## What is Mathematics and Statistics about?

*Subject-specific terms can be found in the glossary.*

Mathematics is the exploration and use of patterns and relationships in quantities, space, and time. Statistics is the exploration and use of patterns and relationships in data. These two disciplines are related, but involve different ways of thinking and solving problems. Both equip ākonga with effective means for modelling, analysing and interpreting the world in which they live.

Mathematicians and statisticians use symbols, graphs, displays, and diagrams to help them find and communicate patterns and relationships. They evaluate information to make informed decisions and create models to represent both real-life and hypothetical situations. These situations are drawn from a wide range of social, cultural, scientific, technological, environmental, and economic contexts.

## Big Ideas and Significant Learning

Big Ideas are derived from the Learning Area essence statement and capture the essence of a subject, ensuring coherence rather than fragmentation of learning. At the subject level, they inform the Significant Learning – learning that is critical for ākonga to know, understand, and do in relation to a subject by the end of each Curriculum Level. This covers knowledge, skills, competencies, and attitudes and also includes level-appropriate contexts ākonga should encounter in senior secondary education.

Significant Learning is collated into a Learning Matrix. Kaiako can use the Learning Matrix as a tool to construct learning programmes that cover all the 'not to be missed' learning in a subject. There is no prescribed order to the Learning Matrix within each level. A programme of learning might begin with a context that is relevant to the local area of the school or an idea that ākonga are particularly interested in. This topic or context has to relate to at least one Big Idea and may also link to other Big Ideas. The Learning Matrix is designed so that kaiako have the freedom to create courses that are both flexible and coherent.

#### Unpacking the Learning Matrix

The Mathematics and Statistics Big Ideas are organised into two categories: 'process' and 'knowledge'. They are key ideas about how to work with, and understand, mathematics and statistics. Within the Learning Matrix, the Significant Learning sit under and alongside the six Big Ideas. Each piece of Significant Learning might relate to any, or all, of the six Big Ideas. This highlights the relevance of the Big Ideas to all learning.

Each piece of Significant Learning is also categorised under a different combination of learning strands (number, algebra and graphs, geometry, measurement, statistics, and probability). It is framed in this way to show that most learning in Mathematics and Statistics is cross-strand. Strands can be taught together, and not unnecessarily compartmentalised. The Learning Matrix provides a starting point for kaiako to weave the strands together; kaiako will have their own ideas about how to do this weaving. This interwoven approach forms a solid base at Curriculum Level 6 which gives school leavers a workable mathematical toolbox, and enables further specialisation at Curriculum Levels 7 and 8.

The process Big Ideas of wānanga, hononga and tāiringa kōrero highlight the importance of Aotearoa’s identity in how ākonga conceptualise the world and solve problems. These processes sit across all the Significant Learning, and the emphasis here on mātauranga Māori recognises the importance of socio-cultural context when learning and applying mathematical and statistical skills. The content of mathematics is universal, but it will be accessed, and engaged with, by different cultures in distinct ways.

The Significant Learning comprises all the skills that ākonga are entitled to leave Curriculum Level 6 with. The Learning Matrix structure reflects that although Mathematics and Statistics involves a broad range of skills, the Big Ideas are the common underlying processes and knowledge that pull these skills together. It is important that learning is firmly embedded in a context which resonates with ākonga and which recognises their cultures. Kaiako are encouraged to match Significant Learning to the interests and needs of ākonga. Course Outlines and Assessment Activities will provide guidance on how teaching can be linked to ākonga particular contexts and future pathways.

#### What is new?

The pieces of Significant Learning are not materially different to what has previously been taught in Mathematics and Statistics. This collection of mathematical learning remains an excellent collection of capabilities for ākonga.

The real change is focused on how the Significant Learning is taught and assessed. Firstly, teaching across strands allows learners to engage with all sides of a problem. Mathematics is a practical skill. Exploring how different mathematical principles apply to anything and everything can empower ākonga to actively use mathematics in all contexts. Secondly, the four new Achievement Standards are designed for flexibility. The standards are wide enough that kaiako will be able to design Assessment Activities that ākonga can see themselves in, and which prepare them for the diverse pathways they follow after school.

#### Critical thinking, and mathematical and statistical generalisations, emerge from te hononga of different observations, knowledges, and processes

As ākonga build critical thinking skills, they move from relying on their intuition, or instincts, to working systematically to solve problems, form generalisations, and reach conclusions. Ākonga critical thinking skills can be developed through engagement with information from varying sources. As ākonga grow to recognise the connections between different observations, knowledges, and processes, their capabilities in making mathematical and statistical generalisations will improve. Hononga is the concept of identifying these connections and links to reach conclusions. Te ao pāngarau is a helpful framework for exploring this Big Idea, as it places emphasis on the interconnectedness of mathematical knowledge and processes.

#### Tāiringa kōrero allows for creativity and exploration, and the discovery of mathematical and statistical concepts, theories, and models

Tāiringa kōrero is a thought put forward by a mathematician or statistician, on the basis of what they have observed, which is yet to be proved. Mathematical and statistical discovery begins with these observations. Tāiringa kōrero is marked by exploration, creativity, discovery, and conjecture. Experimentation and exploration are the mechanisms through which mathematical and statistical change unfolds. Ākonga can participate directly in these processes to enrich their mathematical comprehension and further open their eyes to the beauty of pāngarau.

#### In Mathematics and Statistics, wānanga stimulates logical argument, investigation, analysis, and justification, supporting critical evaluation and reasoned conclusions

Wānanga is a process that values time and discourse as integral factors to support learning. In mathematics and statistics, wānanga allows questions, answers and critical thought to be transformed into knowledge and understanding. Mathematics and statistics are not only processes or strategies for thinking. Through mathematics and statistics, we can reach informed conclusions about the world, understand widely applicable concepts, and test claims against our understanding. As ākonga develop their own mathematical and statistical knowledge, they will grow in their capacity to evaluate information, assess situations, and respond to problems.

#### Numbers, measures, geometric representations, numerical or algebraic expressions, and equations can be represented in multiple ways

The focus of this Big Idea is equivalency. Understanding the various ways in which a mathematical concept can be represented is an essential foundation for problem solving and manipulation. Accessing a mathematical concept through different strands of learning encourages open-minded thinking, teaching ākonga to look at problems from new perspectives and angles. When ākonga move beyond a compartmentalised understanding, they can see more ways into problems, and more fully understand the fluid nature of mathematics.

#### Patterns and relationships can be represented numerically, algebraically, graphically, and geometrically

Patterns and relationships are visible everywhere in the world around us. Patterns and relationships can also reveal how numbers, shapes and data relate to each other within a wider context. Ākonga will need to work with representations of different patterns and relationships in order to engage fluently with the wider implications of their work in mathematics and statistics. Utilising patterns can also be a powerful tool for learning, as ākonga can solve complex problems by working first with their simpler versions.

#### Mathematical and statistical methods can be used to explore, solve, or model problems while recognising variation, certainty, and uncertainty

Ākonga should understand how their mathematical and statistical literacy can apply to tangible problems outside the classroom. This involves correctly identifying when to explore (considering variation and uncertainty), solve with certainty, or create and use a mathematical or statistical model.

## Key Competencies in Mathematics and Statistics

#### Developing Key Competencies through Mathematics and Statistics

Learning in Mathematics and Statistics provides meaningful contexts for developing Key Competencies from the New Zealand Curriculum. These Key Competencies are woven through, and embedded in, the Big Ideas and Significant Learning. Students will engage with critical thinking and analysis, explore different perspectives through mathematics and statistics and develop their understanding of the role of mathematics and statistics in society.

#### Thinking

Students in Mathematics and Statistics will:

- develop mathematical and statistical reasoning, critical-thinking skills, and the capability to work through problems systematically
- develop mathematical and statistical literacy for the purpose of interpreting and evaluating mathematical and statistical data
- use creative thinking and experimentation to further mathematical and statistical comprehension
- understand how to apply mathematical methods and concepts to material problems and contexts, within the world of work.

#### Using language, symbols, and texts

Students in Mathematics and Statistics will:

- develop their ability to make meaning of mathematical and statistical symbols, equations, expressions and graphs
- explain working and reasoning when solving mathematical or statistical problems
- interpret and communicate mathematical and statistical ideas for varied purposes and to solve problems.

#### Relating to others

Students in Mathematics and Statistics will:

- understand how to express mathematical and statistical information for different purposes and audiences
- collect and explore mathematical and statistical data to enhance their understanding of problems and situations which relate to life in Aotearoa.

#### Managing self

Students in Mathematics and Statistics will:

- become capable learners as they develop confidence to apply mathematical concepts to material problems and contexts, within the world of work
- make increasingly appropriate selection of mathematical and statistical methods and processes in appropriate circumstances.

#### Participating and contributing

Students in Mathematics and Statistics will:

- be actively involved in communities through analysing local mathematical and statistical information, and building upon their knowledge to participate in discussion and discourse
- apply mathematical and statistical skills to problems outside of the classroom.

This section of *New Zealand Curriculum online* offers specific guidance to school leaders and teachers on integrating the Key Competencies into the daily activities of the school and its Teaching and Learning Programmes.

## Introduction to sample course outlines

Sample Course Outlines are being produced to help teachers and schools understand the new NCEA Learning Matrix and Achievement Standards. The draft Course Outlines that were published at the end of Phase 1, Level 1 product development are now being taken down. Work will continue on these, reflecting the changes noted in the SEG responses, and the additional detail that will be provided in Phase 2 products. They will be re-published for the next cycle of feedback on the Phase 2 products in early August 2021. Exemplars of student work will be provided after the Pilot phase in 2022.

## Assessment Matrix

## Unpacking The Standards

These statements help to unpack the ways in which the Achievement Standards assess the Significant Learning in the Learning Matrix.

#### 1.1 (Internal) Investigate data for a purpose, using statistical methods

The purpose of this standard is to build in ākonga the capabilities required to use primary data to conduct investigations and explore situations in a variety of ways, for specific purposes.

Ākonga will use the statistical or probability enquiry cycles to collect primary data and carry out three different investigation types.

#### 1.2 (Internal) Solve mathematical and statistical problems that relate to life in Aotearoa and the Pacific

The purpose of this standard is to build in ākonga the capabilities required to apply mathematics and statistics to real world problems. These problems will be approached and solved differently according to ākonga backgrounds, interests, and pathways, in recognition of the breadth of diversity in Aotearoa and the Pacific.

Ākonga will make connections to form or use a mathematical or statistical model, while integrating information from a variety of sources and curriculum strands, to make predictions and generalisations.

#### 1.3 (External) Evaluate mathematical and statistical information

The purpose of this standard is to develop in ākonga the capabilities required to use mathematics and statistics to evaluate, critique and predict in a given context.

Ākonga will recognise the mathematics and statistics needed to make well founded judgements and decisions about evidentiary claims, graphs and other represented information. Ākonga will be given the opportunity to interpret media, make informed life-skill related choices, and work with rates.

#### 1.4 (External) Demonstrate reasoning in mathematical contexts

The purpose of this standard is to develop in ākonga the capabilities required to use mathematics and probability to demonstrate reasoning and assess the truth of statements.

Ākonga will reason mathematically to prove or disprove, manipulate, solve, model and show equivalency, across curriculum strands. This Standard will focus on mathematical theories, not how mathematics and probability apply to real-life contexts.