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NZC - Mathematics and statistics (Years 0-8)

Year 0-8 Mathematics and statistics Learning Area. From 1 January 2025 this content is part of the statement of official policy relating to teaching, learning, and assessment of Mathematics and statistics in all English medium state and state-integrated schools in New Zealand.

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NZC – Mathematics and statistics

Years 0‑8

  

NZC – Mathematics and statistics

Phase 1 – Years 0‑3

  

NZC – Mathematics and statistics

Phase 2 – Years 4‑6

  

NZC – Mathematics and statistics

Phase 3 – Years 7‑8

About this resource

This page provides the Year 0-8 part of the Mathematics and statistics learning area of the New Zealand Curriculum, the official document that sets the direction for teaching, learning, and assessment in all English medium state and state-integrated schools in New Zealand. In mathematics and statistics, students explore relationships in quantities, space, and data and learn to express these relationships in ways that help them to make sense of the world around them. It comes into effect on 1 January 2025. The Year 9 to 13 content is provided on a companion page.

We have also provided the Maths Years 0-8 curriculum in PDF format. There are different versions available for printing (spreads), viewing online (single page), and to view by phase. You can access these using the icons below. Use your mouse and hover over each icon to see the document description.

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The following material on the New Zealand Curriculum, whakapapa of Te Mātaiaho and overview of the learning areas provides context when using the Mathematics and statistics Year 0-8 Learning Area. It is not part of the statement of official policy.

The New Zealand Curriculum – knowledge-rich, informed by the science of learning, and framed within the whakapapa of Te Mātaiaho

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Mātai aho tāhūnui,
Mātai aho tāhūroa,
Hei takapau wānanga
E hora nei.

Lay the kaupapa down
And sustain it,
The learning here
Laid out before us.

 

The New Zealand curriculum is knowledge-rich. It prioritises and explicitly describes what must be taught each year and is deliberately sequenced to enable students to build knowledge, skills, and competencies systematically over time. It supports teachers to design teaching programmes that bring learning to life in the classroom, using local, national, and global contexts.

The science of learning informs curriculum sequencing and teaching practice. The curriculum builds on scientific understanding to identify five characteristics of how we learn:

We learn best when we experience a sense of belonging in the learning environment and feel valued and supported.

Students bring with them different cultural identities, knowledge, belief systems, and experiences. They need to see that these are valued and reflected in a school environment characterised by strong relationships and mutual respect. Students’ sense of belonging is enhanced by sensitivity to their individual needs, emotions, cultures, and beliefs.

A new idea or concept is always interpreted through, and learned in association with, existing knowledge.

The amount of existing knowledge students have, and the degree to which that knowledge is interconnected in long-term memory, influence both the quality and ease with which they can build on that knowledge. Recognising and drawing on students’ prior knowledge therefore improves their learning.

Establishing knowledge in a well-organised way in long-term memory reduces students’ cognitive load when building on that knowledge. It also enables them to apply and transfer the knowledge.

Establishing new knowledge and skill in long-term memory requires active engagement and multiple opportunities to engage with them, practise them, and connect them to existing knowledge structures. When knowledge is well organised in long-term memory, students are more likely to be able to build on it and apply it in novel ways. If knowledge is not well established in long-term memory, students’ working memory is likely to be overloaded when they attempt to build on or apply it. This cognitive overload can cause confusion, anxiety, and disengagement.

Our social and emotional wellbeing directly impacts on our ability to learn new knowledge.

Social and emotional wellbeing reduces anxiety, which frees cognitive capacity to learn new knowledge and skills, leading to deeper, more durable learning. Conversely, anxiety and negative emotions inhibit students’ ability to learn. The factors that impact positively or negatively on social and emotional wellbeing vary between students. The influence of these factors is dynamic – it fluctuates over time, even during the course of a single day.

Motivation is critical for wellbeing and engagement in learning.

Motivation develops when students feel that three basic needs are met: autonomy – developing increasing self-direction in learning; competence – experiencing success in learning and seeing oneself as a successful learner; social connection – belonging and contributing to a group from which one learns. Success in learning helps to build motivation.


The whakapapa of Te Mātaiaho

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The design of this framework encompasses seven curriculum components. Te Mātaiaho as a whole weaves together these components, all of which begin with the word ‘mātai’, meaning to observe, examine, and deliberately consider.

Te Mātaiaho graphic showing the seven components; Mātairangi - The guiding kaupapa, Mātainuku - Creating a foundation, Mātaitipu - Vision for young people, Mātairea - Supporting progress, Mātaiaho - Weaving learning within and across learning areas, Mātaioho - School curriculum design and review, and Mātaiahikā - Connecting to place and community.

Mātairangi | The guiding kaupapa
The overarching kaupapa guiding the curriculum, based on the science of learning and ensuring excellent and equitable outcomes for students.
Mātai ki te rangi, homai te kauhau wānanga ki uta, ka whiti he ora. | Look beyond the horizon, and draw near the bodies of knowledge that will take us into the future.
The outer rings represent our guiding kaupapa.

Mātainuku | Creating a foundation
The curriculum principles (e.g., holding high expectations, and enabling all students to access the full scope of the curriculum).
Mātai ki te whenua, ka tiritiria, ka poupoua. | Ground and nurture the learning.
The centre rings represent the foundation and calls to action.

Mātaitipu | Vision of young people
The educational vision of young people, as conceived by young people.
Mātaitipu hei papa whenuakura. | Grow and nourish a thriving community.
The inner rings and circular space represent the vision and students at the centre.

Mātairea | Supporting progress
The whole schooling pathway and the overarching focus for year-by-year learning and progress.
Mātai ka rea, ka pihi hei māhuri. | Build and support progress.
Niho kurī lines represent building and supporting the development of students.

Mātaiahikā | Relationships with tangata whenua and local community
Learning through relationships with tangata whenua and local communities.
Mātai kōrero ahiahi. | Keep the hearth occupied, maintain the stories by firelight.
Poutama curves represent relationships with tangata whenua and the community.

Mātaioho | National curriculum – contextualised
The process by which schools bring the national curriculum to life through local, national, and global contexts.
Mātai oho, mātai ara, whītiki, whakatika. | Awaken, arise, and prepare for action.
Unaunahi scales represent wealth of knowledge, purpose, and know-how.

Mātaiaho | Learning areas
The eight learning areas, which each include a purpose, big ideas, knowledge, and practices, year-by-year.
Mātai rangaranga te aho tū, te aho pae. | Weave the learning strands together.
Taratara-a-kae niho notches represent diversity, resilience, and mana.

 

Learning Areas

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The curriculum has eight learning areas: English, the arts, health and physical education, learning languages, mathematics and statistics, science, social sciences, and technology. Together they provide the basis for a broad, general education for the first four phases of learning (years 0–10) and collectively lay a foundation for specialisation in phase 5 (years 11–13). Each learning area is knowledge-rich. This knowledge has been carefully chosen to support all students in their schooling pathway and is framed using Understand, Know, and Do:

  • Understand – the deep and enduring big ideas and themes that students develop understanding of over the phases 
  • Know – the meaningful and important content, concepts, and topics at each phase that enrich students’ understanding of the big ideas and themes and that students study using the practices 
  • Do – the practices (skills, strategies, and processes) that bring rigour to learning and support the development of the key competencies.

A progression model provides the structure that sequences the knowledge. It supports all students to develop greater:

  • breadth and depth of knowledge and understanding, through engaging with increasingly complex and ambiguous contexts 
  • refinement and sophistication in their use of competencies, practices, strategies, processes, and skills 
  • ability to connect, transfer, and apply new learning in meaningful contexts 
  • knowledge and awareness of themselves as learners 
  • effectiveness when working with others.

Content of the learning areas

Knowledge and progression are reflected in how the learning areas are organised. Each learning area has the following main sections:

Purpose statement and UKD overview

A purpose statement describes the learning area’s contribution to the lives of students. It is followed by an overview of Understand, Know, and Do. This gives a view of the big ideas, themes, concepts, topics, and practices that underpin the learning area.

Teachers use the purpose statement and UKD overview to develop an understanding of the learning area, so that they can share its benefits with students.

Learning area structure

For each learning area, this section outlines its structure and the changes it undergoes over five phases of learning, particularly in the final phase, where students specialise and choose from a range of subjects.

There are five phases of learning, spanning years 0–13. Each phase covers two to three years of schooling, which reflects how most schools organise learning across year levels.

A critical focus for each phase establishes a sustained, strengths-based, focus on the student and their social, emotional, and cognitive learning at this stage of their schooling journey. Each critical focus builds on the phase before and is reflected in the content of the learning area for the phase.

The critical focuses are: 

  • Phase 1 (years 0–3): Thriving in environments rich in literacy and maths 
  • Phase 2 (years 4–6): Expanding horizons of knowledge, and collaboration 
  • Phase 3 (years 7–8): Seeing ourselves in the wider world and advocating with and for others 
  • Phase 4 (years 9–10): Having a purpose and being empathetic and resilient 
  • Phase 5 (years 11–13): Navigating pathways and developing agency to help shape the future.

Teachers use the critical focus of each phase in their selection and design of topics and activities.

Teaching guidance

Each learning area also draws from the science of learning and wider education theory to provide a knowledge base and guidance for teachers. Teachers use this to help them make purposeful decisions about how to teach the learning area’s content in ways that are inclusive of all students.

The guidance is organised under three headings:

  • Designing a comprehensive teaching and learning programme 
  • Using assessment to inform teaching 
  • Planning.

Progress outcomes

In each learning area, there is one comprehensive progress outcome for each phase.

The progress outcomes act as signposts that describe expectations for what students should sufficiently understand, know, and be able to do at key points in the schooling pathway.

The content of each progress outcome is organised using the Understand–Know–Do framework. While the Understand statements repeat across the five phases, students’ depth of understanding increases as their knowledge of the learning area’s content (Know) grows and their use of the practices (Do) develops.

When read alongside the progress outcomes for prior and subsequent phases, the progress outcome for a phase helps teachers maintain an overview of the learning they are building on and the learning they are preparing students for. Progress outcomes are therefore key for planning, along with the more detailed teaching sequences (described below).

Teachers also use the progress outcomes to help them form a comprehensive view of each student's progress, achievement, learning needs, and strengths. Schools can use information from twice-yearly, standardised assessment tools to help develop this view, which can also be used to report to parents.

In forming a view of progress and achievement, teachers should ask themselves:

  • Are students using learning from the progress outcome of the previous phase to make sense of new learning in the current phase? This demonstrates how well they can connect new learning to what they already know. It generally occurs in the first year of a phase. 
  • Are students consolidating the learning expressed in the progress outcome in a wide range of contexts? This demonstrates how well and confidently they are using their new learning. This generally occurs in the second year of the phase. 
  • Are students secure in the learning described in the progress outcome within an increasingly complex range of contexts? Are they showing greater depth of knowledge, understanding, and application as they use their new learning and prepare for the challenges of the next phase? This generally occurs towards the end of the final year of the phase. 
  • Are there gaps in learning that are going to restrict students' ability to make progress in the next phase of their learning? This is a question teachers should ask across all years of the phase, drawing on the section Using assessment to inform teaching to consider how to adapt their practices to meet students' learning needs.

Leaders must have a mechanism and strategies for prioritising and closely monitoring urgent action, when required, to support classroom teaching. Where teaching needs to be targeted and intensified to meet specific needs for finite periods, leaders draw on a breadth of available supports, as required.

Teaching sequences

Each phase has a year-by-year teaching sequence. These sequences support teachers to know what to teach and when and how to teach it as students work towards the progress outcome for the phase. They have been organised to support students to revisit ideas, knowledge, and practices in ways that deepen their learning and enable them to use it at the next phase.

There are two parts in a teaching sequence: statements of what to teach, and ‘teaching considerations’ for how to teach:

  • the ‘what to teach’ statements are preceded by the stem ‘Informed by prior learning ...’, which reminds teachers to use their professional judgment and assessment information when selecting what content to teach 
  • the teaching considerations help teachers to know ‘how to teach’ this content in response to students’ prior knowledge, strengths, and experiences.

The teaching sequence tables should be viewed both vertically and horizontally. Looking down the columns helps teachers know what to plan for in a year’s programme. Looking across the rows at the statements for the same concept in the preceding and following years helps teachers to recognise prior learning that students may come with and to consider how they might extend this year’s learning. It also helps teachers to form a more detailed view of their students’ progress, and it is a strong support when planning for mixed-level classes.

The approach of the year-by-year teaching sequences changes in phase 5, as the content becomes more discipline-focused.

Te Mātaiaho | The New Zealand Curriculum

Mathematics and statistics – Years 0‑8

(statement of official government policy)

Ānō me he whare pūngāwerewere.
Behold, it is like the web of a spider.

This whakataukī celebrates intricacy, complexity, interconnectedness, and strength. The learning area of mathematics and statistics weaves together the effort and creativity of many cultures that over time have used mathematical and statistical ideas to understand their world.

 

Board requirements


Schools and kura must give effect to the learning area Mathematics and statistics Years 0–8.

Mathematics and Statistics Years 0–8 is published by the Minister of Education under section 90(1) of the Education and Training Act 2020 (the Act) as a foundation curriculum policy statement and a national curriculum statement. These are the statements of official policy in relation to the teaching of mathematics and statistics that give direction to each school’s curriculum and assessment responsibilities (section 127 of the Act), teaching and learning programmes (section 164 of the Act), and monitoring and reporting of student performance (section 165 of the Act and associated Regulations). School boards must ensure that they and their principal and staff give effect to these statements.

The sections of Mathematics and Statistics Years 0–8 that are published as a national curriculum statement are the Understand–Know–Do (UKD) progress outcomes for each phase (UKD for phase 1, UKD for phase 2, and UKD for phase 3). These set out what students are expected to learn over their time at school, including the desirable levels of knowledge, understanding, and skill to be achieved in mathematics and statistics.

The rest is published as a foundation curriculum policy statement. This sets out expectations for teaching, learning, and assessment that underpin the national curriculum statement and give direction for effective mathematics and statistics (or maths, including numeracy) teaching and learning programmes.

The statements come into effect on 1 January 2025. They replace curriculum levels 1–4 of the existing mathematics and statistics national curriculum statement (learning area). The remainder of the existing mathematics and statistics national curriculum statement remains in force. Apart from those for English Years 0–6, other existing foundation curriculum policy statements and national curriculum statements for the New Zealand Curriculum remain in place.

Schools should choose the appropriate mathematics and statistics statements for their students’ needs. This means that intermediate and secondary schools may choose to make use of the new statements for some students if they are currently working below curriculum level 5, or that primary and intermediate schools may choose to make use of the existing statements for some students if they are already working above phase 3.

Reading, writing, and maths teaching time requirements

The teaching and learning of reading, writing,1 and maths2 is a priority for all schools. So that all students are getting sufficient teaching and learning time for reading, writing, and maths, each school board with students in years 0–8 must, through its principal and staff, structure their teaching and learning programmes and/or timetables to provide:

  • 10 hours per week of teaching and learning focused on supporting students’ progress and achievement in reading and writing, and recognising the important contribution oral language development makes, particularly in the early phases of learning 
  • 5 hours per week of teaching and learning focused on supporting students’ progress and achievement in maths.

Where reading, writing, and/or maths teaching and learning time is occurring within the context of national curriculum statements other than English or mathematics and statistics, the progression of students’ reading, writing, and/or maths dispositions, knowledge, and skills at the appropriate level must be explicitly and intentionally planned for and attended to.

Purpose Statement

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In the mathematics and statistics learning area, students learn about and appreciate the power of symbolic representation, reasoning, and abstraction. They learn to investigate, interpret, and explain patterns and relationships in quantity, space, time, data, and uncertainty. As they achieve deep conceptual understanding and procedural fluency in the learning area, students can accurately and efficiently use mathematics and statistics as a foundation for new learning and to solve problems.

Students engage with mathematics and statistics through the exploration of problems, patterns, and trends and appreciate the everyday value of this learning in many areas of their lives, such as personal finance, health, dance, and design. Every student in New Zealand can engage in mathematics and statistics and discover personal enjoyment and curiosity in their learning.

Throughout their learning, students engage with diverse perspectives as they apply their mathematical and statistical understandings. They also learn that mathematics and statistics has an evolving history; many cultures have contributed to, and continue to contribute to, innovations that shape our current thinking.

As they move through the phases of the learning area, students come to understand the value of mathematical and statistical investigation as a lens for collective local, national, and global challenges. Mathematics and statistics allow us to engage with important societal matters, such as the robust and ethical gathering, interpretation, and communication of data, and the use of valid and reliable data to challenge misinformation and disinformation.

Learning in mathematics and statistics builds literacy by developing students’ skills in oral and written communication, reasoning, and comprehension. The learning area opens pathways into a wide range of industries that rely on mathematical and statistical knowledge and reasoning. Learning how to use this knowledge purposefully and flexibly allows students to participate fully in an increasingly technology- and information-rich world of work.


Understand-Know-Do Overview

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NZC - Mathematics and statistics Understand-Know-Do diagram showing the three strands weaving together into the learning that matters. Understand is described as big ideas and themes. Know is described as content, concepts, and topics. Do is described as practices (skills, strategies, and processes).

Understand

Understand describes the deep and enduring mathematical and statistical big ideas that students develop over phases 1–5.

Patterns and variation | Ngā ia auau me ngā rerekētanga

The world is full of patterns and is defined by a multitude of relationships in which change and variation occur. Mathematics and statistics provide structures that are useful for noticing, exploring, and describing different types of patterns and relationships, enabling us to generate insights or make conjectures.

Logic and reasoning | Te whakaaro arorau me te whakaaroaro

By engaging with mathematical concepts, we develop logical reasoning and critical thinking skills that enable us to evaluate information, question assumptions, and present arguments with clarity. Statistical reasoning from observation and theory allows us to differentiate what is probable from what is possible and to draw reliable conclusions about what is reasonable.

Visualisation and application | Te whakakite me te whakatinana

The visualisation of mathematical and statistical ideas profoundly influences how we perceive, understand, and interact with abstract concepts. Application in mathematics and statistics involves creating structures and processes that help us understand complex situations, enabling better decision making and communication of ideas.

Know

Know describes the meaningful and important mathematical and statistical concepts and procedures through which students develop understanding of the big ideas.

Number | Mātauranga tau

Number focuses on the study of numerical concepts. People use numbers to represent quantities, estimate, and measure. We perform operations on numbers to calculate or compare. Throughout history, different number systems have been developed, reflecting practical and social needs.

Algebra | Taurangi

Algebra focuses on making and using generalisations to reason mathematically. It allows us to identify patterns and underlying mathematical relationships. These generalisations, patterns, and relationships can be represented and communicated using diagrams, graphs, and symbols (including variables). The algebra we use today was created and refined over thousands of years.

Measurement | Ine

Measurement focuses on the concepts and techniques that allow us to quantify phenomena, using appropriate units and systems of measurement. Countries around the world use both standard and non-standard units to measure tangible and intangible objects and quantities.

Geometry | Āhuahanga

Geometry focuses on visualising, representing, and reasoning about the shape, position, orientation, and transformation of objects. Many cultures use tools and techniques derived from the natural world when exploring and describing objects and space.

Statistics | Tauanga

Statistics focuses on tools, concepts, and systematic processes for interpreting situations, using data and its context to understand uncertainty, make conjectures, and inform decision making. Statistical practices include considering the ethics of data collection and the responsibility of safely and securely handling data in different contexts.

Probability | Tūponotanga

Probability focuses on tools and concepts for quantifying chance, dealing with expectation, and using evidence to identify how likely events are to occur. People around the world have relied on and continue to rely on probabilistic thinking when making decisions.

Do

Do describes the processes that are fundamental to all mathematical and statistical activities and that underpin students’ learning of the big ideas, concepts, and procedures.

Investigating situations | Te tūhura pūāhua

When we investigate situations using mathematics and statistics, we describe and explore them to build our understanding of them. When investigating, we need to decide which approaches, concepts, and tools to use and how to use them. We often begin with a question or focus of interest and proceed in systematic but flexible ways, using mathematical and statistical concepts and procedures to solve problems and make sense of findings in context. We conclude by evaluating the investigation, which involves reflecting on the solutions and outcomes and our approaches and choices to determine whether they were reasonable, made sense in context, and could be improved on in future investigations.

Representing situations | Te whakaata pūāhua

When we represent situations mathematically and statistically, we use words or symbols and mental, oral, physical, digital, graphical, or diagrammatic ways to show concepts and findings. We can use representations to compare, explore, simplify, illustrate, prove, and justify, as well as to look for patterns, variations, and trends. Representing a situation in multiple ways enables a deeper and more flexible understanding and allows us to communicate with different audiences.

Connecting situations | Te tūhono pūāhua

When we connect situations using mathematics and statistics, we recognise and make links by noticing similarities and differences. Connecting helps us to understand the relationships between concepts and procedures in mathematics and statistics. This is important because number, algebra, measurement, geometry, statistics, and probability form a web of interconnected ideas and approaches that can be easier to remember and understand if the connections between them are clear. Connecting also involves linking mathematics and statistics to other learning areas and to a range of contexts.

Generalising findings | Te whakatauwhānui i ngā kitenga

When we generalise mathematical and statistical findings, we move from specific examples to general principles. We use the patterns, regularities, and structures that we find to make conjectures that might apply to other situations. Further investigation can test and refine these conjectures and determine if they apply in all cases. In statistics, we generalise by using trends and variation in data to make inferences and conjectures and to articulate and evaluate claims about similar situations.

Explaining and justifying findings | Te whakamārama me te parahau i ngā kitenga

When we explain and justify, we use mathematical and statistical ways of communicating and reasoning to share our ideas and to respond to the ideas, reasoning, and inferences of others. Explaining is how we communicate our inferences and conjectures, build arguments, and unpack stories from data. Justifying involves describing why decisions and findings are reasonable, taking into account limitations arising from assumptions and choices and the evidence on which findings are based.

Phases 1, 2, and 3: Mathematics and statistics learning area structure

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This section describes the structure of the mathematics and statistics learning area and how it changes over the five phases of learning. (See the Content of the learning areas section for the general structure of each learning area in the New Zealand curriculum.) Each phase has:

  • a progress outcome describing what students understand, know, and can do by the end of the phase 
  • an introduction to the teaching sequence highlighting how to teach during this particular phase 
  • a year-by-year teaching sequence highlighting what to teach in the phase, along with teaching considerations for particular aspects of content.

Progress outcomes

The progress outcomes (one per phase) describe what students will understand, know, and be able to do by the end of the phase.

  • Understand describes the big ideas that students develop from learning mathematics and statistics over phases 1 to 5. They help connect school mathematics and statistics with the wider world and represent the critical big-picture concepts of mathematics and statistics. 
  • Know describes the meaningful and important concepts and procedures in mathematics and statistics. They are broken down into six strands: number, algebra, measurement, geometry, statistics, and probability. 
  • Do describes the processes students use to represent and work with what they know and understand in mathematics and statistics. These processes are central to how students learn and apply mathematical and statistical knowledge. While there are small progressions in the processes from phase to phase, in general the increasing sophistication of their use comes from applying them to more advanced concepts and procedures.

It is through the interweaving of Understand, Know, and Do that students develop their conceptual understandings and procedural fluency, supporting success and bringing richness and meaning to mathematics and statistics for them.

NZC - Mathematics and statistics Understand-Know-Do diagram showing the three strands weaving together into the learning that matters. Understand is described as patterns and variation, logic and reasoning, visualisation, and application. Know is described as number, algebra, measurement, geometry, statistics, and probability. Do is described as investigating situations, representing situations, connecting situations, generalising findings, and explaining and justifying findings.

As students progress through the phases, the focus of their learning shifts. In phase 1, the focus is on developing foundational skills across all strands. In phases 2 and 3, students expand their range of representations and their reasoning to work with increasingly complex concepts across all strands.

This change in focus is seen in how the Understand, Know, and Do progress outcomes are reflected in the year-by-year teaching sequences. The descriptors of what to teach each year have the stem ‘Informed by prior learning ...’ in order to reinforce that teachers will use their professional judgment about what content to teach and how to teach it. They will make these judgments in response to the prior knowledge, strengths, and experiences that students bring to their learning.

Teaching sequences

The year-by-year teaching sequences are organised in line with the strands from Know. They describe the incremental teaching required each year as students work towards the progress outcome.

Some statements in the teaching sequences are repeated across multiple years, allowing more time for progression and consolidation. Not all statements are progressed each year; some topics start and others end, reflecting what is developmentally appropriate in learning in mathematics and statistics.

Each statement in a sequence varies in the amount of teaching time it requires. The learning area is designed to enable knowledge and procedures to be connected and taught together, so individual statements in a year sequence should be combined in ways that enhance learning.

The year-by-year content can be viewed both vertically and horizontally. The vertical view helps teachers know what to plan for the next year. The horizontal view allows teachers to follow the statements for one concept across several stages. This helps them understand the prior knowledge students may bring to their learning and helps them decide how to extend this learning. The horizontal view also helps teachers plan for mixed-level classes.

The teaching sequence statements are supported by ‘teaching considerations’. These describe evidence-based practices and show how teachers can integrate the processes of Do to help their students develop conceptual and procedural knowledge.


Phases 1, 2, and 3: Teaching guidance

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Key characteristics of how people learn have informed the development of the mathematics and statistics learning area. These characteristics are:

  • We learn best when we experience a sense of belonging in the learning environment and feel valued and supported. 
  • A new idea or concept is always interpreted through, and learned in association with, existing knowledge. 
  • Establishing knowledge in a well-organised way in long-term memory reduces students’ cognitive load when building on that knowledge. It also enables them to apply and transfer the knowledge. 
  • Our social and emotional wellbeing directly impacts on our ability to learn new knowledge. 
  • Motivation is critical for wellbeing and engagement in learning.3

All five characteristics are interconnected in a dynamic way. They are always only pieces of the whole, so it is critical to consider them all together. The dynamic and individual nature of learning explains why we see individual learners develop along different paths and at different rates.

The implications of these characteristics for teaching mathematics and statistics are described in this section, with more detail in the introduction to each phase and the ‘teaching considerations’ in the year-by-year teaching sequences.

The remainder of this section focuses on three key areas of teacher decision making: 

  • developing a comprehensive teaching and learning programme 
  • using assessment to inform teaching 
  • planning.

Developing a comprehensive teaching and learning programme

A comprehensive mathematics and statistics programme needs the following components: 

  • explicit teaching 
  • positive relationships with mathematics and statistics 
  • rich tasks 
  • communication in mathematics and statistics.

Explicit teaching

Explicit teaching is a structured, carefully sequenced approach to teaching. The sequencing of content is thought out and broken down into manageable steps, each of which is clearly and concisely explained and modelled by the teacher. Explicit teaching requires a high level of teacher-student interaction, guided student practice, and, when proficiency is achieved, independent practice.

Explicit teaching supports cumulative learning as new knowledge is built on what students already know. Teachers provide multiple opportunities for practising, reviewing, consolidating, and using previous learning alongside new learning.

Explicit teaching takes account of cognitive overload. With sufficient practice, new learning is transferred to long-term memory. This frees up working memory, opening up opportunities for extension, enrichment, and new learning.

Explicit teaching is strongly interactive – it is not simply teacher talk. It includes rich discussions between teachers and students and amongst students, to check on understanding. Teachers adapt the pace of their teaching in response to students’ progress. They engage students in creative and challenging tasks to foster motivation and engagement.

Using materials and visual representations throughout explicit teaching supports students to develop conceptual understandings as they move towards more abstract forms of representation, such as equations. Teachers can reduce students’ cognitive load by carefully considering the ways in which visual and written information are presented (e.g., how working and explanations are laid out) and by removing unnecessary information to focus on the key teaching and learning points.

Explicit teaching involves: 

  • connecting the current focus to previous learning 
  • providing concise, step-by-step explanations, accompanied by student input and discussion 
  • explaining, modelling, and demonstrating 
  • regularly checking for understanding and providing feedback 
  • providing opportunities for collaborative and independent practice.

Positive relationships with mathematics and statistics

Learning is enhanced when students succeed in and feel positive about their learning. If students feel anxious, they have fewer cognitive resources available for learning.

Positive relationships with mathematics and statistics are supported by teachers through: 

  • setting high expectations 
  • planning experiences that are accessible to every student and provide daily opportunities for success 
  • incorporating students’ interests, cultures, and prior knowledge 
  • planning opportunities for students to explore and think critically 
  • supporting students to use mathematics and statistics to make sense of their world and address local, national, and global issues 
  • providing manageable challenges that encourage students to develop perseverance, reinforcing that conceptual understanding and procedural fluency develop with consistent effort 
  • increasing scaffolding and supports in response to anxiety as a result of cognitive overload 
  • valuing mistakes as an important part of the learning process.

Involving families in students’ learning journeys and offering opportunities for collaboration support positive relationships with mathematics and statistics. Teachers also model such relationships by showing curiosity, persistence, and enjoyment, and by engaging in mathematics and statistics themselves.

Rich tasks

Rich tasks are meaningful problem-solving and investigation experiences, designed to invoke curiosity and engagement. They should relate both to mathematical contexts and wider contexts relevant to the communities, cultures, interests, and aspirations of students.

Rich tasks provide a motivational hook when exploring new concepts and procedures. They can also be used to consolidate concepts and procedures that have already been taught, to develop the mathematical and statistical processeses of Do, and to facilitate the transfer and application of learning to new situations. These experiences often allow students to decide how to approach the task, developing their agency, confidence, and motivation.

Teachers design rich tasks that are accessible to all students and offer different levels of challenge. They ensure that students are clear about the purpose of learning, and they consider the core requirements of the task as well as the range of possible responses. As students work on rich tasks, teachers plan opportunities for discussion, collaboration, and feedback. They are actively involved in monitoring, prompting, and questioning during the task, to encourage students to ask questions, test conjectures, make generalisations, and form connections.

Communication in mathematics and statistics

Students communicate throughout the learning process, both to develop conceptual understanding and to share their thinking and reasoning. Rich, extended interactions are pivotal to students’ development of knowledge, processes, and dispositions in mathematics and statistics. Effective discussions build knowledge through sharing, active listening or attending, critiquing, questioning, and extending thinking and reasoning.

Rich interactions make students’ reasoning visible. This helps teachers recognise how well students are developing mathematical and statistical processes and concepts, and it provides opportunities for teachers to identify misconceptions and correct them. These interactions also allow teachers to develop students’ use of mathematical and statistical language, vocabulary, symbols, representations, and reasoning.

Using assessment to inform teaching

Assessment that informs decisions about adapting teaching practice is moment-by-moment and ongoing. Teachers use observation, conversations, and low-stakes testing to continuously monitor students’ progress in relation to their year level in the teaching sequence. They ensure that they notice and recognise the development, consolidation, and use of learning-area knowledge by students within daily lessons, and that they provide timely feedback. They respond by adapting their practice accordingly. For example, they reduce or increase scaffolding and supports, paying particular attention to anxiety caused by cognitive overload. Formative assessment information can also be collected through self and peer assessment, with students reflecting on goals and identifying next steps.

In addition to daily monitoring, teachers use purposefully designed, formative assessment tasks at different points throughout a unit or topic to highlight the concepts and reasoning students use and understand. Teachers ensure such tasks are valid by addressing barriers to learning, so that every student is able to demonstrate what they know and can do.

When planning next steps for teaching and learning, teachers consider students’ strengths and responses along with potential opportunities for further consolidation. Next steps could include: 

  • designing scaffolds to support students to access and enrich their learning 
  • providing opportunities for students to apply new learning 
  • planning lessons focused on revisiting, reteaching, or consolidating learning.

Providing timely feedback throughout the learning process and identifying and addressing misconceptions as they arise lead to the efficient and accurate development of learning-area concepts and promote further learning. Teachers can use feedback to prompt students to recall previous learning, make connections, and extend their understanding.

Planning

This section provides guidance on what to pay attention to when planning mathematics and statistics teaching and learning programmes. In every classroom, there are many ways in which students engage in learning and show what they know and can do. Using assessment information and designing inclusive experiences, teachers plan an ‘entry point’ to a new concept or procedure that every student can access. Students’ interests and the school culture and community shape the planning, adding richness, creativity, and meaning to the programme.

Teaching and learning plans are developed for each year, topic or unit, week, and lesson and make optimal use of instructional time. The following considerations are critical when planning and designing learning:

  • Develop plans using the teaching sequence statements for the year and knowledge of students’ prior learning. Plan for all students to experience all the statements in the sequence for their year level. 
  • Map out a year-long programme composed of ‘units’ by looking for opportunities where statements from the teaching sequence can be taught together. These may be in the same strand or across several strands (e.g., statistics and measurement; algebra and geometry). Plan to weave together learning under Know and Do across the unit to build understanding of the big ideas.
  • Order the units so that new learning builds on students’ previous learning and connects over the course of the year. Consider the length of time allocated to specific strands and concepts across the year – some concepts may require more teaching time than others. Ensure the year’s programme includes opportunities to retrieve, consolidate, and extend learning around previously taught concepts and processes. Regular opportunities to revisit learning within and across units and years supports students to develop procedural fluency with mathematics and statistics concepts. The shape of these opportunities will vary, depending on students’ learning needs. 
  • Within unit or weekly plans, break down new concepts and procedures into a series of manageable learning experiences, so that students have several opportunities to develop understanding and fluency. Teach mathematics and statistics for an hour a day. Plan for a balance of explicit teaching (to introduce and reinforce learning) and rich tasks (to investigate a concept, support consolidation of previously taught concepts or procedures, and apply learning to new situations).
  • Plan for inclusive teaching and learning at all times. Consider offering multiple methods of participating to all students so that they can engage in a variety of learning experiences and have multiple ways to show their progress. Design for equitable access in all learning opportunities. Identify and reduce barriers to learning, and plan universal supports that are available to all students. 
  • Use flexible groups within a lesson, based on the learning purpose for the lesson (e.g., working as a whole class for demonstration and discussion, in smaller groups to investigate a situation or solve a problem, in pairs to explain thinking and findings). Provide opportunities for both individual and collaborative work, and enable students to determine when they need to work with others and when they need time and space to work independently. 
  • Teach students to use digital tools accurately, appropriately, and efficiently to support their purpose. Enhance teaching and learning with tools for calculating, representing graphs and shapes, and analysing data. While using digital technology is an important skill, students still need the ability to estimate, visualise, and reason, so that they can evaluate whether findings generated by a digital tool are reasonable and effective.

To support students who have not developed the prior knowledge needed for teaching sequence statements for their year or have not learnt everything they have been explicitly taught, teachers can use accelerative approaches. These are approaches that make year-level concepts and procedures accessible to students. They can include additional, targeted small-group teaching, the use of verbal and visual prompts, carefully chosen representations, and explicit teaching of problem-solving strategies.

Teachers can extend students who have developed deep conceptual understanding and procedural fluency for their year by using more challenging rich tasks and problem solving that allow the students to apply their understanding to unfamiliar situations. This also encourages the students to develop further generalisations and to strengthen their mathematical and statistical communication and reasoning.

Dedicated mathematics and statistics lessons

Depending on the purpose of the lesson, plan to include one or more experiences in each part (Getting started, Working, and Connecting and reflecting). As students are working, take time to notice, recognise, and respond to their learning.

Getting started
  • Recall and connect to prior learning to provide a starting point for all students to access and understand new concepts or processes. 
  • Introduce new concepts using a focus activity, group challenge, or task that activates prior knowledge and interests.

Working
  • Provide whole-class, small-group, paired, or individual work opportunities for students to develop or apply concepts and procedures through investigations, tasks, or games. 
  • Explicitly teach concepts and procedures by leading interactions that include explanations, demonstrations, questioning, short tasks, and discussion. Use clear and concise language, including correct mathematical and statistical vocabulary, and clear working layouts and notation. 
  • Provide additional explicit teaching based on the learning needs of individual students. 
  • Help students organise new knowledge in ways that connect with their prior learning – for example, by discussing connections, using graphic organisers, or carefully ordering concepts and procedures in relation to prior learning. 
  • Support consolidation of knowledge with targeted practice and activities. For students early in the process of consolidation, these activities should be scaffolded and guided. As students develop understanding and fluency, they complete the activities with increasing independence. 
  • Support students to retrieve and use previously taught concepts and procedures in connected ways, such as applying them while investigating situations.

Connecting and reflecting
  • Clearly summarise and connect to the purpose of the lesson. 
  • Review learning by discussing, sharing, and analysing the experiences of the lesson. 
  • Make connections with prior learning, between mathematics and statistics concepts, with other learning areas, and with situations outside of the classroom. 
  • Pre-teach to prepare students for the next lesson. 
  • Highlight progress and examples of curiosity, resilience, and persevering through challenge.

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Heading

References

1. While the terms reading and writing are used, these expectations are inclusive of alternative methods of communication, including New Zealand Sign Language, augmentative and alternative communication (AAC), and Braille.

2. For simplicity, ‘maths’ is used as an all-encompassing term to refer to the grouping of subject matter, dispositions, skills, competencies, and understandings that encompasses all aspects of numeracy, mathematics, and statistics.

3. A description of each characteristic is found here.

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Abstract

Symbolic representation of a concept.

Acceleration

Acceleration or accelerated learning is where students are enabled to learn concepts and procedures more rapidly than the expected rate of progress. Accelerated learning approaches involve teaching students year-level content and experiences and supporting them with appropriate scaffolds to make this work accessible.

Additive identity

Zero will not change the value when added to a number. For example, 16 + 0 = 16.

Algorithm

A set of step-by-step instructions to complete a task or solve a problem.

Algorithmic thinking

Defining a sequence of clear steps to solve a problem.

Argument

Providing an idea or finding that is based on reasoning and evidence.

Associative property

A property of operations when three numbers can be calculated (addition or multiplication) in any order without changing the result. For example, (4 + 3) + 7 = 4 + (3 + 7) because 7 + 7 = 4 + 10, and (4 x 3) x 5 = 4 x (3 x 5).

Assumptions

A proposition (a statement or assertion) which is taken as being true with respect to a given context.

Attribute

A characteristic or feature of an object or common feature of a group of objects —such as size, shape, colour, number of sides.

Base 10

Our number value system with ten digit symbols (0-9); the place value of a digit in a number depends on its position; as we move to the left, each column is worth ten times more, with zero used as a placeholder; to the right, the system continues past the ones’ column, to create decimals (tenths, hundredths, thousandths); the decimal point marks the column immediately to the right as the tenths column.

Benchmarks

A reference point that we can use for comparison or estimation. For example, “My finger is about one centimetre wide.”

Categorical variables

A variable that classifies objects or individuals into groups or categories. For example, hair colour, breed of dog.

Chance

The likelihood that an outcome will occur.

Claim

A statement of what the student believes to be correct.

Commutative property

In addition and multiplication, each number can be operated in any order. For example, 5 + 6 = 6 + 5.

Compose and decompose

Compose is to make a shape using other shapes. Decompose is to break a shape into other shapes. 

Comparison investigative question

An investigative question that compares a variable across two clearly identified populations or groups. For example, “I wonder if girls in our class tend to be taller than boys in our class?”

Conceptual understanding

The comprehension of mathematical concepts, operations, and relations by connecting related ideas, representing concepts in different ways, identifying commonalities and differences between aspects of content, communicating their mathematical thinking, and interpreting mathematical information.

Conjecture

A statement whose truth or otherwise is not yet determined but is open to further investigation.

Constant

A constant term is a fixed value that will not change.

Continuous materials

Models based on relative length or area, such as a number line, fraction wall, bar model.

Data

A collection of facts, numbers, or information; the individual values of which are often the results of an experiment or observations.

Data ethics

The principles behind how data is gathered, protected, and used; at the core of ethical practice is the need to do good and to do no harm.

Data visualisations

A graphical, tabular, or pictorial representation of information or data.

Deduction

Make a conclusion based on knowledge, definitions, and rules.

Digital tools

Digital applications, calculators.

Discrete materials

Separate objects that can be counted and grouped. For example, counters, ice block sticks.

Discrete numerical variables

Variables that can be counted and have a limited range of possibilities. For example, number of students in each team, the result of rolling a die.

Disinformation

False information spread deliberately to deceive others.

Distribution

In mathematics, distribution describes spreading terms out equally across an expression; in statistics, distribution describes how data values are spread across the range of values collected.

Distributive property

An operation is said to be distributive over another operation if it can take priority over the operation used for combination within brackets. For example, 6 × 17 = 6 × (10 + 7) = (6 × 10) + (6 × 7) = 60 + 42 = 102.

Efficient

A procedure is said to be efficient when it is carried out in the most simple and effective way.

Element in a pattern

In a repeating pattern, an element is the repeating core.
In a growing pattern, an element is a section of the pattern. For example, in the Fibonacci sequence 1, 1, 2, 3, 5, 8, the next element is the sum of the previous two elements (5 + 8 = 13).

Equation

A number statement that contains an equal sign. The expressions on either side of the equal sign have the same value (are equal).

Equivalent fraction

Two different fractions that represent the same number are referred to as equivalent fractions. For example, 1—2, 2—4, 3—6, and 4—8 are equivalent fractions because they represent the same number.

Estimate

A rough judgement of quantity, value, or number. In statistics, an assessment of the value of an existing, but unknown, quantity. In probability, the probability of an outcome in an experiment.

Event

One or more outcomes from a probability activity, situation, or experiment.

Evidence

Information, findings, data that support (prove) a statement or argument.

Expression

Two or more numbers, operations or variables connected by operations. Expressions do not include an equal or inequality sign.

Families of facts

A group of equations that use the same numbers. For example, 3 x 2 = 6, 2 x 3 = 6, 6 ÷3 = 2, 6 ÷ 2 = 3.

Generalisation

To recognise and describe patterns in relationships.

Growing pattern

A pattern where there is a constant increase or decrease between each term. For example, 5, 10, 15, 20.

Horizontal method

Representing the operation as an equation across the page, often partitioning one of the numbers into tens and units. For example, 16 + 23 = 16 + 20 + 3.                                                                                                                                          

Inequality

A statement in which one number or expression is greater or less than another.

Inference

Making a conclusion based on evidence and reasoning.

Informal unit

A non-standard unit used to measure. For example, blocks, pens, fingers. The informal units used should all be the same size.

Interpret

To make meaning from something.

Inverse operations

The opposite operation, so addition is inverse to subtraction, and multiplication is inverse to division. They are useful to check calculations. For example, to check 4 x 5 = 20, we can see if 20 ÷5 = 4.

Limitations

Possible missing evidence or information.

Mathematical modelling

An investigation of the relationships and behaviours of quantities in physical, economic, social, and everyday contexts; used to analyse applied situations and make informed decisions, starting the model with forming assumptions.

Misconception

A misunderstanding about a procedure, method or definition in mathematics or statistics. This could be about which procedure is needed, how the procedure or method should be followed or an incorrect definition. For example, thinking that 2.5 is the decimal form of 2/5.

Misinformation

Incorrect information (mistakes).

Multiplicative identity

When a number is multiplied by 1, it does not change its value. For example, 15 x 1 = 15.

Number sentence

An equation or inequality expressed using numbers and mathematical symbols. For example, 10 = 3 + 7 + 5 + 5.

One step transformation

One change in a shape's position or size. For example, a triangle is flipped (reflected).

Ordinal

The numerical position of the element in the sequence. For example, first, second, third.

Orientation

The angle that an object is positioned.

Outcomes

A possible result of a trial of a probability activity or a situation involving an element of chance; could also refer to a result or a finding.

Partition and regroup

Partitioning is the process of "breaking up" numbers. For example, 55 = 50 + 5.
Regroup means to rearrange the formation of the group. For example, 55 = 40 + 15 or 55 = 30 + 25.

Primary data

Data collected first-hand for a specific purpose. For example, a survey, experiment, or interview.

Probabilistic thinking

Considering the likelihood (chance) of an outcome occurring; this is based on logic and reasoning.

Probability experiment

A test that can be carried out multiple times in the same way (trials). The outcome of each trial is recorded.

Procedural fluency

Choosing procedures appropriately and carrying them out flexibly, accurately, and efficiently. It is not the same as memorisation of facts and steps; rather it is being able to activate what you know and when to use it.

Procedure

A sequence of operations carried out in a specific order, such as the procedure to multiply two numbers, or the procedure to measure an object.

Quantifying

Expressing a quantity using numbers.

Rational number

All integers, fractions, and decimals.

Reasoning

Analysing a situation and thinking and working mathematically to arrive at a finding.
Inductive reasoning is where an explanation is made based on observations and data. For example, “18 out of 22 people in our class like grapes, so grapes are the favourite fruit of our class.”
Deductive reasoning is where we apply a known rule or fact to solve a problem. For example, prime numbers have two factors (itself and one), 4 has three factors, so it is not a prime number.

Relational

Symbols that show relationships between elements (terms) in an expression. For example, =, <, >

Repeating pattern

A pattern containing a 'unit of repeat'. For example, red, green, blue, red, green, blue.

Secondary data

Data collected by someone else, or a process, and/or obtained from another source. For example, online, books, other researchers.

Subitise

Instantly recognise the number of items in an arrangement without counting.

Summary investigative question

A question that asks about the overall distribution of the data or what is typical and reflects the population or group. For example, “I wonder how many pets are in our class? I wonder what the heights of the students in the class are?”

Tangible and intangible

Tangible is an object that can be touched. For example, a group of blocks. Intangible is a quality or measurement that cannot be touched. For example, colour or length.

Term in a pattern

One of the numbers in a pattern or sequence. For example, for 2, 4, 6, 8, the second term is 4.

Theoretical probability

A calculation of how likely an event is to occur in a situation involving chance.

Uncertainty

In probability, when the chance of an event occurring is unknown.

Unit of repeat

The part of a repeating pattern that repeats. The part is made up of several elements

Variables (statistics and algebra)

A property that may have different values for different individuals (statistics) or that may have different values at different times (statistics and algebra).

Variation

The differences seen in the values of a property for different individuals or at different times.

Vertical column method

The method for solving an operation by recording numbers in columns according to place value, working down the page.

Visualisation

To mentally represent and manipulate.