## Get to know Mathematics and Numeracy

This overview should be read together with A guide to Curriculum for Wales 2022

## Introduction

Mathematics is an international discipline, and numeracy, the application of mathematics, plays a critical part in our private, social and civic lives, and in the economic health of the nation.

It is imperative that mathematics and numeracy experiences are as engaging, exciting and accessible as possible for learners, and that they ensure that learners develop mathematical resilience (the ability to embrace challenge asa positive aspect of learning). Developing mathematical resilience contributes to the development of ambitious and capable learners.

In the early years, play forms an important part in the development of mathematics and numeracy, enabling learners to solve problems, explore ideas, establish connections and collaborate with others. In later years, learners need to have opportunities to work both independently and collaboratively to build on the foundations established in the early years.

For learners of all ages, real-life examples drawn from the local, national and international environment help them make connections between the concrete and the abstract. Real-life contexts can be used to introduce and explore mathematical concepts, as well as to consolidate them. Indeed, teaching that introduces a reasoning and problem-solving approach to all mathematics and numeracy experiences supports the development both of positive dispositions and of the four purposes of the curriculum, as well as the development of the mathematical proficiencies.

### A transformational curriculum

The White Paper Our National Mission: A Transformational Curriculum set out the detailed legislative proposals for Curriculum for Wales 2022.

The proposal is that funded non-maintained settings and schools will be required to provide a broad and balanced curriculum that meets the four purposes of the curriculum, and comprises the six areas of learning and experience. There will be statutory duties to teach Welsh, English, religious education, relationships and sexuality education, and the three cross-curricular responsibilities of literacy, numeracy and digital competence. Further information on how the Mathematics and Numeracy Area of Learning and Experience can support this is provided in the ‘Developing a broad and balanced curriculum’ section of this document.

Funded non-maintained settings and schools will have discretion as to how they design their school-level curriculum to meet their curriculum duties. However, in considering the exercise of that discretion, they must have regard to statutory guidance issued by Welsh Ministers. In practice, that means they should follow the statutory guidance unless they have good reason not to.

This statutory guidance for the Mathematics and Numeracy Area of Learning and Experience, which forms part of the wider Curriculum for Wales 2022 statutory guidance, is intended to provide a national framework that funded non‑maintained settings and schools can build on to develop their own curriculum. It is not intended to be a comprehensive or exhaustive syllabus, nor a guide for organising timetables. It sets out:

- what funded non-maintained settings and schools should take into account in designing their curriculum and how it could be structured
- the broad expectations for learners for the Mathematics and Numeracy Area of Learning and Experience at each progression step.

## Supporting the four purposes of the curriculum

The development of mathematics has always gone hand in hand with the development of civilisation itself. Mathematics surrounds us and underpins so many aspects of our daily lives, such as architecture, art, music, money and engineering. And while it is creative and beautiful, both in its own right and in its applications, it is also essential for progress in other areas of learning and experience, not least in Science and Technology, which would be virtually impossible without it. What is more, numeracy – the use of mathematics to solve problems in real-world contexts – is required in almost all areas of life.

Formal mathematics is founded on basic truths and develops through rigorous logical reasoning. It involves inventing or discovering abstract objects and establishing the relationships between them. It also teaches us the difference between conjecture, likelihood and proof.

Mathematical thinking involves applying similarly logical reasoning, this time to the investigation of relations within and between concepts, along with justifying and proving findings. Indeed, understanding mathematical concepts and being able to apply and reason with the abstract representations of concepts is central to learning mathematics. And essential to this is comprehension of, and proficiency with, the symbols and symbol systems used in mathematics.

Applying mathematics requires strategic competence in the use of abstraction and modelling, and learners also develop resilience, as well as a sense of achievement and enjoyment, as they overcome the challenges involved. Subsequently, mathematical activities teach learners not to be afraid of unfamiliar or complex problems, as they can be reduced to a succession of simpler problems and, eventually, to basic computations. And as they reflect on the approaches used, and on their own mathematics and numeracy learning, learners develop metacognitive skills which help them know which steps to take to improve performance. Thus they become **ambitious, capable learners, ready to learn throughout their lives**.

Mathematics also contributes to developing **enterprising, creative contributors, ready to play a full part in life and work**. It encourages learners to be creative because it requires them to play, experiment, take risks and be flexible in tackling mathematical problems. Because mathematics is essentially abstract, it teaches learners to operate with objects that do not physically exist, using and developing their creativity to imagine and discover new realities.It also supports numerical modelling and forecasting to encourage entrepreneurial thinking.

Mathematics promotes **ethical, informed citizens of Wales and the world** by providing learners with tools to analyse data critically, enabling them to develop informed views on social, political, economic and environmental issues. It encourages clarity of thinking, allowing learners to understand and make reasoned decisions.

In mathematics and numeracy, learners encounter contexts involving health and personal finance, and develop the skills needed to manage their own finances, make informed decisions and become critical consumers. They learn to interpret information and data to assess risk, and they use their numeracy skills across the curriculum to make effective choices, becoming **healthy, confident individuals, ready to lead fulfilling lives as valued members of society**.

## Relationships between what matters statements

The different areas of mathematics are highly interconnected and dependent on one another, and, as any teacher of mathematics knows, concepts are built up over time, drawing on prior knowledge and learning, often from more than one area of mathematics. What is important when planning to teach any specific topic is to work out the prior knowledge the learners need in order to be able to access and understand the new topic.

Algebra, geometry and statistics cannot be understood without a prior understanding of number and consistent reference to numbers, calculations and the number system. As learners progress, they learn to see numerical expressions as relational rather than computational, e.g. a computation such as 2 + 8 = 10, and that this is the basis for deriving other facts, e.g. 8 + 2 = 10, 8 = 10 – 2, and so on. This lays the foundations for using algebraic symbolisation successfully. Making connections between arithmetic and algebra helps to develop tools and skills for abstract reasoning from an early age.

Measure is an aspect of geometrical thinking which is closely connected to number, and much of the development of understanding of number can emerge through increasingly sophisticated measuring. Geometric thinking involves reasoning with proportion, which connects with development in number work; it also involves transforming shapes, which relates to the use of functions and mapping in algebra.

Probability is expressed through number in various ways, using percentages, fractions and decimals, and an understanding of the different representations, and the connections between them, is necessary for effective expression of probability. Statistics involves manipulation, representation and interpretation of data, which in turn require numerical and geometric thinking.

## Progression

Progression in Mathematics and Numeracy involves the development of the following interconnected and interdependent proficiencies.

- Conceptual understanding; having the knowledge to understand and explain a mathematical concept.
- Communication with symbols.
- Strategic competence (i.e. formulating problems mathematically in order to solve them).
- Logical reasoning.
- Fluency.

A key point to note in any planning and assessment of Mathematics and Numeracy is that the proficiencies are interconnected and interdependent; they cannot be seen as hierarchical (e.g. strategic competence does not come after learning facts and techniques) and they can be developed alongside each other. However, it may be helpful to target certain proficiencies at certain points, if this is appropriate to ensure progression.

### Contextual example 1: Learning about fractions – the case of ½

#### Conceptual understanding

Understanding that a half is a result of dividing something into two equal parts. This could be through connecting their concrete and/or real-life experiences of partitioning objects and numbers into equal parts with images (e.g. pictures and images on the number line) and the abstract representation of a ½ using the symbolic notation. A learner who understands what a half is might be able to give real-life or visual examples, and would also be able to explain why something might not be a half (e.g. a pizza cut into two parts which are not equal).

#### Communicating with symbols

Understanding the convention of how a half is written and what the symbols mean. This could also involve linking division and fractions (i.e. ½ = 1 ÷ 2) and using terms such as numerator and denominator.

#### Strategic competence

Being able to recognise real-life situations which involve a half; being able to represent these mathematically; being able to model situations involving halving mathematically; using pictures/images and language and symbols.

#### Fluency

Being able to count in steps of a half and being able to begin to recall halves of numbers.

#### Logical reasoning

Being able to understand the relationship between a half and a whole; being able to justify why 2/4_{ }is also a half. Being able to reason that ½ + ½ = 1, ½ x 2 = 1 and 1 ÷ 2 = ½. Being able to justify why there may be many ways of splitting a shape in half.

Through connecting these proficiencies within a learner’s experience of ½, learners should develop a deep understanding of a half as an example of a fraction. The way in which these foci are introduced could vary, but ultimately, having opportunities to explore and connect these proficiencies should ensure learner progression within this idea.

## Developing a broad and balanced curriculum

### Literacy, numeracy and digital competence

The cross-curricular responsibilities of literacy, numeracy and digital competence support almost all learning and are essential for learners to be able to participate successfully and confidently in the modern world.

#### Literacy

Literacy can be developed through engaging and accessible experiences where learners are given regular opportunities to describe, explain and justify their understanding of various mathematical concepts, using appropriate mathematical vocabulary. The development of these skills can be seen in rhymes and songs through to discussions around abstract concepts. Learners will also use and develop their literacy skills in a written form in order to describe mathematical processes, such as reasoning.

Learners will increasingly use literacy skills in order to understand a range of calculation strategies, describing visualisation of shapes, studying and interpreting information in statistics and comparing alternative methods before arriving at a solution to a mathematical problem. They should use these literacy skills as they encounter practical, real-world problems.

#### Numeracy

Mathematics and Numeracy, by definition, has numeracy at its heart.

Numeracy involves applying and connecting the five mathematical proficiencies in a range of real-life contexts, within the Mathematics and Numeracy Area of Learning and Experience, and the wider curriculum.

Real-life contexts can be used to introduce and explore mathematical concepts, as well as to consolidate them. For example, the use of percentages can be applied to annual percentage rates (APRs) to demonstrate their application to financial literacy.

#### Digital competence

Digital approaches will enhance learners’ mathematical and numeracy skills across a range of situations that will naturally occur within the area of learning and experience. Digital competence is more than the interaction with technology. For example, collaborating to solve a problem and the development of algorithms to support the understanding of patterns will provide the opportunity to support the development of learner’s digital skills. Creating a graph by using a spreadsheet, for example, will enhance digital understanding and also strengthen learners’ mathematical and numerical skills.

Naturally, as learners develop and progress, they will increasingly use more complex digital skills, and processes, techniques and systems to create solutions to address specific problems, opportunities or needs. Aspects of collection, representation and analysis, for example, will become more sophisticated as learners progress.

### Welsh language

The number system in Welsh (Cymraeg) is an area that highlights the distinctive nature of mathematics in a Welsh context. In the early years, in particular, it would be helpful if teachers could consider teaching learners the traditional (vigesimal) way of counting in Welsh, as well as the more modern way. The decision as to when to introduce it, however, should be left to individual schools, to reflect the context of the school. Children from homes/communities where Welsh is widely spoken, for example, may already know these from an early age, whereas for English-medium schools and some Welsh-medium schools, it could be taught when learning, to tell the time or date.

The Welsh number system, as with other Celtic and some European languages, traditionally used the vigesimal system, using a base of 20, instead of the modern (and common) base of 10. In comparatively recent times a decimal system, in common with other Indo-European languages, was set up to simplify the teaching of number. (For example, in the old system 11 was ‘un ar ddeg’ [1+10] and 14 was pedwar ar ddeg [4+10]. In the ‘new’ system, 11 is ‘un deg un’ [10+1] and 14 is ‘un deg pedwar’ [10+4]).

### Welsh dimension and international perspective

The Welsh dimension and international perspective elements will enable learners to understand the connection between mathematics (and numeracy) and authentic real-world contexts that span across both Wales and the world. Mathematics is a universal language, but in order for learners to make sense of this language, and to understand mathematical concepts, it helps to provide examples rooted in everyday life. Learners, especially young learners, are often unable to think abstractly, and find it easier to learn mathematics in more concrete terms.

Each school in Wales will have a unique environment in which they work, and schools should explore local sources and resources that might have a mathematics application. The Learned Society of Wales refers to Wales providing ‘practical contexts through which procedure, theories, and/or principles are given concrete form through examples, case study, and illustrations of real-life applications’ (The Learned Society of Wales, 2018). Emeritus Professor Gareth Ffowc Roberts, in his submission to the Learned Society paper, stated that ‘while there may be no such thing as ‘Welsh mathematics/numeracy’ there are very particular Welsh ways of experiencing mathematics/numeracy’. The challenge is to teach and learn mathematics and numeracy in Wales so that children gain ownership of them, and perceive them as being a natural part of their culture rather than believing that they are copying another community’s culture. Various local and national organisations (e.g. the National Museum of Wales and CADW) have already developed mathematics teaching resources. Since April 2019 the Welsh Government has had responsibility for deciding the rates of Income Tax paid by Welsh taxpayers. Schools are encouraged to use Welsh examples when teaching financial matters, and specifically about forms of taxation, highlighting the link between Mathematics and Numeracy, the real world and the responsibility Welsh Government has for determining Income Tax levels in Wales.

There are particular contributions to the field of mathematics from Wales and internationally that could be used to support understanding of mathematical concepts and conventions, and also to support understanding of the development of mathematics as a body of knowledge. Wales has a proud history of producing outstanding mathematicians such as Robert Recorde and William Jones. Schools should consider every opportunity to highlight their achievements and, hopefully, inspire learners to become mathematicians themselves. Using international examples from a range of cultures could enable learners to understand the history of mathematics and its development into an international and universally applicable language. This could also promote and support cross-curricular learning.

The Mathematics and Numeracy Area of Learning and Experience has developed a set of mathematical proficiencies which permeate every aspect of what matters statements and have shaped progression in the area of learning and experience. These proficiencies build on the work of Kilpatrick et al. and elsewhere (such as the Australian curriculum), but have been adapted to the Welsh context by the pioneer group following considerable work with experts. These proficiencies are fundamental to learning within the discipline and will be part of a recognisably Welsh identity of the emerging Mathematics and Numeracy curriculum.

### Wider skills

The Mathematics and Numeracy Area of Learning and Experience provides opportunities for learners to develop all four of the following wider skills.

#### Critical thinking and problem-solving

The development of logical and critical thinking underpins learning in mathematics. Mathematics teaches us problem-solving skills, which transfer to all areas of the curriculum, to life in general and to the world of work. The work of mathematics involves solving problems, beginning by analysing requirements, asking questions and evaluating information. In the development of solutions, learners identify potential approaches and develop arguments, justifying their decisions.

#### Planning and organisation

Mathematical thinking requires learners to be organised and, as they progress through school, their organisational skills will develop, particularly as they plan and implement the data-handling cycle. In their mathematical problem-solving, learners are encouraged to predict and estimate solutions and then to check their answers, reflect on their results and evaluate their approaches. Increasing confidence in decision-making for mathematical problem-solving supports learners to be more aspirational in setting goals and challenges for themselves.

#### Creativity and innovation

Mathematical working requires and develops creativity and curiosity which also transfer to other aspects of life. Frequently in mathematical problem-solving the learner does not immediately know how to approach the problem; it takes creativity and courage to explore different approaches before deciding how to proceed. Planning and modelling tasks within mathematics develops learners’ ability to turn ideas into action.

#### Personal effectiveness

Studying mathematics develops personal effectiveness. Everyone encounters challenges in studying mathematics at some point, and overcoming the challenges requires and develops resourcefulness and resilience.

Communicating about mathematical thinking and solving problems is a core aspect of mathematics. Mathematical communication is precise and logical and transfers to other areas of life.

### Careers and work-related experiences

#### Learning from careers and labour market information

Mathematics and numeracy are an essential part of every aspect of our lives, whether at work or undertaking practical everyday activities. We use mathematics when we go shopping, plan holidays, decide on mortgages, or evaluate and plan career pathways. Decisions in life are so often based on numerical information that, to make the best choices, we need to be numerate.

Mathematics helps to develop skills such as problem-solving, interpreting data and information, attention to detail, and accurate measuring and reasoning – all of which are highly desirable to employers. The ability and knowledge developed through mathematics and numeracy are a fundamental part of most professions. As competency increases, a wider range of opportunities can be accessed. There is substantial evidence that a lack of numeracy skills leads to poorer outcomes. Organisation for Economic Co-operation and Development (OECD) data indicates that those with poor numeracy skills are more than twice as likely to be unemployed. It also indicates a direct relationship between wage distribution and numeracy skills.

#### Linking the area of learning and experience to careers and work-related experiences

A number of careers specifically require the use of mathematics, including air traffic controller, surveyor, accountant, psychologist, teacher and carpenter or joiner. It is essential for learners to be aware of the wide range of careers available to them, and to be shown how the experiences, knowledge and skills made available in mathematics and numeracy can be applied in the world of work, whether in employment or as an entrepreneur, where these skills are increasingly crucial.

Incorporating careers and work-related experiences into mathematics can help contextualise learning and increase engagement. For instance, statistical analysis and probability can be used in the evaluation of labour market intelligence and information. Learners will therefore be able to effectively identify where jobs are, the qualifications and skills required, and whether this is an area of job increase or decline, thereby applying mathematics in a practical, constructive manner.

Careers and work-related experiences can help to raise aspirations and support financial planning by adding realism and context. Learners can be encouraged to plan for the future and to consider the implications of finance on career pathway decisions, for example, in considering the pay scale of various jobs. This engages and motivates learners and can be further enhanced by the incorporation of valuable lifestyle budgeting exercises and money management for independent living.

Learner progression relating to careers and work-related experiences is part of a continuum of learning for learners aged 3 to 16. Success for a young primary school learner could include:

- acting a variety of different jobs through role play
- belief that they can do any job – tackling gender stereotyping
- communicating with people in their community about the different jobs they do and the rewards that a job can bring.

By progressing learning, success for 16-year-old learners could include:

- demonstrating and applying the skills learned in relation to the world of work
- identifying interests, strengths and skills to make informed post-16 choices
- understanding and demonstrating the behaviours an employer looks for in a good employee
- evaluating risks when developing a business idea and exploring different methods of setting up and sustaining an enterprise.

#### Work-related experiences

Learners develop interests, strengths, skills and aspirations through experiences as part of the curriculum and in life beyond school. Employer engagement, whether by employer talks, visits to workplaces, events or practical activities are essential in adding realism to the world of work and underlining the importance of mathematics and numeracy. It can further support learner progression to enable an understanding of the labour market and trends locally, pan Wales, nationally and globally. Employers raise awareness of roles within the workplace to challenge stereotypes and preconceptions and to raise awareness of all learning and training routes available.

Incorporating careers and work-related experiences within Mathematics and Numeracy allows learners to understand, evaluate and engage with the world of work and enables greater engagement, raised aspirations, and informed, effective decision-making. In addition, it creates an environment where stereotypes can be challenged and the merits of diversity expressed.

Effective careers guidance is essential in securing the most appropriate route for learners’ aspirations, informing them of the diversity of entry points and pathways into the world of work. Schools should offer opportunities to foster entrepreneurial skills and learners should be aware of the benefits of setting up enterprises.

#### Understanding post-16 and higher-education opportunities

It is essential for learners to be aware of all opportunities available to them post-16. Therefore, as well as understanding about employment, training and apprenticeships, learners should be provided with information and the opportunity to engage with a range of learning providers. Opportunities for engagement should include attending careers and skills fairs, talks from and visits to further and higher education providers, as well as presentations from students in further or higher education. Learners should be directed to online research tools that provide course and progression information to support their understanding of the range of learning opportunities available, to help raise their aspirations and form a basis on which informed decisions can be made.

### Relationships and sexuality education

When preparing mathematics sessions we must be aware that reinforcing any stereotype around gender is potentially restricting.

Effective mathematics instruction must be conducted in an environment that encourages equal achievement in mathematics for boys and girls. Teachers must ensure that there are equal opportunities for participation and that the classroom setting is a positive factor in achievement.

When introducing famous mathematicians, care should be taken to ensure that both female and male mathematicians are presented to learners, in order to promote equality and gender positivity. Teachers should choose activities connecting mathematics activities to careers in ways that do not reinforce existing gender stereotypes and select activities that spark initial curiosity about mathematics.

Teachers of all age-groups should be encouraging gender equality within mathematics and promoting career choices which show no bias.

### Enrichment and experiences

Throughout the Mathematics and Numeracy Area of Learning and Experience, there is an emphasis on actively engaging learners in their mathematical learning, by reference to, for example, exploring, investigating, playing and deducing. The approach suggested through the experiences outlined in the descriptions of learning is to engage learners in meaningful problem-solving in all areas of mathematics, set in the contexts of real-world local and global situations. This will promote the learners’ own desire for learning, by making its value clearer and more relevant.

The achievement outcomes refer in multiple places to the resources that could and should be used to enhance understanding and enrich the mathematical experiences of learners. The appropriate use of a range of digital technologies, manipulables, everyday objects, and concrete and abstract representations of mathematical objects helps the learners to engage with mathematical concepts from more than one perspective.

In mathematics, learners sometimes collaborate to solve problems together in groups. For example, some mathematics depends on generating several sources of data, forms of representation, and separate steps of multi-step tasks which can be shared within a group. All can, therefore, contribute to completion of something beyond that which any individual might have done alone. As they work together, they engage actively in mathematical thinking and working, communicating about mathematics. This can aid the development of their own understanding, and enhance their own experience of mathematics. However, it is also important in mathematics to encourage independent and individual work, with learners taking responsibility for their own learning. They need time to think without being swamped by others’ talk; time to internalise methods, facts, etc. A balanced diet of group and individual working will enrich the learners’ experience.

The learning of mathematics will be further enriched by the use of meaningful and substantial tasks, from modelling real-life numerical problems, perhaps involving financial calculations, to relating patterns in nature to mathematical sequences, to using the data-handling cycle to investigate their own research questions.

## Putting the area of learning and experience into practice

The Mathematics and Numeracy curriculum will need to be used by schools to design school-level provision, with the four purposes of the curriculum central to the planning of experiences that learners will encounter. Important decisions will need to be taken about the structure and sequence of mathematics and numeracy topics, and these decisions should be informed by the hierarchic and connected nature of mathematics concepts, in order to ensure foundations are built upon and experiences are connected.

Deep understanding in mathematics and numeracy should develop through planning for all five of the mathematical proficiencies, and the connection and application of these in a range of contexts. Schools will have the autonomy to determine the length of time spent on particular aspects of mathematics and numeracy, and formative assessment should drive decisions about planning in the short term. In planning over longer periods of time, schools will need to ensure that systems of mapping and tracking provision across year groups are robust.

The Mathematics and Numeracy curriculum does not prescribe one particular pedagogy, rather it requires practitioners to collaborate and make informed decisions to ensure that progress is achieved.