectors.

When a student with SEND does not hold a GCSE grade 9 to 4, A* to C or equivalent qualification in maths and/or English, every effort should be made to enable them to achieve these valuable qualifications. Students with SEND and who have achieved a GCSE grade 3, grade D or equivalent qualification in maths and/or English should be capable of improving their GCSE grade with the right teaching and support.

When a student is not able to study a GCSE, institutions should support them in studying a stepping stone qualification. There are a range of qualifications available for students from entry level upwards. Institutions can enrol a student on a course for up to 3 years, or longer if the student is in learning to age 25. This range of qualifications gives institutions scope to offer the student opportunities to progress to the appropriate level in maths and English.

There are a small number of students who are not able to take any qualifications. These students are exempt from studying qualifications but appropriate literacy and numeracy should still be included in their programme at an appropriate level.

We expect institutions to use the flexibilities available to tailor programmes to enable each student to improve in maths and/or English to the best of their ability. Institutions should recognise and record students' progress and achievement.

Students with overseas qualifications exemption

Students who have studied in a country that does not provide GCSEs are potentially exempt from the condition of funding where the overseas qualifications they hold are equivalent to GCSE grade 4 or grade C.

Institutions must get confirmation from the National Information Centre (UK ENIC) that the qualification is equivalent to GCSE grade 9 to 4 or A* to C. ENIC is the UK's national agency responsible for providing information and opinion on academic, vocational and professional qualifications from across the world.

As well as confirmation from ENIC that an overseas qualification is equivalent to GCSE grade 4 or grade C, institutions also have to ensure that the student possesses the necessary competence in maths and English to achieve their programme. If the student needs additional teaching or support to function on their programme, institutions must include it in the programme. This kind of activity is eligible for funding and should be included in the planned hours.

Students who have achieved the ICCE do not have to study maths and English, as the ICCE is equivalent to a standard pass in maths and English. When students have achieved an ICCE, institutions can record an exemption for holding an overseas equivalent qualification. They do not have to get the equivalence certified by ENIC.

Recording and monitoring of exemptions and prior attainment

To avoid funding being withdrawn unnecessarily, institutions must record exemptions in their data returns. It is also good practice for institutions to note exemptions in their own records.

There is no need to inform ESFA that a student is out of scope for the maths and English condition of funding because they are studying for less than 150 hours in the academic year.

ILR returns use funding and monitoring (FAM) codes to show when a student qualifies for one of 2 exemptions. The same FAM field shows when the student meets the condition of funding by holding a UK qualification or by studying maths/English at another institution.

School census returns use a set of letter codes to record the same information.

Collaboration between institutions to ensure students are meeting the maths and English condition of funding

When institutions collaborate to deliver maths and/or English, one institution must be responsible for claiming funding for a student's whole programme, and making sure that they meet the condition of funding. Wherever possible, institutions must avoid enrolling and claiming funding for part of a student's programme.

However, in a small number of cases, students study for maths and/or English with another institution. In these circumstances, the home institution must make sure that students meet the condition of funding.

For institutions that return the ILR

To avoid being penalised for students not meeting the condition of funding, institutions must use the FAM code ECF/MCF 4. Institutions must only use this code when students are studying maths and/or English qualifications at another institution.

For institutions that return the school census

The school census does not have the capacity to show that students are studying maths and/or English qualifications at another institution. Therefore, schools must submit a business case for an adjustment to their funding after they have received their allocation statement.

The business case must include this information:

the numbers of students who have studied or retaken a maths and/or English qualification at another institution (we do not need individual student data such as unique pupil numbers (UPNs) and names)

the name of the other institution(s) where they have studied

the maths and/or English qualifications studied

a statement that collaboration has taken place with brief details

accounting officer confirmation that the information is correct

Demonstrating progression

Institutions must be able to demonstrate progression for students enrolled on maths and/or English qualifications. In most cases, we expect that it will be appropriate for students to study for a qualification at a higher level than they have already attained, rather than just improving their grade at the same level as their previous achievement.

When a student is studying for a qualification at the same level as a previous achievement, institutions must have evidence of why the qualification is the best choice for the student.

Audit and inspection

Institutions will be subject to risk-based compliance audit to ensure the data submitted for maths and English participation and prior attainment and evidence for exemptions are sound. On inspection, Ofsted will establish that students are on the most suitable study programme and that any exemptions from studying maths and English are appropriate.

Auditors appointed to test the use of ESFA funding will check when institutions claim a student is exempt to ensure the exemption reason is valid. Institutions must ensure that they keep adequate evidence when they apply either of the 2 exemptions. ESFA will monitor how institutions use exemptions.

Applying a tolerance to allocations

We have decided to continue to apply a tolerance level of 5% until further notice, in recognition of the continued efforts of institutions to deliver the 16 to 19 maths and English policy. We expect that the clarity over the level of tolerance in future years will provide institutions with increased certainty on which to plan.

We will apply funding reductions for non-compliance with the condition of funding to institutions where more than 5% of students (calculated by value) without a GCSE grade 9 to 4 or A* to C in maths and/or English did not enrol on an approved qualification in these subjects. The funding reductions for these institutions will be applied at half the national funding rate above the tolerance.

An example of how the funding deduction is calculated

We will apply funding reductions to institutions above a tolerance of 5% of total students (calculated by value) for students who do not meet the condition of funding. We will remove funding for these institutions for each student above the tolerance level at half the national funding rate.

The 5% tolerance is calculated by determining the value of the non-compliant students and expressing this as a percentage of the total value of all students. The value for a student is the national rate for their funding band.

The below example shows the calculation of the funding reduction for the 2022 to 2023 year, which is based on data for 2020 to 2021. For students who are non-compliant above the tolerance, we will remove funding at a reduced rate of 50% of the 2020 to 2021 national funding rate for the band. We will only remove funding above the 5% tolerance.

Example

Stage 1

To calculate the value of non-compliance (E), the number of non-compliant students from end year 2020 to 2021 data (C) was multiplied by the appropriate national funding rate (A) according to their total planned hours. Values for students in the 0 to 279 band was calculated as a proportion of the full-time national rate based on their total planned hours divided by 600.

Stage 2

To calculate the total value of 2020 to 2021 students (D), the total number of students in each band (B) was multiplied by the appropriate national funding rate (A). Data was from the final return of the ILR or the autumn census for the 2020 to 2021 academic year.

In the example below the value of non-compliant students (E) is £707,886; the value of all students (D) is £8,655,437. This is non-compliance of 8.18%.

Stage 3

We calculated 5% of the total value for all students at the national funding rate (D) to calculate the financial value of the tolerance (F). The tolerance is then discounted from (E) the condition of funding reduction to calculate the in-scope reduction (G). The final reduction (H) is then calculated by taking 50% of (G) of the in-scope reduction.

In this example, the tolerance (F) is £432,772, the reduction in-scope (G) is £275,114, and the final reduction is £137,557. If the value of non-compliant students in this example (E) was less than £432,772 the institution would have no withdrawal of funding.

Other useful information

Resources

We have published an interactive tool for the school census to help users to understand how the condition of funding works within the census. It provides specific examples of how to check qualifications that do not meet the condition of funding and the impact this will have when non-compliant or out of date qualifications are included.

When you return your autumn census, which includes the post-16 module, 2 reports on the condition of funding are available through COLLECT (one for the current funding year, and one for the previous year). We have published guidance on the reports.

The Funding Information System (FIS) will help further education (FE) institutions validate their ILR data. The reports available through FIS gives the maths and English status of every student, showing whether they have achieved at least a GCSE grade 4/or grade C in each subject and whether they are currently studying for a valid qualification in the subject. The reports also show those students recorded as exempt from the condition of funding.

The Education Training Foundation (ETF) offers a range of opportunities and resources for teachers, trainers and leaders to support practitioners to confidently teach maths and English.

The Excellence Gateway provides a wide range of resources for teaching maths and English.

Enquiries

If you have any queries on the condition of funding, visit the ESFA help centre.

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16 to 19 funding: maths and English condition of funding

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Information for institutions on maths and English condition of funding.

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Government skills training and employment schemes for employers considering hiring or offering placements including Apprenticeships, Traineeships, T Levels.

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Poverty in Bradford District

Indices of Deprivation 2019

The most recent edition of the English Indices of Deprivation was published in 2019. The indices include the Index of Multiple Deprivation (IMD) 2019.

Bradford District is ranked 5th most income deprived and 6th most employment deprived local authority in England.

15% of the District's households are in fuel poverty.

22% of children are living below the poverty line.

13% of working age people have no qualifications.

11% of the working age population claim an out-of-work benefit.

How the indices are calculated

IMD 2019 uses 39 separate indicators, organised across seven distinct domains of deprivation (income, employment, education skills and training, health deprivation and disability, crime, barriers to housing and and services and living environment) which are weighted and combined to calculate the Index of Multiple Deprivation 2019.

What this means for Bradford

Bradford District ranks 13th most deprived local authority in England (where 1 is the most deprived local authority and 317 is the least deprived) and 2nd most deprived in the Yorkshire and Humber region (after the City of Kingston upon Hull). When the previous indices were published in 2015, Bradford was ranked 19th most deprived.

34% of the District's LSOAs fall within the 10% most deprived neighbourhoods and 16 of the District's LSOAs fall within the 10% least deprived neighbourhoods.

When the LSOA data is aggregated to ward level it shows that 14 of the District's wards are amongst the 10% most deprived wards in England. Manningham is consistently the most deprived ward in the District. Wharfedale is consistently the least deprived ward in the District.

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Indices of Deprivation 2019 The most recent edition of the English Indices of Deprivation was published in 2019. The indices include the Index of Multiple Deprivation (IMD) 2019. Bradford District is ranked 5th most income deprived and 6th most employment deprived local authority in England. 15% of the District's households are in fuel poverty.

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Guidance for primary and secondary schools to meet existing expectations for teaching reading.

Applies to England

Documents

Details

This guidance is for:

teachers

school leaders

reading and literacy leads

governing bodies

initial teacher training partnerships

specialist provision

This guidance:

sets out the research underpinning the importance of talk, stories and systematic synthetic phonics (SSP) in reception, the importance of fidelity in phonics, and the Department for Education's evidence-informed position on the best way to teach reading

supports schools to evaluate their teaching of reading from reception to year 9, and to identify how to improve provision if weaknesses are found

provides practical support for high-quality teaching of SSP, fluency and comprehension, including assessment

explains the importance of systematic phonics teaching for older pupils who are at risk of failing to learn to read because they cannot decode well enough

supports schools in motivating pupils to develop a love of reading

supports schools working with parents to help their children learn to read

helps schools to meet existing expectations for teaching early reading, as set out in:

This document has been developed through the contributions of experts from across the literacy sector and of school leaders. It is based on the valued experience and knowledge of teachers, reading and language experts, educational organisations, English Hubs council members, and our 34 English Hubs.

Schools considering purchasing a new SSP programme can find more information at choosing a phonics programme.

Further support for schools is available through our English Hubs Programme.

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Guidance for primary and secondary schools to meet existing expectations for teaching reading.

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A series of rapid evidence reviews exploring current research and technology applicable to learners with Specific Learning Difficulties (SpLDs).

Documents

Details

In January 2019, the Government Office for Science commissioned a series of 4 rapid evidence reviews to explore how technology and research can help improve educational outcomes for learners with Specific Learning Difficulties ( SpLDs).

This review examined:

current understanding of the causes and identification of SpLDs

the support system for learners with SpLDs

technology-based interventions for SpLDs

a case study approach focusing on dyscalculia to explore all 3 themes

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Specific Learning Difficulties: current understanding, support systems, and technology-led interventions

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Purpose of study

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 most forms of employment. A high-quality mathematics education 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.

Aims

The national curriculum for mathematics aims to ensure that all pupils:

Mathematics is an interconnected subject in which pupils need to be able to move fluently between representations of mathematical ideas. The programmes of study are, by necessity, organised into apparently distinct domains, but pupils should make rich connections across mathematical ideas to develop fluency, mathematical reasoning and competence in solving increasingly sophisticated problems. They should also apply their mathematical knowledge to science and other subjects.

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.

Information and communication technology (ICT)

Calculators should not be used as a substitute for good written and mental arithmetic. They should therefore only be introduced near the end of key stage 2 to support pupils' conceptual understanding and exploration of more complex number problems, if written and mental arithmetic are secure. In both primary and secondary schools, teachers should use their judgement about when ICT tools should be used.

Spoken language

The national curriculum for mathematics reflects the importance of spoken language in pupils' development across the whole curriculum - cognitively, socially and linguistically. The quality and variety of language that pupils hear and speak are key factors in developing their mathematical vocabulary and presenting a mathematical justification, argument or proof. They must be assisted in making their thinking clear to themselves as well as others, and teachers should ensure that pupils build secure foundations by using discussion to probe and remedy their misconceptions.

School curriculum

The programmes of study for mathematics are set out year-by-year for key stages 1 and 2. Schools are, however, only required to teach the relevant programme of study by the end of the key stage. Within each key stage, schools therefore have the flexibility to introduce content earlier or later than set out in the programme of study. In addition, schools can introduce key stage content during an earlier key stage, if appropriate. All schools are also required to set out their school curriculum for mathematics on a year-by-year basis and make this information available online.

Attainment targets

By the end of each key stage, pupils are expected to know, apply and understand the matters, skills and processes specified in the relevant programme of study.

Schools are not required by law to teach the example content in [square brackets] or the content indicated as being 'non-statutory'.

Key stage 1 - years 1 and 2

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 4 operations, including with practical resources [for example, 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.

Year 1 programme of study

Number - number and place value

Pupils should be taught to:

Number - addition and subtraction

Pupils should be taught to:

Number - multiplication and division

Pupils should be taught to:

solve one-step problems involving multiplication and division, by calculating the answer using concrete objects, pictorial representations and arrays with the support of the teacher

Number - fractions

Pupils should be taught to:

recognise, find and name a half as 1 of 2 equal parts of an object, shape or quantity

recognise, find and name a quarter as 1 of 4 equal parts of an object, shape or quantity

Notes and guidance (non-statutory)

Pupils are taught half and quarter as 'fractions of' discrete and continuous quantities by solving problems using shapes, objects and quantities. For example, they could recognise and find half a length, quantity, set of objects or shape. Pupils connect halves and quarters to the equal sharing and grouping of sets of objects and to measures, as well as recognising and combining halves and quarters as parts of a whole.

Measurement

Pupils should be taught to:

compare, describe and solve practical problems for:

lengths and heights [for example, long/short, longer/shorter, tall/short, double/half]

mass/weight [for example, heavy/light, heavier than, lighter than]

capacity and volume [for example, full/empty, more than, less than, half, half full, quarter]

time [for example, quicker, slower, earlier, later]

measure and begin to record the following:

lengths and heights

mass/weight

capacity and volume

time (hours, minutes, seconds)

recognise and know the value of different denominations of coins and notes

sequence events in chronological order using language [for example, before and after, next, first, today, yesterday, tomorrow, morning, afternoon and evening]

recognise and use language relating to dates, including days of the week, weeks, months and years

tell the time to the hour and half past the hour and draw the hands on a clock face to show these times

Geometry - properties of shapes

Pupils should be taught to:

recognise and name common 2-D and 3-D shapes, including:

2-D shapes [for example, rectangles (including squares), circles and triangles]

3-D shapes [for example, cuboids (including cubes), pyramids and spheres]

Geometry - position and direction

Pupils should be taught to:

describe position, direction and movement, including whole, half, quarter and three-quarter turns

Year 2 programme of study

Number - number and place value

Pupils should be taught to:

Number - addition and subtraction

Pupils should be taught to:

solve problems with addition and subtraction:

using concrete objects and pictorial representations, including those involving numbers, quantities and measures

applying their increasing knowledge of mental and written methods

recall and use addition and subtraction facts to 20 fluently, and derive and use related facts up to 100

add and subtract numbers using concrete objects, pictorial representations, and mentally, including:

a two-digit number and 1s

a two-digit number and 10s

2 two-digit numbers

adding 3 one-digit numbers

show that addition of 2 numbers can be done in any order (commutative) and subtraction of 1 number from another cannot

recognise and use the inverse relationship between addition and subtraction and use this to check calculations and solve missing number problems

Number - multiplication and division

Pupils should be taught to:

Number - fractions

Pupils should be taught to:

recognise, find, name and write fractions , , and of a length, shape, set of objects or quantity

write simple fractions, for example of 6 = 3 and recognise the equivalence of and

Notes and guidance (non-statutory)

Pupils use fractions as 'fractions of' discrete and continuous quantities by solving problems using shapes, objects and quantities. They connect unit fractions to equal sharing and grouping, to numbers when they can be calculated, and to measures, finding fractions of lengths, quantities, sets of objects or shapes. They meet as the first example of a non-unit fraction.

Pupils should count in fractions up to 10, starting from any number and using the and equivalence on the number line (for example, 1 , 1 (or 1 ), 1 , 2). This reinforces the concept of fractions as numbers and that they can add up to more than 1.

Measurement

Pupils should be taught to:

choose and use appropriate standard units to estimate and measure length/height in any direction (m/cm); mass (kg/g); temperature (°C); capacity (litres/ml) to the nearest appropriate unit, using rulers, scales, thermometers and measuring vessels

compare and order lengths, mass, volume/capacity and record the results using >, < and =

recognise and use symbols for pounds (£) and pence (p); combine amounts to make a particular value

find different combinations of coins that equal the same amounts of money

solve simple problems in a practical context involving addition and subtraction of money of the same unit, including giving change

compare and sequence intervals of time

tell and write the time to five minutes, including quarter past/to the hour and draw the hands on a clock face to show these times

know the number of minutes in an hour and the number of hours in a day

Geometry - properties of shapes

Pupils should be taught to:

Geometry - position and direction

Pupils should be taught to:

order and arrange combinations of mathematical objects in patterns and sequences

use mathematical vocabulary to describe position, direction and movement, including movement in a straight line and distinguishing between rotation as a turn and in terms of right angles for quarter, half and three-quarter turns (clockwise and anti-clockwise)

Statistics

Pupils should be taught to:

Notes and guidance (non-statutory)

Pupils record, interpret, collate, organise and compare information (for example, using many-to-one correspondence in pictograms with simple ratios 2, 5,10).

Lower key stage 2 - years 3 and 4

The principal focus of mathematics teaching in lower key stage 2 is to ensure that pupils become increasingly fluent with whole numbers and the 4 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.

Year 3 programme of study

Number - number and place value

Pupils should be taught to:

Number - addition and subtraction

Pupils should be taught to:

add and subtract numbers mentally, including:

a three-digit number and 1s

a three-digit number and 10s

a three-digit number and 100s

add and subtract numbers with up to 3 digits, using formal written methods of columnar addition and subtraction

estimate the answer to a calculation and use inverse operations to check answers

solve problems, including missing number problems, using number facts, place value, and more complex addition and subtraction

Notes and guidance (non-statutory)

Pupils practise solving varied addition and subtraction questions. For mental calculations with two-digit numbers, the answers could exceed 100.

Pupils use their understanding of place value and partitioning, and practise using columnar addition and subtraction with increasingly large numbers up to 3 digits to become fluent (see Mathematics appendix 1 (PDF, 248KB)).

Number - multiplication and division

Pupils should be taught to:

Number - fractions

Pupils should be taught to:

count up and down in tenths; recognise that tenths arise from dividing an object into 10 equal parts and in dividing one-digit numbers or quantities by 10

recognise, find and write fractions of a discrete set of objects: unit fractions and non-unit fractions with small denominators

recognise and use fractions as numbers: unit fractions and non-unit fractions with small denominators

recognise and show, using diagrams, equivalent fractions with small denominators

add and subtract fractions with the same denominator within one whole [for example, + = ]

compare and order unit fractions, and fractions with the same denominators

solve problems that involve all of the above

Notes and guidance (non-statutory)

Pupils connect tenths to place value, decimal measures and to division by 10.

They begin to understand unit and non-unit fractions as numbers on the number line, and deduce relations between them, such as size and equivalence. They should go beyond the [0, 1] interval, including relating this to measure.

Pupils understand the relation between unit fractions as operators (fractions of), and division by integers.

They continue to recognise fractions in the context of parts of a whole, numbers, measurements, a shape, and unit fractions as a division of a quantity.

Pupils practise adding and subtracting fractions with the same denominator through a variety of increasingly complex problems to improve fluency.

Measurement

Pupils should be taught to:

Geometry - properties of shapes

Pupils should be taught to:

Statistics

Pupils should be taught to:

interpret and present data using bar charts, pictograms and tables

solve one-step and two-step questions [for example 'How many more?' and 'How many fewer?'] using information presented in scaled bar charts and pictograms and tables

Year 4 programme of study

Number - number and place value

Pupils should be taught to:

count in multiples of 6, 7, 9, 25 and 1,000

find 1,000 more or less than a given number

count backwards through 0 to include negative numbers

recognise the place value of each digit in a four-digit number (1,000s, 100s, 10s, and 1s)

order and compare numbers beyond 1,000

identify, represent and estimate numbers using different representations

round any number to the nearest 10, 100 or 1,000

solve number and practical problems that involve all of the above and with increasingly large positive numbers

read Roman numerals to 100 (I to C) and know that over time, the numeral system changed to include the concept of 0 and place value

Number - addition and subtraction

Pupils should be taught to:

Notes and guidance (non-statutory)

Pupils continue to practise both mental methods and columnar addition and subtraction with increasingly large numbers to aid fluency (see Mathematics appendix 1 (PDF, 248KB)).

Number - multiplication and division

Pupils should be taught to:

Notes and guidance (non-statutory)

Pupils continue to practise recalling and using multiplication tables and related division facts to aid fluency.

Pupils practise mental methods and extend this to 3-digit numbers to derive facts, (for example 600 ÷ 3 = 200 can be derived from 2 x 3 = 6).

Pupils practise to become fluent in the formal written method of short multiplication and short division with exact answers (see Mathematics appendix 1 (PDF, 248KB)).

Pupils write statements about the equality of expressions (for example, use the distributive law 39 × 7 = 30 × 7 + 9 × 7 and associative law (2 × 3) × 4 = 2 × (3 × 4)). They combine their knowledge of number facts and rules of arithmetic to solve mental and written calculations for example, 2 x 6 x 5 = 10 x 6 = 60.

Pupils solve two-step problems in contexts, choosing the appropriate operation, working with increasingly harder numbers. This should include correspondence questions such as the numbers of choices of a meal on a menu, or 3 cakes shared equally between 10 children.

Number - fractions (including decimals)

Pupils should be taught to:

Measurement

Pupils should be taught to:

Geometry - properties of shapes

Pupils should be taught to:

Geometry - position and direction

Pupils should be taught to:

Notes and guidance (non-statutory)

Pupils draw a pair of axes in one quadrant, with equal scales and integer labels. They read, write and use pairs of co-ordinates, for example (2, 5), including using co-ordinate-plotting ICT tools.

Statistics

Pupils should be taught to:

interpret and present discrete and continuous data using appropriate graphical methods, including bar charts and time graphs

solve comparison, sum and difference problems using information presented in bar charts, pictograms, tables and other graphs

Upper key stage 2 - years 5 and 6

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 4 operations, including long multiplication and division, and in working with fractions, decimals and percentages.

Pupils should read, spell and pronounce mathematical vocabulary correctly.

Year 5 programme of study

Number - number and place value

Pupils should be taught to:

Number - addition and subtraction

Pupils should be taught to:

Notes and guidance (non-statutory)

Pupils practise using the formal written methods of columnar addition and subtraction with increasingly large numbers to aid fluency (see Mathematics appendix 1 (PDF, 248KB)).

They practise mental calculations with increasingly large numbers to aid fluency (for example, 12,462 - 2,300 = 10,162).

Number - multiplication and division

Pupils should be taught to:

identify multiples and factors, including finding all factor pairs of a number, and common factors of 2 numbers

know and use the vocabulary of prime numbers, prime factors and composite (non-prime) numbers

establish whether a number up to 100 is prime and recall prime numbers up to 19

multiply numbers up to 4 digits by a one- or two-digit number using a formal written method, including long multiplication for two-digit numbers

multiply and divide numbers mentally, drawing upon known facts

divide numbers up to 4 digits by a one-digit number using the formal written method of short division and interpret remainders appropriately for the context

multiply and divide whole numbers and those involving decimals by 10, 100 and 1,000

recognise and use square numbers and cube numbers, and the notation for squared (²) and cubed (³)

solve problems involving multiplication and division, including using their knowledge of factors and multiples, squares and cubes

solve problems involving addition, subtraction, multiplication and division and a combination of these, including understanding the meaning of the equals sign

solve problems involving multiplication and division, including scaling by simple fractions and problems involving simple rates

Notes and guidance (non-statutory)

Pupils practise and extend their use of the formal written methods of short multiplication and short division (see Mathematics appendix 1 (PDF, 248KB)). They apply all the multiplication tables and related division facts frequently, commit them to memory and use them confidently to make larger calculations.

They use and understand the terms factor, multiple and prime, square and cube numbers.

Pupils interpret non-integer answers to division by expressing results in different ways according to the context, including with remainders, as fractions, as decimals or by rounding (for example, 98 ÷ 4 = = 24 r 2 = 24 = 24.5 ≈ 25).

Pupils use multiplication and division as inverses to support the introduction of ratio in year 6, for example, by multiplying and dividing by powers of 10 in scale drawings or by multiplying and dividing by powers of a 1,000 in converting between units such as kilometres and metres.

Distributivity can be expressed as a(b + c) = ab + ac.

They understand the terms factor, multiple and prime, square and cube numbers and use them to construct equivalence statements (for example, 4 x 35 = 2 x 2 x 35; 3 x 270 = 3 x 3 x 9 x 10 = 9² x 10).

Pupils use and explain the equals sign to indicate equivalence, including in missing number problems (for example 13 + 24 = 12 + 25; 33 = 5 x ?).

Number - fractions (including decimals and percentages)

Pupils should be taught to:

compare and order fractions whose denominators are all multiples of the same number

identify, name and write equivalent fractions of a given fraction, represented visually, including tenths and hundredths

recognise mixed numbers and improper fractions and convert from one form to the other and write mathematical statements > 1 as a mixed number [for example, + = = 1 ]

add and subtract fractions with the same denominator, and denominators that are multiples of the same number

multiply proper fractions and mixed numbers by whole numbers, supported by materials and diagrams

read and write decimal numbers as fractions [for example, 0.71 = ]

recognise and use thousandths and relate them to tenths, hundredths and decimal equivalents

round decimals with 2 decimal places to the nearest whole number and to 1 decimal place

read, write, order and compare numbers with up to 3 decimal places

solve problems involving number up to 3 decimal places

recognise the per cent symbol (%) and understand that per cent relates to 'number of parts per 100', and write percentages as a fraction with denominator 100, and as a decimal fraction

solve problems which require knowing percentage and decimal equivalents of , , , , and those fractions with a denominator of a multiple of 10 or 25

Notes and guidance (non-statutory)

Pupils should be taught throughout that percentages, decimals and fractions are different ways of expressing proportions.

They extend their knowledge of fractions to thousandths and connect to decimals and measures.

Pupils connect equivalent fractions > 1 that simplify to integers with division and other fractions > 1 to division with remainders, using the number line and other models, and hence move from these to improper and mixed fractions.

Pupils connect multiplication by a fraction to using fractions as operators (fractions of), and to division, building on work from previous years. This relates to scaling by simple fractions, including fractions > 1.

Pupils practise adding and subtracting fractions to become fluent through a variety of increasingly complex problems. They extend their understanding of adding and subtracting fractions to calculations that exceed 1 as a mixed number.

Pupils continue to practise counting forwards and backwards in simple fractions.

Pupils continue to develop their understanding of fractions as numbers, measures and operators by finding fractions of numbers and quantities. Pupils extend counting from year 4, using decimals and fractions including bridging 0, for example on a number line.

Pupils say, read and write decimal fractions and related tenths, hundredths and thousandths accurately and are confident in checking the reasonableness of their answers to problems.

They mentally add and subtract tenths, and one-digit whole numbers and tenths.

They practise adding and subtracting decimals, including a mix of whole numbers and decimals, decimals with different numbers of decimal places, and complements of 1 (for example, 0.83 + 0.17 = 1).

Pupils should go beyond the measurement and money models of decimals, for example, by solving puzzles involving decimals.

Pupils should make connections between percentages, fractions and decimals (for example, 100% represents a whole quantity and 1% is , 50% is , 25% is ) and relate this to finding 'fractions of'.

Measurement

Pupils should be taught to:

Geometry - properties of shapes

Pupils should be taught to:

identify 3-D shapes, including cubes and other cuboids, from 2-D representations

know angles are measured in degrees: estimate and compare acute, obtuse and reflex angles

draw given angles, and measure them in degrees (°)

identify:

angles at a point and 1 whole turn (total 360°)

angles at a point on a straight line and half a turn (total 180°)

other multiples of 90°

use the properties of rectangles to deduce related facts and find missing lengths and angles

distinguish between regular and irregular polygons based on reasoning about equal sides and angles

Geometry - position and direction

Pupils should be taught to:

identify, describe and represent the position of a shape following a reflection or translation, using the appropriate language, and know that the shape has not changed

Notes and guidance (non-statutory)

Pupils recognise and use reflection and translation in a variety of diagrams, including continuing to use a 2-D grid and coordinates in the first quadrant. Reflection should be in lines that are parallel to the axes.

Statistics

Pupils should be taught to:

solve comparison, sum and difference problems using information presented in a line graph

complete, read and interpret information in tables, including timetables

Year 6 programme of study

Number - number and place value

Pupils should be taught to:

Notes and guidance (non-statutory)

Pupils use the whole number system, including saying, reading and writing numbers accurately.

Number - addition, subtraction, multiplication and division

Pupils should be taught to:

multiply multi-digit numbers up to 4 digits by a two-digit whole number using the formal written method of long multiplication

divide numbers up to 4 digits by a two-digit whole number using the formal written method of long division, and interpret remainders as whole number remainders, fractions, or by rounding, as appropriate for the context

divide numbers up to 4 digits by a two-digit number using the formal written method of short division where appropriate, interpreting remainders according to the context

perform mental calculations, including with mixed operations and large numbers

identify common factors, common multiples and prime numbers

use their knowledge of the order of operations to carry out calculations involving the 4 operations

solve addition and subtraction multi-step problems in contexts, deciding which operations and methods to use and why

solve problems involving addition, subtraction, multiplication and division

use estimation to check answers to calculations and determine, in the context of a problem, an appropriate degree of accuracy

Notes and guidance (non-statutory)

Pupils practise addition, subtraction, multiplication and division for larger numbers, using the formal written methods of columnar addition and subtraction, short and long multiplication, and short and long division (see Mathematics appendix 1 (PDF, 248KB)).

They undertake mental calculations with increasingly large numbers and more complex calculations.

Pupils continue to use all the multiplication tables to calculate mathematical statements in order to maintain their fluency.

Pupils round answers to a specified degree of accuracy, for example, to the nearest 10, 20, 50, etc, but not to a specified number of significant figures.

Pupils explore the order of operations using brackets; for example, 2 + 1 x 3 = 5 and (2 + 1) x 3 = 9.

Common factors can be related to finding equivalent fractions.

Number - Fractions (including decimals and percentages)

Pupils should be taught to:

use common factors to simplify fractions; use common multiples to express fractions in the same denomination

compare and order fractions, including fractions >1

add and subtract fractions with different denominators and mixed numbers, using the concept of equivalent fractions

multiply simple pairs of proper fractions, writing the answer in its simplest form [for example, × = ]

divide proper fractions by whole numbers [for example, ÷ 2 = ]

associate a fraction with division and calculate decimal fraction equivalents [for example, 0.375] for a simple fraction [for example, ]

identify the value of each digit in numbers given to 3 decimal places and multiply and divide numbers by 10, 100 and 1,000 giving answers up to 3 decimal places

multiply one-digit numbers with up to 2 decimal places by whole numbers

use written division methods in cases where the answer has up to 2 decimal places

solve problems which require answers to be rounded to specified degrees of accuracy

recall and use equivalences between simple fractions, decimals and percentages, including in different contexts

Notes and guidance (non-statutory)

Pupils should practise, use and understand the addition and subtraction of fractions with different denominators by identifying equivalent fractions with the same denominator. They should start with fractions where the denominator of one fraction is a multiple of the other (for example, + = ] and progress to varied and increasingly complex problems.

Pupils should use a variety of images to support their understanding of multiplication with fractions. This follows earlier work about fractions as operators (fractions of), as numbers, and as equal parts of objects, for example as parts of a rectangle.

Pupils use their understanding of the relationship between unit fractions and division to work backwards by multiplying a quantity that represents a unit fraction to find the whole quantity (for example, if quarter of a length is 36cm, then the whole length is 36 × 4 = 144cm).

They practise calculations with simple fractions and decimal fraction equivalents to aid fluency, including listing equivalent fractions to identify fractions with common denominators.

Pupils can explore and make conjectures about converting a simple fraction to a decimal fraction (for example, 3 ÷ 8 = 0.375). For simple fractions with recurring decimal equivalents, pupils learn about rounding the decimal to three decimal places, or other appropriate approximations depending on the context. Pupils multiply and divide numbers with up to 2 decimal places by one-digit and two-digit whole numbers. Pupils multiply decimals by whole numbers, starting with the simplest cases, such as 0.4 × 2 = 0.8, and in practical contexts, such as measures and money.

Pupils are introduced to the division of decimal numbers by one-digit whole numbers, initially, in practical contexts involving measures and money. They recognise division calculations as the inverse of multiplication.

Pupils also develop their skills of rounding and estimating as a means of predicting and checking the order of magnitude of their answers to decimal calculations. This includes rounding answers to a specified degree of accuracy and checking the reasonableness of their answers.

Ratio and proportion

Pupils should be taught to:

Notes and guidance (non-statutory)

Pupils recognise proportionality in contexts when the relations between quantities are in the same ratio (for example, similar shapes and recipes).

Pupils link percentages or 360° to calculating angles of pie charts.

Pupils should consolidate their understanding of ratio when comparing quantities, sizes and scale drawings by solving a variety of problems. They might use the notation a:b to record their work.

Pupils solve problems involving unequal quantities, for example, 'for every egg you need 3 spoonfuls of flour', ' of the class are boys'. These problems are the foundation for later formal approaches to ratio and proportion.

Algebra

Pupils should be taught to:

Notes and guidance (non-statutory)

Pupils should be introduced to the use of symbols and letters to represent variables and unknowns in mathematical situations that they already understand, such as:

missing numbers, lengths, coordinates and angles

formulae in mathematics and science

equivalent expressions (for example, a + b = b + a)

generalisations of number patterns

number puzzles (for example, what 2 numbers can add up to)

Measurement

Pupils should be taught to:

Notes and guidance (non-statutory)

Pupils connect conversion (for example, from kilometres to miles) to a graphical representation as preparation for understanding linear/proportional graphs.

They know approximate conversions and are able to tell if an answer is sensible.

Using the number line, pupils use, add and subtract positive and negative integers for measures such as temperature.

They relate the area of rectangles to parallelograms and triangles, for example, by dissection, and calculate their areas, understanding and using the formulae (in words or symbols) to do this.

Pupils could be introduced to compound units for speed, such as miles per hour, and apply their knowledge in science or other subjects as appropriate.

Geometry - properties of shapes

Pupils should be taught to:

Geometry - position and direction

Pupils should be taught to:

describe positions on the full coordinate grid (all 4 quadrants)

draw and translate simple shapes on the coordinate plane, and reflect them in the axes

Statistics

Pupils should be taught to:

interpret and construct pie charts and line graphs and use these to solve problems

calculate and interpret the mean as an average

Key stage 3

Introduction

Mathematics is an interconnected subject in which pupils need to be able to move fluently between representations of mathematical ideas. The programme of study for key stage 3 is organised into apparently distinct domains, but pupils should build on key stage 2 and connections across mathematical ideas to develop fluency, mathematical reasoning and competence in solving increasingly sophisticated problems. They should also apply their mathematical knowledge in science, geography, computing and other subjects.

Decisions about progression should 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 in preparation for key stage 4. Those who are not sufficiently fluent should consolidate their understanding, including through additional practice, before moving on.

Working mathematically

Through the mathematics content, pupils should be taught to:

Develop fluency

Reason mathematically

Solve problems

Subject content

Number

Pupils should be taught to:

understand and use place value for decimals, measures and integers of any size

order positive and negative integers, decimals and fractions; use the number line as a model for ordering of the real numbers; use the symbols =, ≠, <, >, ≤, ≥

use the concepts and vocabulary of prime numbers, factors (or divisors), multiples, common factors, common multiples, highest common factor, lowest common multiple, prime factorisation, including using product notation and the unique factorisation property

use the 4 operations, including formal written methods, applied to integers, decimals, proper and improper fractions, and mixed numbers, all both positive and negative

use conventional notation for the priority of operations, including brackets, powers, roots and reciprocals

recognise and use relationships between operations including inverse operations

use integer powers and associated real roots (square, cube and higher), recognise powers of 2, 3, 4, 5 and distinguish between exact representations of roots and their decimal approximations

interpret and compare numbers in standard form A x 10n 1≤A<10, where n is a positive or negative integer or 0

work interchangeably with terminating decimals and their corresponding fractions (such as 3.5 and or 0.375 and )

define percentage as 'number of parts per hundred', interpret percentages and percentage changes as a fraction or a decimal, interpret these multiplicatively, express 1 quantity as a percentage of another, compare 2 quantities using percentages, and work with percentages greater than 100%

interpret fractions and percentages as operators

use standard units of mass, length, time, money and other measures, including with decimal quantities

round numbers and measures to an appropriate degree of accuracy [for example, to a number of decimal places or significant figures]

use approximation through rounding to estimate answers and calculate possible resulting errors expressed using inequality notation a<x≤b

use a calculator and other technologies to calculate results accurately and then interpret them appropriately

appreciate the infinite nature of the sets of integers, real and rational numbers

Algebra

Pupils should be taught to:

use and interpret algebraic notation, including:

ab in place of a × b

3y in place of y + y + y and 3 × y

a² in place of a × a, a³ in place of a × a × a; a²b in place of a × a × b

in place of a ÷ b

coefficients written as fractions rather than as decimals

brackets

substitute numerical values into formulae and expressions, including scientific formulae

understand and use the concepts and vocabulary of expressions, equations, inequalities, terms and factors

simplify and manipulate algebraic expressions to maintain equivalence by:

collecting like terms

multiplying a single term over a bracket

taking out common factors

expanding products of 2 or more binomials

understand and use standard mathematical formulae; rearrange formulae to change the subject

model situations or procedures by translating them into algebraic expressions or formulae and by using graphs

use algebraic methods to solve linear equations in 1 variable (including all forms that require rearrangement)

work with coordinates in all 4 quadrants

recognise, sketch and produce graphs of linear and quadratic functions of 1 variable with appropriate scaling, using equations in x and y and the Cartesian plane

interpret mathematical relationships both algebraically and graphically

reduce a given linear equation in 2 variables to the standard form y = mx + c; calculate and interpret gradients and intercepts of graphs of such linear equations numerically, graphically and algebraically

use linear and quadratic graphs to estimate values of y for given values of x and vice versa and to find approximate solutions of simultaneous linear equations

find approximate solutions to contextual problems from given graphs of a variety of functions, including piece-wise linear, exponential and reciprocal graphs

generate terms of a sequence from either a term-to-term or a position-to-term rule

recognise arithmetic sequences and find the nth term

recognise geometric sequences and appreciate other sequences that arise

Ratio, proportion and rates of change

Pupils should be taught to:

change freely between related standard units [for example time, length, area, volume/capacity, mass]

use scale factors, scale diagrams and maps

express 1 quantity as a fraction of another, where the fraction is less than 1 and greater than 1

use ratio notation, including reduction to simplest form

divide a given quantity into 2 parts in a given part:part or part:whole ratio; express the division of a quantity into 2 parts as a ratio

understand that a multiplicative relationship between 2 quantities can be expressed as a ratio or a fraction

relate the language of ratios and the associated calculations to the arithmetic of fractions and to linear functions

solve problems involving percentage change, including: percentage increase, decrease and original value problems and simple interest in financial mathematics

solve problems involving direct and inverse proportion, including graphical and algebraic representations

use compound units such as speed, unit pricing and density to solve problems

Geometry and measures

Pupils should be taught to:

derive and apply formulae to calculate and solve problems involving: perimeter and area of triangles, parallelograms, trapezia, volume of cuboids (including cubes) and other prisms (including cylinders)

calculate and solve problems involving: perimeters of 2-D shapes (including circles), areas of circles and composite shapes

draw and measure line segments and angles in geometric figures, including interpreting scale drawings

derive and use the standard ruler and compass constructions (perpendicular bisector of a line segment, constructing a perpendicular to a given line from/at a given point, bisecting a given angle); recognise and use the perpendicular distance from a point to a line as the shortest distance to the line

describe, sketch and draw using conventional terms and notations: points, lines, parallel lines, perpendicular lines, right angles, regular polygons, and other polygons that are reflectively and rotationally symmetric

use the standard conventions for labelling the sides and angles of triangle ABC, and know and use the criteria for congruence of triangles

derive and illustrate properties of triangles, quadrilaterals, circles, and other plane figures [for example, equal lengths and angles] using appropriate language and technologies

identify properties of, and describe the results of, translations, rotations and reflections applied to given figures

identify and construct congruent triangles, and construct similar shapes by enlargement, with and without coordinate grids

apply the properties of angles at a point, angles at a point on a straight line, vertically opposite angles

understand and use the relationship between parallel lines and alternate and corresponding angles

derive and use the sum of angles in a triangle and use it to deduce the angle sum in any polygon, and to derive properties of regular polygons

apply angle facts, triangle congruence, similarity and properties of quadrilaterals to derive results about angles and sides, including Pythagoras' Theorem, and use known results to obtain simple proofs

use Pythagoras' Theorem and trigonometric ratios in similar triangles to solve problems involving right-angled triangles

use the properties of faces, surfaces, edges and vertices of cubes, cuboids, prisms, cylinders, pyramids, cones and spheres to solve problems in 3-D

interpret mathematical relationships both algebraically and geometrically

Probability

Pupils should be taught to:

Statistics

Pupils should be taught to:

Key stage 4

This programme of study specifies:

the mathematical content that should be taught to all pupils, in standard type

additional mathematical content to be taught to more highly attaining pupils, in braces { }

Together, the mathematical content set out in the key stage 3 and key stage 4 programmes of study covers the full range of material contained in the GCSE Mathematics qualification. Wherever it is appropriate, given pupils' security of understanding and readiness to progress, pupils should be taught the full content set out in this programme of study.

Working mathematically

Through the mathematics content pupils should be taught to:

Develop fluency

Reason mathematically

Solve problems

Subject content

Number

In addition to consolidating subject content from key stage 3, pupils should be taught to:

apply systematic listing strategies, {including use of the product rule for counting}

{estimate powers and roots of any given positive number}

calculate with roots, and with integer {and fractional} indices

calculate exactly with fractions, {surds} and multiples of π {simplify surd expressions involving squares [for example √12 = √(4 × 3) = √4 × √3 = 2√3] and rationalise denominators}

calculate with numbers in standard form A × 10n, where 1 ≤ A < 10 and n is an integer

{change recurring decimals into their corresponding fractions and vice versa}

identify and work with fractions in ratio problems

apply and interpret limits of accuracy when rounding or truncating, {including upper and lower bounds}

Algebra

In addition to consolidating subject content from key stage 3, pupils should be taught to:

simplify and manipulate algebraic expressions (including those involving surds {and algebraic fractions}) by:

factorising quadratic expressions of the form x² + bx + c, including the difference of 2 squares; {factorising quadratic expressions of the form ax² + bx + c}

simplifying expressions involving sums, products and powers, including the laws of indices

know the difference between an equation and an identity; argue mathematically to show algebraic expressions are equivalent, and use algebra to support and construct arguments {and proofs}

where appropriate, interpret simple expressions as functions with inputs and outputs; {interpret the reverse process as the 'inverse function'; interpret the succession of 2 functions as a 'composite function'}

use the form y = mx + c to identify parallel {and perpendicular} lines; find the equation of the line through 2 given points, or through 1 point with a given gradient

identify and interpret roots, intercepts and turning points of quadratic functions graphically; deduce roots algebraically {and turning points by completing the square}

recognise, sketch and interpret graphs of linear functions, quadratic functions, simple cubic functions, the reciprocal function y = with x ≠ 0, {the exponential function y = kx for positive values of k, and the trigonometric functions (with arguments in degrees) y = sin x, y = cos x and y = tan x for angles of any size}

{sketch translations and reflections of the graph of a given function}

plot and interpret graphs (including reciprocal graphs {and exponential graphs}) and graphs of non-standard functions in real contexts, to find approximate solutions to problems such as simple kinematic problems involving distance, speed and acceleration

{calculate or estimate gradients of graphs and areas under graphs (including quadratic and other non-linear graphs), and interpret results in cases such as distance-time graphs, velocity-time graphs and graphs in financial contexts}

{recognise and use the equation of a circle with centre at the origin; find the equation of a tangent to a circle at a given point}

solve quadratic equations {including those that require rearrangement} algebraically by factorising, {by completing the square and by using the quadratic formula}; find approximate solutions using a graph

solve 2 simultaneous equations in 2 variables (linear/linear {or linear/quadratic}) algebraically; find approximate solutions using a graph

{find approximate solutions to equations numerically using iteration}

translate simple situations or procedures into algebraic expressions or formulae; derive an equation (or 2 simultaneous equations), solve the equation(s) and interpret the solution

solve linear inequalities in 1 {or 2} variable {s}, {and quadratic inequalities in 1 variable}; represent the solution set on a number line, {using set notation and on a graph}

recognise and use sequences of triangular, square and cube numbers, simple arithmetic progressions, Fibonacci type sequences, quadratic sequences, and simple geometric progressions (rn where n is an integer, and r is a positive rational number {or a surd}) {and other sequences}

deduce expressions to calculate the nth term of linear {and quadratic} sequences.

Ratio, proportion and rates of change

In addition to consolidating subject content from key stage 3, pupils should be taught to:

compare lengths, areas and volumes using ratio notation and/or scale factors; make links to similarity (including trigonometric ratios)

convert between related compound units (speed, rates of pay, prices, density, pressure) in numerical and algebraic contexts

understand that X is inversely proportional to Y is equivalent to X is proportional to ; {construct and} interpret equations that describe direct and inverse proportion

interpret the gradient of a straight line graph as a rate of change; recognise and interpret graphs that illustrate direct and inverse proportion

{interpret the gradient at a point on a curve as the instantaneous rate of change; apply the concepts of instantaneous and average rate of change (gradients of tangents and chords) in numerical, algebraic and graphical contexts}

set up, solve and interpret the answers in growth and decay problems, including compound interest {and work with general iterative processes}

Geometry and measures

In addition to consolidating subject content from key stage 3, pupils should be taught to:

interpret and use fractional {and negative} scale factors for enlargements

{describe the changes and invariance achieved by combinations of rotations, reflections and translations}

identify and apply circle definitions and properties, including: centre, radius, chord, diameter, circumference, tangent, arc, sector and segment

{apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results}

construct and interpret plans and elevations of 3D shapes

interpret and use bearings

calculate arc lengths, angles and areas of sectors of circles

calculate surface areas and volumes of spheres, pyramids, cones and composite solids

apply the concepts of congruence and similarity, including the relationships between lengths, {areas and volumes} in similar figures

apply Pythagoras' Theorem and trigonometric ratios to find angles and lengths in right-angled triangles {and, where possible, general triangles} in 2 {and 3} dimensional figures

know the exact values of sin θ and cos θ for θ = 0°, 30°, 45°, 60° and 90°; know the exact value of tan θ for θ = 0°, 30°, 45°, 60°

{know and apply the sine rule, = = , and cosine rule, a² = b² + c² - 2bc cos A, to find unknown lengths and angles}

{know and apply Area = ab sin C to calculate the area, sides or angles of any triangle}

describe translations as 2D vectors

apply addition and subtraction of vectors, multiplication of vectors by a scalar, and diagrammatic and column representations of vectors; {use vectors to construct geometric arguments and proofs}

Probability

In addition to consolidating subject content from key stage 3, pupils should be taught to:

Statistics

In addition to consolidating subject content from key stage 3, pupils should be taught to:

infer properties of populations or distributions from a sample, whilst knowing the limitations of sampling

interpret and construct tables and line graphs for time series data

{construct and interpret diagrams for grouped discrete data and continuous data, ie, histograms with equal and unequal class intervals and cumulative frequency graphs, and know their appropriate use}

interpret, analyse and compare the distributions of data sets from univariate empirical distributions through:

appropriate graphical representation involving discrete, continuous and grouped data, {including box plots}

appropriate measures of central tendency (including modal class) and spread {including quartiles and inter-quartile range}

apply statistics to describe a population

use and interpret scatter graphs of bivariate data; recognise correlation and know that it does not indicate causation; draw estimated lines of best fit; make predictions; interpolate and extrapolate apparent trends whilst knowing the dangers of so doing.

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National curriculum in England: mathematics programmes of study

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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 most forms of employment.

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Purpose of study

English has a pre-eminent place in education and in society. A high-quality education in English will teach pupils to speak and write fluently so that they can communicate their ideas and emotions to others, and through their reading and listening, others can communicate with them. Through reading in particular, pupils have a chance to develop culturally, emotionally, intellectually, socially and spiritually. Literature, especially, plays a key role in such development. Reading also enables pupils both to acquire knowledge and to build on what they already know. All the skills of language are essential to participating fully as a member of society; pupils who do not learn to speak, read and write fluently and confidently are effectively disenfranchised.

Aims

The overarching aim for English in the national curriculum is to promote high standards of language and literacy by equipping pupils with a strong command of the spoken and written language, and to develop their love of literature through widespread reading for enjoyment. The national curriculum for English aims to ensure that all pupils:

read easily, fluently and with good understanding

develop the habit of reading widely and often, for both pleasure and information

acquire a wide vocabulary, an understanding of grammar and knowledge of linguistic conventions for reading, writing and spoken language

appreciate our rich and varied literary heritage

write clearly, accurately and coherently, adapting their language and style in and for a range of contexts, purposes and audiences

use discussion in order to learn; they should be able to elaborate and explain clearly their understanding and ideas

are competent in the arts of speaking and listening, making formal presentations, demonstrating to others and participating in debate

Spoken language

The national curriculum for English reflects the importance of spoken language in pupils' development across the whole curriculum - cognitively, socially and linguistically. Spoken language underpins the development of reading and writing. The quality and variety of language that pupils hear and speak are vital for developing their vocabulary and grammar and their understanding for reading and writing. Teachers should therefore ensure the continual development of pupils' confidence and competence in spoken language and listening skills. Pupils should develop a capacity to explain their understanding of books and other reading, and to prepare their ideas before they write. They must be assisted in making their thinking clear to themselves as well as to others, and teachers should ensure that pupils build secure foundations by using discussion to probe and remedy their misconceptions. Pupils should also be taught to understand and use the conventions for discussion and debate.

All pupils should be enabled to participate in and gain knowledge, skills and understanding associated with the artistic practice of drama. Pupils should be able to adopt, create and sustain a range of roles, responding appropriately to others in role. They should have opportunities to improvise, devise and script drama for one another and a range of audiences, as well as to rehearse, refine, share and respond thoughtfully to drama and theatre performances.

Statutory requirements which underpin all aspects of spoken language across the 6 years of primary education form part of the national curriculum. These are reflected and contextualised within the reading and writing domains which follow.

Reading

The programmes of study for reading at key stages 1 and 2 consist of 2 dimensions:

word reading

comprehension (both listening and reading)

It is essential that teaching focuses on developing pupils' competence in both dimensions; different kinds of teaching are needed for each.

Skilled word reading involves both the speedy working out of the pronunciation of unfamiliar printed words (decoding) and the speedy recognition of familiar printed words. Underpinning both is the understanding that the letters on the page represent the sounds in spoken words. This is why phonics should be emphasised in the early teaching of reading to beginners (ie unskilled readers) when they start school.

Good comprehension draws from linguistic knowledge (in particular of vocabulary and grammar) and on knowledge of the world. Comprehension skills develop through pupils' experience of high-quality discussion with the teacher, as well as from reading and discussing a range of stories, poems and non-fiction. All pupils must be encouraged to read widely across both fiction and non-fiction to develop their knowledge of themselves and the world they live in, to establish an appreciation and love of reading, and to gain knowledge across the curriculum. Reading widely and often increases pupils' vocabulary because they encounter words they would rarely hear or use in everyday speech. Reading also feeds pupils' imagination and opens up a treasure house of wonder and joy for curious young minds.

It is essential that, by the end of their primary education, all pupils are able to read fluently, and with confidence, in any subject in their forthcoming secondary education.

Writing

The programmes of study for writing at key stages 1 and 2 are constructed similarly to those for reading:

transcription (spelling and handwriting)

composition (articulating ideas and structuring them in speech and writing)

It is essential that teaching develops pupils' competence in these 2 dimensions. In addition, pupils should be taught how to plan, revise and evaluate their writing. These aspects of writing have been incorporated into the programmes of study for composition.

Writing down ideas fluently depends on effective transcription: that is, on spelling quickly and accurately through knowing the relationship between sounds and letters (phonics) and understanding the morphology (word structure) and orthography (spelling structure) of words. Effective composition involves articulating and communicating ideas, and then organising them coherently for a reader. This requires clarity, awareness of the audience, purpose and context, and an increasingly wide knowledge of vocabulary and grammar. Writing also depends on fluent, legible and, eventually, speedy handwriting.

Spelling, vocabulary, grammar, punctuation and glossary

The 2 statutory appendices - on spelling and on vocabulary, grammar and punctuation - give an overview of the specific features that should be included in teaching the programmes of study.

Opportunities for teachers to enhance pupils' vocabulary arise naturally from their reading and writing. As vocabulary increases, teachers should show pupils how to understand the relationships between words, how to understand nuances in meaning, and how to develop their understanding of, and ability to use, figurative language. They should also teach pupils how to work out and clarify the meanings of unknown words and words with more than 1 meaning. References to developing pupils' vocabulary are also included in the appendices.

Pupils should be taught to control their speaking and writing consciously and to use Standard English. They should be taught to use the elements of spelling, grammar, punctuation and 'language about language' listed. This is not intended to constrain or restrict teachers' creativity, but simply to provide the structure on which they can construct exciting lessons. A non-statutory glossary is provided for teachers.
Throughout the programmes of study, teachers should teach pupils the vocabulary they need to discuss their reading, writing and spoken language. It is important that pupils learn the correct grammatical terms in English and that these terms are integrated within teaching.

School curriculum

The programmes of study for English are set out year-by-year for key stage 1 and two-yearly for key stage 2. The single year blocks at key stage 1 reflect the rapid pace of development in word reading during these 2 years. Schools are, however, only required to teach the relevant programme of study by the end of the key stage. Within each key stage, schools therefore have the flexibility to introduce content earlier or later than set out in the programme of study. In addition, schools can introduce key stage content during an earlier key stage if appropriate. All schools are also required to set out their school curriculum for English on a year-by-year basis and make this information available online.

Attainment targets

By the end of each key stage, pupils are expected to know, apply and understand the matters, skills and processes specified in the relevant programme of study.

Schools are not required by law to teach the example content in [square brackets] or the content indicated as being 'non-statutory'.

Spoken language - years 1 to 6

Spoken language

Pupils should be taught to:

listen and respond appropriately to adults and their peers

ask relevant questions to extend their understanding and knowledge

use relevant strategies to build their vocabulary

articulate and justify answers, arguments and opinions

give well-structured descriptions, explanations and narratives for different purposes, including for expressing feelings

maintain attention and participate actively in collaborative conversations, staying on topic and initiating and responding to comments

use spoken language to develop understanding through speculating, hypothesising, imagining and exploring ideas

speak audibly and fluently with an increasing command of Standard English

participate in discussions, presentations, performances, role play/improvisations and debates

gain, maintain and monitor the interest of the listener(s)

consider and evaluate different viewpoints, attending to and building on the contributions of others

select and use appropriate registers for effective communication

Notes and guidance (non-statutory)

These statements apply to all years. The content should be taught at a level appropriate to the age of the pupils. Pupils should build on the oral language skills that have been taught in preceding years.

Pupils should be taught to develop their competence in spoken language and listening to enhance the effectiveness of their communication across a range of contexts and to a range of audiences. They should therefore have opportunities to work in groups of different sizes - in pairs, small groups, large groups and as a whole class. Pupils should understand how to take turns and when and how to participate constructively in conversations and debates.

Teachers should also pay attention to increasing pupils' vocabulary, ranging from describing their immediate world and feelings to developing a broader, deeper and richer vocabulary to discuss abstract concepts and a wider range of topics, and enhancing their knowledge about language as a whole.

Pupils should receive constructive feedback on their spoken language and listening, not only to improve their knowledge and skills but also to establish secure foundations for effective spoken language in their studies at primary school, helping them to achieve in secondary education and beyond.

Key stage 1 - year 1

During year 1, teachers should build on work from the early years foundation stage, making sure that pupils can sound and blend unfamiliar printed words quickly and accurately using the phonic knowledge and skills that they have already learnt. Teachers should also ensure that pupils continue to learn new grapheme-phoneme correspondences ( GPCs) and revise and consolidate those learnt earlier. The understanding that the letter(s) on the page represent the sounds in spoken words should underpin pupils' reading and spelling of all words. This includes common words containing unusual GPCs. The term 'common exception words' is used throughout the programmes of study for such words.

Alongside this knowledge of GPCs, pupils need to develop the skill of blending the sounds into words for reading and establish the habit of applying this skill whenever they encounter new words. This will be supported by practice in reading books consistent with their developing phonic knowledge and skill and their knowledge of common exception words. At the same time they will need to hear, share and discuss a wide range of high-quality books to develop a love of reading and broaden their vocabulary.

Pupils should be helped to read words without overt sounding and blending after a few encounters. Those who are slow to develop this skill should have extra practice.

Pupils' writing during year 1 will generally develop at a slower pace than their reading. This is because they need to encode the sounds they hear in words (spelling skills), develop the physical skill needed for handwriting, and learn how to organise their ideas in writing.

Pupils entering year 1 who have not yet met the early learning goals for literacy should continue to follow their school's curriculum for the Early Years Foundation Stage to develop their word reading, spelling and language skills. However, these pupils should follow the year 1 programme of study in terms of the books they listen to and discuss, so that they develop their vocabulary and understanding of grammar, as well as their knowledge more generally across the curriculum. If they are still struggling to decode and spell, they need to be taught to do this urgently through a rigorous and systematic phonics programme so that they catch up rapidly.

Teachers should ensure that their teaching develops pupils' oral vocabulary as well as their ability to understand and use a variety of grammatical structures, giving particular support to pupils whose oral language skills are insufficiently developed.

Year 1 programme of study

Reading - word reading

Pupils should be taught to:

apply phonic knowledge and skills as the route to decode words

respond speedily with the correct sound to graphemes (letters or groups of letters) for all 40+ phonemes, including, where applicable, alternative sounds for graphemes

read accurately by blending sounds in unfamiliar words containing GPCs that have been taught

read common exception words, noting unusual correspondences between spelling and sound and where these occur in the word

read words containing taught GPCs and -s, -es, -ing, -ed, -er and -est endings

read other words of more than one syllable that contain taught GPCs

read words with contractions [for example, I'm, I'll, we'll], and understand that the apostrophe represents the omitted letter(s)

read books aloud, accurately, that are consistent with their developing phonic knowledge and that do not require them to use other strategies to work out words

reread these books to build up their fluency and confidence in word reading

Notes and guidance (non-statutory)

Pupils should revise and consolidate the GPCs and the common exception words taught in reception year. As soon as they can read words comprising the year 1 GPCs accurately and speedily, they should move on to the year 2 programme of study for word reading.

The number, order and choice of exception words taught will vary according to the phonics programme being used. Ensuring that pupils are aware of the GPCs they contain, however unusual these are, supports spelling later.

Young readers encounter words that they have not seen before much more frequently than experienced readers do, and they may not know the meaning of some of these. Practice at reading such words by sounding and blending can provide opportunities not only for pupils to develop confidence in their decoding skills, but also for teachers to explain the meaning and thus develop pupils' vocabulary.

Pupils should be taught how to read words with suffixes by being helped to build on the root words that they can read already. Pupils' reading and rereading of books that are closely matched to their developing phonic knowledge and knowledge of common exception words supports their fluency, as well as increasing their confidence in their reading skills. Fluent word reading greatly assists comprehension, especially when pupils come to read longer books.

Reading - comprehension

Pupils should be taught to:

understand both the books they can already read accurately and fluently and those they listen to by:

drawing on what they already know or on background information and vocabulary provided by the teacher

checking that the text makes sense to them as they read, and correcting inaccurate reading

discussing the significance of the title and events

making inferences on the basis of what is being said and done

predicting what might happen on the basis of what has been read so far

participate in discussion about what is read to them, taking turns and listening to what others say

explain clearly their understanding of what is read to them

Writing - transcription

Pupils should be taught to:

Notes and guidance (non-statutory)

Reading should be taught alongside spelling, so that pupils understand that they can read back words they have spelt.

Pupils should be shown how to segment spoken words into individual phonemes and then how to represent the phonemes by the appropriate grapheme(s). It is important to recognise that phoneme-grapheme correspondences (which underpin spelling) are more variable than grapheme-phoneme correspondences (which underpin reading). For this reason, pupils need to do much more word-specific rehearsal for spelling than for reading.

At this stage pupils will be spelling some words in a phonically plausible way, even if sometimes incorrectly. Misspellings of words that pupils have been taught to spell should be corrected; other misspelt words should be used to teach pupils about alternative ways of representing those sounds.

Writing simple dictated sentences that include words taught so far gives pupils opportunities to apply and practise their spelling.

Handwriting

Pupils should be taught to:

sit correctly at a table, holding a pencil comfortably and correctly

begin to form lower-case letters in the correct direction, starting and finishing in the right place

form capital letters

form digits 0-9

understand which letters belong to which handwriting 'families' (ie letters that are formed in similar ways) and to practise these

Notes and guidance (non-statutory)

Handwriting requires frequent and discrete, direct teaching. Pupils should be able to form letters correctly and confidently. The size of the writing implement (pencil, pen) should not be too large for a young pupil's hand. Whatever is being used should allow the pupil to hold it easily and correctly so that bad habits are avoided.

Left-handed pupils should receive specific teaching to meet their needs.

Writing - composition

Pupils should be taught to:

discuss what they have written with the teacher or other pupils

read their writing aloud, clearly enough to be heard by their peers and the teacher

Notes and guidance (non-statutory)

At the beginning of year 1, not all pupils will have the spelling and handwriting skills they need to write down everything that they can compose out loud.

Pupils should understand, through demonstration, the skills and processes essential to writing: that is, thinking aloud as they collect ideas, drafting, and rereading to check their meaning is clear.

Writing - vocabulary, grammar and punctuation

Pupils should be taught to:

develop their understanding of the concepts set out in English appendix 2 by:

leaving spaces between words

joining words and joining clauses using 'and'

beginning to punctuate sentences using a capital letter and a full stop, question mark or exclamation mark

using a capital letter for names of people, places, the days of the week, and the personal pronoun 'I'

learning the grammar for year 1 in English appendix 2

use the grammatical terminology in English English appendix 2 in discussing their writing

Notes and guidance (non-statutory)

Pupils should be taught to recognise sentence boundaries in spoken sentences and to use the vocabulary listed in English appendix 2 ('Terminology for pupils') when their writing is discussed.

Pupils should begin to use some of the distinctive features of Standard English in their writing. 'Standard English' is defined in the glossary.

Key stage 1 - year 2

By the beginning of year 2, pupils should be able to read all common graphemes. They should be able to read unfamiliar words containing these graphemes, accurately and without undue hesitation, by sounding them out in books that are matched closely to each pupil's level of word-reading knowledge. They should also be able to read many common words containing GPCs taught so far [for example, shout, hand, stop, or dream], without needing to blend the sounds out loud first. Pupils' reading of common exception words [for example, you, could, many, or people], should be secure. Pupils will increase their fluency by being able to read these words easily and automatically. Finally, pupils should be able to retell some familiar stories that have been read to and discussed with them or that they have acted out during year 1.

During year 2, teachers should continue to focus on establishing pupils' accurate and speedy word-reading skills. They should also make sure that pupils listen to and discuss a wide range of stories, poems, plays and information books; this should include whole books. The sooner that pupils can read well and do so frequently, the sooner they will be able to increase their vocabulary, comprehension and their knowledge across the wider curriculum.

In writing, pupils at the beginning of year 2 should be able to compose individual sentences orally and then write them down. They should be able to spell many of the words covered in year 1 correctly - see English appendix 1. They should also be able to make phonically plausible attempts to spell words they have not yet learnt. Finally, they should be able to form individual letters correctly, establishing good handwriting habits from the beginning.

It is important to recognise that pupils begin to meet extra challenges in terms of spelling during year 2. Increasingly, they should learn that there is not always an obvious connection between the way a word is said and the way it is spelt. Variations include different ways of spelling the same sound, the use of so-called silent letters and groups of letters in some words and, sometimes, spelling that has become separated from the way that words are now pronounced, such as the 'le' ending in table. Pupils' motor skills also need to be sufficiently advanced for them to write down ideas that they may be able to compose orally. In addition, writing is intrinsically harder than reading: pupils are likely to be able to read and understand more complex writing (in terms of its vocabulary and structure) than they are capable of producing themselves.

For pupils who do not have the phonic knowledge and skills they need for year 2, teachers should use the year 1 programmes of study for word reading and spelling so that pupils' word-reading skills catch up. However, teachers should use the year 2 programme of study for comprehension so that these pupils hear and talk about new books, poems, other writing, and vocabulary with the rest of the class.

Year 2 programme of study

Reading - word reading

Pupils should be taught to:

continue to apply phonic knowledge and skills as the route to decode words until automatic decoding has become embedded and reading is fluent

read accurately by blending the sounds in words that contain the graphemes taught so far, especially recognising alternative sounds for graphemes

read accurately words of two or more syllables that contain the same graphemes as above

read words containing common suffixes

read further common exception words, noting unusual correspondences between spelling and sound and where these occur in the word

read most words quickly and accurately, without overt sounding and blending, when they have been frequently encountered

read aloud books closely matched to their improving phonic knowledge, sounding out unfamiliar words accurately, automatically and without undue hesitation

reread these books to build up their fluency and confidence in word reading

Notes and guidance (non-statutory)

Pupils should revise and consolidate the GPCs and the common exception words taught in year 1. The exception words taught will vary slightly, depending on the phonics programme being used. As soon as pupils can read words comprising the year 2 GPCs accurately and speedily, they should move on to the years 3 and 4 programme of study for word reading.

When pupils are taught how to read longer words, they should be shown syllable boundaries and how to read each syllable separately before they combine them to read the word.

Pupils should be taught how to read suffixes by building on the root words that they have already learnt. The whole suffix should be taught as well as the letters that make it up.

Pupils who are still at the early stages of learning to read should have ample practice in reading books that are closely matched to their developing phonic knowledge and knowledge of common exception words. As soon as the decoding of most regular words and common exception words is embedded fully, the range of books that pupils can read independently will expand rapidly. Pupils should have opportunities to exercise choice in selecting books and be taught how to do so.

Reading - comprehension

Pupils should be taught to:

develop pleasure in reading, motivation to read, vocabulary and understanding by:

understand both the books that they can already read accurately and fluently and those that they listen to by:

drawing on what they already know or on background information and vocabulary provided by the teacher

checking that the text makes sense to them as they read, and correcting inaccurate reading

making inferences on the basis of what is being said and done

answering and asking questions

predicting what might happen on the basis of what has been read so far

participate in discussion about books, poems and other works that are read to them and those that they can read for themselves, taking turns and listening to what others say

explain and discuss their understanding of books, poems and other material, both those that they listen to and those that they read for themselves

Writing - transcription

Pupils should be taught to:

add suffixes to spell longer words including -ment, -ness, -ful, -less, -ly

apply spelling rules and guidance, as listed in English appendix 1

write from memory simple sentences dictated by the teacher that include words using the GPCs, common exception words and punctuation taught so far

Notes and guidance (non-statutory)

In year 2, pupils move towards more word-specific knowledge of spelling, including homophones. The process of spelling should be emphasised: that is, that spelling involves segmenting spoken words into phonemes and then representing all the phonemes by graphemes in the right order. Pupils should do this both for single-syllable and polysyllabic words.

At this stage, children's spelling should be phonically plausible, even if not always correct. Misspellings of words that pupils have been taught to spell should be corrected; other misspelt words can be used as an opportunity to teach pupils about alternative ways of representing those sounds.

Pupils should be encouraged to apply their knowledge of suffixes from their word reading to their spelling. They should also draw from and apply their growing knowledge of word and spelling structure, as well as their knowledge of root words.

Handwriting

Pupils should be taught to:

form lower-case letters of the correct size relative to one another

start using some of the diagonal and horizontal strokes needed to join letters and understand which letters, when adjacent to one another, are best left unjoined

write capital letters and digits of the correct size, orientation and relationship to one another and to lower-case letters

use spacing between words that reflects the size of the letters

Writing - composition

Pupils should be taught to:

read aloud what they have written with appropriate intonation to make the meaning clear

Notes and guidance (non-statutory)

Reading and listening to whole books, not simply extracts, helps pupils to increase their vocabulary and grammatical knowledge, including their knowledge of the vocabulary and grammar of Standard English. These activities also help them to understand how different types of writing, including narratives, are structured. All these can be drawn on for their writing.

Pupils should understand, through being shown these, the skills and processes essential to writing: that is, thinking aloud as they collect ideas, drafting, and rereading to check their meaning is clear.

Drama and role play can contribute to the quality of pupils' writing by providing opportunities for pupils to develop and order their ideas through playing roles and improvising scenes in various settings.

Pupils might draw on and use new vocabulary from their reading, their discussions about it (one-to-one and as a whole class) and from their wider experiences.

Writing - vocabulary, grammar and punctuation

Pupils should be taught to:

develop their understanding of the concepts set out in English appendix 2 by:

learning how to use both familiar and new punctuation correctly - see English appendix 2, including full stops, capital letters, exclamation marks, question marks, commas for lists and apostrophes for contracted forms and the possessive (singular)

learn how to use:

sentences with different forms: statement, question, exclamation, command

expanded noun phrases to describe and specify [for example, the blue butterfly]

the present and past tenses correctly and consistently, including the progressive form

subordination (using when, if, that, or because) and co-ordination (using or, and, or but)

the grammar for year 2 in English appendix 2

some features of written Standard English

use and understand the grammatical terminology in English appendix 2 in discussing their writing

Lower key stage 2 - years 3 and 4

By the beginning of year 3, pupils should be able to read books written at an age-appropriate interest level. They should be able to read them accurately and at a speed that is sufficient for them to focus on understanding what they read rather than on decoding individual words. They should be able to decode most new words outside their spoken vocabulary, making a good approximation to the word's pronunciation. As their decoding skills become increasingly secure, teaching should be directed more towards developing their vocabulary and the breadth and depth of their reading, making sure that they become independent, fluent and enthusiastic readers who read widely and frequently. They should be developing their understanding and enjoyment of stories, poetry, plays and non-fiction, and learning to read silently. They should also be developing their knowledge and skills in reading non-fiction about a wide range of subjects. They should be learning to justify their views about what they have read: with support at the start of year 3 and increasingly independently by the end of year 4.

Pupils should be able to write down their ideas with a reasonable degree of accuracy and with good sentence punctuation. Teachers should therefore be consolidating pupils' writing skills, their vocabulary, their grasp of sentence structure and their knowledge of linguistic terminology. Teaching them to develop as writers involves teaching them to enhance the effectiveness of what they write as well as increasing their competence. Teachers should make sure that pupils build on what they have learnt, particularly in terms of the range of their writing and the more varied grammar, vocabulary and narrative structures from which they can draw to express their ideas. Pupils should be beginning to understand how writing can be different from speech. Joined handwriting should be the norm; pupils should be able to use it fast enough to keep pace with what they want to say.

Pupils' spelling of common words should be correct, including common exception words and other words that they have learnt - see English appendix 1. Pupils should spell words as accurately as possible using their phonic knowledge and other knowledge of spelling, such as morphology and etymology.

Most pupils will not need further direct teaching of word reading skills: they are able to decode unfamiliar words accurately, and need very few repeated experiences of this before the word is stored in such a way that they can read it without overt sound-blending. They should demonstrate understanding of figurative language, distinguish shades of meaning among related words and use age-appropriate, academic vocabulary.

As in key stage 1, however, pupils who are still struggling to decode need to be taught to do this urgently through a rigorous and systematic phonics programme so that they catch up rapidly with their peers. If they cannot decode independently and fluently, they will find it increasingly difficult to understand what they read and to write down what they want to say. As far as possible, however, these pupils should follow the year 3 and 4 programme of study in terms of listening to new books, hearing and learning new vocabulary and grammatical structures, and discussing these.

Specific requirements for pupils to discuss what they are learning and to develop their wider skills in spoken language form part of this programme of study. In years 3 and 4, pupils should become more familiar with and confident in using language in a greater variety of situations, for a variety of audiences and purposes, including through drama, formal presentations and debate.

Years 3 and 4 programme of study

Reading - word reading

Pupils should be taught to:

apply their growing knowledge of root words, prefixes and suffixes (etymology and morphology) as listed in - see English appendix 1 , both to read aloud and to understand the meaning of new words they meet

read further exception words, noting the unusual correspondences between spelling and sound, and where these occur in the word

Notes and guidance (non-statutory)

At this stage, teaching comprehension should be taking precedence over teaching word reading directly. Any focus on word reading should support the development of vocabulary.

When pupils are taught to read longer words, they should be supported to test out different pronunciations. They will attempt to match what they decode to words they may have already heard but may not have seen in print (for example, in reading technical, the pronunciation /tɛtʃnɪkəl/ ('tetchnical') might not sound familiar, but /tɛknɪkəl/ ('teknical') should).

Reading - comprehension

Pupils should be taught to:

develop positive attitudes to reading, and an understanding of what they read, by:

retrieve and record information from non-fiction

participate in discussion about both books that are read to them and those they can read for themselves, taking turns and listening to what others say

Writing - transcription

Pupils should be taught to:

use further prefixes and suffixes and understand how to add them - see English appendix 1

spell further homophones

spell words that are often misspelt - see English appendix 1

place the possessive apostrophe accurately in words with regular plurals [for example, girls', boys'] and in words with irregular plurals [for example, children's]

use the first 2 or 3 letters of a word to check its spelling in a dictionary

write from memory simple sentences, dictated by the teacher, that include words and punctuation taught so far

Notes and guidance (non-statutory)

Pupils should learn to spell new words correctly and have plenty of practice in spelling them.

As in years 1 and 2, pupils should continue to be supported in understanding and applying the concepts of word structure - see English appendix 2.

Pupils need sufficient knowledge of spelling in order to use dictionaries efficiently.

Handwriting

Pupils should be taught to:

use the diagonal and horizontal strokes that are needed to join letters and understand which letters, when adjacent to one another, are best left unjoined

increase the legibility, consistency and quality of their handwriting, [for example, by ensuring that the downstrokes of letters are parallel and equidistant, and that lines of writing are spaced sufficiently so that the ascenders and descenders of letters do not touch]

Writing - composition

Pupils should be taught to:

plan their writing by:

discussing writing similar to that which they are planning to write in order to understand and learn from its structure, vocabulary and grammar

discussing and recording ideas

draft and write by:

composing and rehearsing sentences orally (including dialogue), progressively building a varied and rich vocabulary and an increasing range of sentence structures English appendix 2

organising paragraphs around a theme

in narratives, creating settings, characters and plot

in non-narrative material, using simple organisational devices [for example, headings and sub-headings]

evaluate and edit by:

assessing the effectiveness of their own and others' writing and suggesting improvements

proposing changes to grammar and vocabulary to improve consistency, including the accurate use of pronouns in sentences

proofread for spelling and punctuation errors

read their own writing aloud to a group or the whole class, using appropriate intonation and controlling the tone and volume so that the meaning is clear

Notes and guidance (non-statutory)

Pupils should continue to have opportunities to write for a range of real purposes and audiences as part of their work across the curriculum. These purposes and audiences should underpin the decisions about the form the writing should take, such as a narrative, an explanation or a description.

Pupils should understand, through being shown these, the skills and processes that are essential for writing: that is, thinking aloud to explore and collect ideas, drafting, and rereading to check their meaning is clear, including doing so as the writing develops. Pupils should be taught to monitor whether their own writing makes sense in the same way that they monitor their reading, checking at different levels.

Writing - vocabulary, grammar and punctuation

Pupils should be taught to:

develop their understanding of the concepts set out in English appendix 2 by:

use and understand the grammatical terminology in English appendix 2 accurately and appropriately when discussing their writing and reading

Notes and guidance (non-statutory)

Grammar should be taught explicitly: pupils should be taught the terminology and concepts set out in English appendix 2, and be able to apply them correctly to examples of real language, such as their own writing or books that they have read.

At this stage, pupils should start to learn about some of the differences between Standard English and non-Standard English and begin to apply what they have learnt, for example, in writing dialogue for characters.

Upper key stage 2 - years 5 and 6

By the beginning of year 5, pupils should be able to read aloud a wider range of poetry and books written at an age-appropriate interest level with accuracy and at a reasonable speaking pace. They should be able to read most words effortlessly and to work out how to pronounce unfamiliar written words with increasing automaticity. If the pronunciation sounds unfamiliar, they should ask for help in determining both the meaning of the word and how to pronounce it correctly.

They should be able to prepare readings, with appropriate intonation to show their understanding, and should be able to summarise and present a familiar story in their own words. They should be reading widely and frequently, outside as well as in school, for pleasure and information. They should be able to read silently, with good understanding, inferring the meanings of unfamiliar words, and then discuss what they have read.

Pupils should be able to write down their ideas quickly. Their grammar and punctuation should be broadly accurate. Pupils' spelling of most words taught so far should be accurate and they should be able to spell words that they have not yet been taught by using what they have learnt about how spelling works in English.

During years 5 and 6, teachers should continue to emphasise pupils' enjoyment and understanding of language, especially vocabulary, to support their reading and writing. Pupils' knowledge of language, gained from stories, plays, poetry, non-fiction and textbooks, will support their increasing fluency as readers, their facility as writers, and their comprehension. As in years 3 and 4, pupils should be taught to enhance the effectiveness of their writing as well as their competence.

It is essential that pupils whose decoding skills are poor are taught through a rigorous and systematic phonics programme so that they catch up rapidly with their peers in terms of their decoding and spelling. However, as far as possible, these pupils should follow the upper key stage 2 programme of study in terms of listening to books and other writing that they have not come across before, hearing and learning new vocabulary and grammatical structures, and having a chance to talk about all of these.

By the end of year 6, pupils' reading and writing should be sufficiently fluent and effortless for them to manage the general demands of the curriculum in year 7, across all subjects and not just in English, but there will continue to be a need for pupils to learn subject-specific vocabulary. They should be able to reflect their understanding of the audience for and purpose of their writing by selecting appropriate vocabulary and grammar. Teachers should prepare pupils for secondary education by ensuring that they can consciously control sentence structure in their writing and understand why sentences are constructed as they are. Pupils should understand nuances in vocabulary choice and age-appropriate, academic vocabulary. This involves consolidation, practice and discussion of language.

Specific requirements for pupils to discuss what they are learning and to develop their wider skills in spoken language form part of this programme of study. In years 5 and 6, pupils' confidence, enjoyment and mastery of language should be extended through public speaking, performance and debate.

Years 5 and 6 programme of study

Reading - word reading

Pupils should be taught to:

apply their growing knowledge of root words, prefixes and suffixes (morphology and etymology), as listed in English appendix 1, both to read aloud and to understand the meaning of new words that they meet

Notes and guidance (non-statutory)

At this stage, there should be no need for further direct teaching of word-reading skills for almost all pupils. If pupils are struggling or failing in this, the reasons for this should be investigated. It is imperative that pupils are taught to read during their last two years at primary school if they enter year 5 not being able to do so.

Pupils should be encouraged to work out any unfamiliar word. They should focus on all the letters in a word so that they do not, for example, read 'invitation' for 'imitation' simply because they might be more familiar with the first word. Accurate reading of individual words, which might be key to the meaning of a sentence or paragraph, improves comprehension.

When teachers are reading with or to pupils, attention should be paid to new vocabulary - both a word's meaning(s) and its correct pronunciation.

Reading - comprehension

Pupils should be taught to:

maintain positive attitudes to reading and an understanding of what they read by:

discuss and evaluate how authors use language, including figurative language, considering the impact on the reader

distinguish between statements of fact and opinion

retrieve, record and present information from non-fiction

participate in discussions about books that are read to them and those they can read for themselves, building on their own and others' ideas and challenging views courteously

explain and discuss their understanding of what they have read, including through formal presentations and debates, maintaining a focus on the topic and using notes where necessary

provide reasoned justifications for their views

Writing - transcription

Pupils should be taught to:

use further prefixes and suffixes and understand the guidance for adding them

spell some words with 'silent' letters [for example, knight, psalm, solemn]

continue to distinguish between homophones and other words which are often confused

use knowledge of morphology and etymology in spelling and understand that the spelling of some words needs to be learnt specifically, as listed in English appendix 1

use dictionaries to check the spelling and meaning of words

use the first 3 or 4 letters of a word to check spelling, meaning or both of these in a dictionary

use a thesaurus

Handwriting and presentation

Pupils should be taught to:

write legibly, fluently and with increasing speed by:

choosing which shape of a letter to use when given choices and deciding whether or not to join specific letters

choosing the writing implement that is best suited for a task

Writing - composition

Pupils should be taught to:

draft and write by:

selecting appropriate grammar and vocabulary, understanding how such choices can change and enhance meaning

in narratives, describing settings, characters and atmosphere and integrating dialogue to convey character and advance the action

précising longer passages

using a wide range of devices to build cohesion within and across paragraphs

using further organisational and presentational devices to structure text and to guide the reader [for example, headings, bullet points, underlining]

proofread for spelling and punctuation errors

perform their own compositions, using appropriate intonation, volume, and movement so that meaning is clear

Writing - vocabulary, grammar and punctuation

Pupils should be taught to:

develop their understanding of the concepts set out in English appendix 2 by:

recognising vocabulary and structures that are appropriate for formal speech and writing, including subjunctive forms

using passive verbs to affect the presentation of information in a sentence

using the perfect form of verbs to mark relationships of time and cause

using expanded noun phrases to convey complicated information concisely

using modal verbs or adverbs to indicate degrees of possibility

using relative clauses beginning with who, which, where, when, whose, that or with an implied (ie omitted) relative pronoun

learning the grammar for years 5 and 6 in English appendix 2

use and understand the grammatical terminology in English appendix 2 accurately and appropriately in discussing their writing and reading

Key stage 3

Subject content

Reading

Pupils should be taught to:

Writing

Pupils should be taught to:

plan, draft, edit and proofread through:

considering how their writing reflects the audiences and purposes for which it was intended

amending the vocabulary, grammar and structure of their writing to improve its coherence and overall effectiveness

paying attention to accurate grammar, punctuation and spelling; applying the spelling patterns and rules set out in English appendix 1 to the key stage 1 and 2 programmes of study for English

Grammar and vocabulary

Pupils should be taught to:

consolidate and build on their knowledge of grammar and vocabulary through:

extending and applying the grammatical knowledge set out in English appendix 2 to the key stage 1 and 2 programmes of study to analyse more challenging texts

studying the effectiveness and impact of the grammatical features of the texts they read

drawing on new vocabulary and grammatical constructions from their reading and listening, and using these consciously in their writing and speech to achieve particular effects

knowing and understanding the differences between spoken and written language, including differences associated with formal and informal registers, and between Standard English and other varieties of English

using Standard English confidently in their own writing and speech

discussing reading, writing and spoken language with precise and confident use of linguistic and literary terminology*

*Teachers should refer to the glossary that accompanies the programmes of study for English for their own information on the range of terms used within the programmes of study as a whole.

Spoken English

Pupils should be taught to:

Key stage 4

Subject content

Purpose of study

English has a pre-eminent place in education and in society. A high-quality education in English will teach pupils to speak and write fluently so that they can communicate their ideas and emotions to others and through their reading and listening, others can communicate with them. Through reading in particular, pupils have a chance to develop culturally, emotionally, intellectually, socially and spiritually. Literature, especially, plays a key role in such development. Reading also enables pupils both to acquire knowledge and to build on what they already know. All the skills of language are essential to participating fully as a member of society; pupils, therefore, who do not learn to speak, read and write fluently and confidently are effectively disenfranchised.

Aims

The overarching aim for English in the national curriculum is to promote high standards of language and literacy by equipping pupils with a strong command of the spoken and written word, and to develop their love of literature through widespread reading for enjoyment. The national curriculum for English aims to ensure that all pupils:

read easily, fluently and with good understanding

develop the habit of reading widely and often, for both pleasure and information

acquire a wide vocabulary, an understanding of grammar and knowledge of linguistic conventions for reading, writing and spoken language

appreciate our rich and varied literary heritage

write clearly, accurately and coherently, adapting their language and style in and for a range of contexts, purposes and audiences

use discussion in order to learn; they should be able to elaborate and explain clearly their understanding and ideas

are competent in the arts of speaking and listening, making formal presentations, demonstrating to others and participating in debate

Spoken language

The national curriculum for English reflects the importance of spoken language in pupils' development across the whole curriculum - cognitively, socially and linguistically. Spoken language continues to underpin the development of pupils' reading and writing during key stage 4 and teachers should therefore ensure pupils' confidence and competence in this area continue to develop. Pupils should be taught to understand and use the conventions for discussion and debate, as well as continuing to develop their skills in working collaboratively with their peers to discuss reading, writing and speech across the curriculum.

Reading and writing

Reading at key stage 4 should be wide, varied and challenging. Pupils should be expected to read whole books, to read in depth and to read for pleasure and information.

Pupils should continue to develop their knowledge of and skills in writing, refining their drafting skills and developing resilience to write at length. They should be taught to write formal and academic essays as well as writing imaginatively. They should be taught to write for a variety of purposes and audiences across a range of contexts. This requires an increasingly wide knowledge of vocabulary and grammar.

Opportunities for teachers to enhance pupils' vocabulary will arise naturally from their reading and writing. Teachers should show pupils how to understand the relationships between words, how to understand nuances in meaning, and how to develop their understanding of, and ability to use, figurative language.

Pupils should be taught to control their speaking and writing consciously, understand why sentences are constructed as they are and to use Standard English. They should understand and use age-appropriate vocabulary, including linguistic and literary terminology, for discussing their reading, writing and spoken language. This involves consolidation, practice and discussion of language. It is important that pupils learn the correct grammatical terms in English and that these terms are integrated within teaching.

Teachers should build on the knowledge and skills that pupils have been taught at key stage 3. Decisions about progression should be based on the security of pupils' linguistic knowledge, skills and understanding and their readiness to progress to the next stage. Pupils whose linguistic development is more advanced should be challenged through being offered opportunities for increased breadth and depth in reading and writing. Those who are less fluent should consolidate their knowledge, understanding and skills, including through additional practice.

Glossary

A non-statutory glossary is provided for teachers.

Reading

Pupils should be taught to:

read and appreciate the depth and power of the English literary heritage through:

reading a wide range of high-quality, challenging, classic literature and extended literary non-fiction, such as essays, reviews and journalism. This writing should include whole texts. The range will include:

at least one play by Shakespeare

works from the 19th, 20th and 21st centuries

poetry since 1789, including representative Romantic poetry

re-reading literature and other writing as a basis for making comparisons

choosing and reading books independently for challenge, interest and enjoyment

understand and critically evaluate texts through:

make an informed personal response, recognising that other responses to a text are possible and evaluating these

Writing

Pupils should be taught to:

write accurately, fluently, effectively and at length for pleasure and information through:

make notes, draft and write, including using information provided by others [e.g. writing a letter from key points provided; drawing on and using information from a presentation]

revise, edit and proof-read through:

Grammar and vocabulary

Pupils should be taught to:

Spoken English

Pupils should be taught to:

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National curriculum in England: English programmes of study

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English has a pre-eminent place in education and in society. A high-quality education in English will teach pupils to speak and write fluently so that they can communicate their ideas and emotions to others, and through their reading and listening, others can communicate with them.

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Our diagram shows a school's journey towards mastery. Use it to determine where your school's journey starts ( click/tap diagram to show a larger version with more detail of the journey).

If your school is ready to start developing a teaching for mastery approach, you can join a Work Group starting in the new academic year.

During the 2022/23 school year, Maths Hubs across England will be working with thousands of primary schools to continue the spread of teaching for mastery across England.

The Teaching for Mastery Programme is a professional development opportunity designed to support teachers like you to develop best practice in maths in your school. It is suitable for schools interested in implementing a teaching for mastery approach to maths.

The video below was made for a previous recruitment campaign. The details still apply, with the exception of the information about matched textbook funding, as this is no longer available.

What is involved?

Two teachers from each participating school join a Work Group, consisting of six or seven local primary schools. Each Work Group is led by a trained Primary Mastery Specialist. Work Groups meet regularly to plan, observe and discuss teaching for mastery. In between meetings, teachers explore mastery approaches in their own classrooms and across their school.

Support is provided from a local classroom-based Mastery Specialist who leads the group. This model of professional development involves hands-on learning and peer-to-peer support. It is evidence-based and designed to support substantial long-term change.

Who can take part?

This programme is for state-funded primary schools in England. Two teachers from each participant school will attend six half-day meetings during the school year, and lead teaching for mastery development in their own school.

NB: Schools can only take part in this particular Work Group once. However, there are other Work Groups dealing with teaching for mastery which are available in the year after a school is in a Development Work Group and beyond. Talk to your Maths Hub about your options.

What is the cost?

The Primary Teaching for Mastery Development Work Groups project is fully funded by the Maths Hubs Programme so is free to participating schools.

What will you learn?

Teachers tell us about what involvement has meant for them and their pupils:

Alana Mead, Year 6 teacher at St Francis C of E Primary School in Swindon

Being involved in a Work Group, you feel much more involved in your own CPD. You are actually doing something, rather than being talked at.

Going away, implementing something and then coming back and discussing how it worked is great. Schools working together is so powerful. We hear what another school has done successfully and think 'Right, we will give that a try.'

Teachers involved really benefit from the time and freedom to spread the ideas of mastery throughout their school. Taking part is a commitment but it benefits the individual and the school, so I would definitely advise anyone to do it.

Calum Boothroyd, Year 5 teacher, Swindon Academy's Primary Phase

As an NQT, it has been so useful for me to speak to other teachers at the TRG about how they teach maths. In an hour and a half this morning, I've already got so many ideas to take back.

I feel like it has been a springboard for me. It gives me further insight and a variety of different ideas which I can take away. It has helped me develop my own subject knowledge too. I've now got more confidence and belief that I can lead on things.

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Primary Teaching for Mastery - Development

# data[].attributes.attachment_link.description
Our diagram shows a school's journey towards mastery. Use it to determine where your school's journey starts ( click/tap diagram to show a larger version with more detail of the journey). If your school is ready to start developing a teaching for mastery approach, you can join a Work Group starting in the new academic year.

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Here is an example - Phonics - guidance for schools

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Here is an example - Phonics - guidance for schools

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Relevance: This Policy underpins my schools approach. We.........

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Teaching phonics: information for schools

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Documents on effective phonics teaching strategies.