## What is Mathematics and Statistics about?

In conversation with

Jim Davis

Liz Sneddon

Katalina Ma

#### Transcript below:

I reckon the biggest change you'll notice is, that the whole subject has been unpacked. It's not siloed anymore. So they actually have that chance to teach across the curriculum in one go, and assess from across the curriculum.

Knowledge of Mathematics and Statistics isn't compartmentalised anymore. I think it's a good thing, because we tend to just focus on one area. Then, when we want students to be able to go back to that area of learning, they thought it was just for the beginning of the year. So yeah, real exciting stuff.

For ākonga, they'll develop their maths more holistically, and realise that it is all connected. But not only that, with only four standards, they are... Their workload is going to go down significantly. So pretty much each term they might have only one assessment. So more time on the teaching and learning, less time on the assessment.

We're still assessing, still learning the same stuff, but the balance has changed. I think there's the opportunity to assist in a different way for our students. Not all our students like the sort of sit-down traditional kind of testing. Sort of being able to look at a possible project way of learning and way of assessing, as well as opportunities to do integrated units, cross-curricular with other subjects.

The idea that people who, or any student, that picks up any of these assessments, will be able to see themselves in it. Therefore will be able to bring their own mātauranga into what they're doing. Into their learning and their products, that they're making. I think that's what we'll see.

I know for me personally, I've got a better understanding than I had. I've still got a long way to go on my journey. But I've got a much better understanding than I had of what it means to be completely inclusive.

Yeah, I think it's really important, that we reiterate the fact, that the Learning Matrix we've put together has nothing new in it. It has nothing omitted from it. It is what was always there for curriculum level six. All we are asking our teachers to do now is maybe approach that differently, and have a different understanding of how they link together. You want students to take these skill sets and be able to build their communities with it, build themselves as individuals, use it later on in life.

That it's a... If we're wanting to produce these lifelong learners, that it's evident in what we're delivering in terms of our curriculum. We've had so many varied experiences from a range of different schools. From a range of different types of students, from all of that kind of stuff. It means that each of us are coming in with our own ideas, and putting it together to make something that is bigger than any one of us by ourselves.

We're able to form something cohesive that takes elements off everyone, and everyone has contributed to in amazing ways. It's not to say it's been a smooth sailing, there's definitely been difficult conversations. But as a teacher and as a person, I know I've grown.

Some of the research readings that we've had, oh my gosh, it has blown my mind. Because it's made me grow, and think, and reflect in ways that I hadn't before. That has been so powerful for me. To have that opportunity has been phenomenal.

I think for me, the biggest takeaway I've had from being in the group is the confidence to enact change. As a head of a faculty, at a fairly large one, with some teachers have been there a while, it's not always easy to make change. But I think for them to see me involved in this, and me to bring the expertise of the group back to our faculty, and help us develop and grow as a team, that's been for me the biggest personal growth. And you're right, there was some sort of uncomfortable, sort of conversations and stuff but conversations that needed to happen.

The biggest piece of advice I'll give is be open-minded. We haven't redesigned maths. We've just redesigned the approach to maths. For the absolute purist out there, nothing's gone, nothing's been removed, nothing's been dumbed down. It's just how we approach it. We want them to be inclusive, we want them to explore, and be free to teach in the way that we love teaching.

Find those problems that we want to tackle, that we enjoy. Pull out from the kids that love of Maths and that love of Stats that we have. We want to be able to get out of them. And that's the real value, let's get back into some of that really amazing, connected stuff. Build those relationships with your students, get to know who they are. Then figure out what Maths might work for them, and let's see how far we can take this.

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

This section outlines the meaning of, and connection between, the Big Ideas and Significant Learning, which together form the Learning Matrix. It then explains each Mathematics and Statistics Big Idea.

The Mathematics and Statistics Learning Area curriculum, including its Whakataukī, inform this subject's Significant Learning – learning that is critical for students to know, understand, and do in a subject by the end of each Curriculum Level. This covers knowledge, skills, competencies, and attitudes. It also includes level-appropriate contexts students should encounter in their education. The Learning Area's Whakataukī is:

Kei hopu tōu ringa ki te aka tāepa, engari kia mau ki te aka matua.

Cling to the main vine, not the loose one.

The subject's Big Ideas and Significant Learning are collated into a Learning Matrix for Curriculum Level 6. Teachers 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 students are particularly interested in. This context or topic must relate to at least one Big Idea and may also link to other Big Ideas.

There are six Big Ideas in Mathematics and Statistics. The nature of this subject as a discipline means aspects of Significant Learning often cross over multiple Big Ideas, and vice versa.

#### 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 topics (number, algebra, geometry, measurement, statistics, and probability). It is framed in this way to show that most learning in Mathematics and Statistics is cross-topic. Topics can be taught together, and not unnecessarily compartmentalised. The Learning Matrix provides a starting point for kaiako to weave the topics 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 New Zealand’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 the particular contexts and future pathways of ākonga.

#### 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 topics 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 discussion, 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

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. Critical thinking includes indigenous ways of acquiring knowledge, for example talanoa and collaboration
- 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 New Zealand.

#### 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.

__Key Competencies __

This section of The *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. Examples of how a year-long Mathematics and Statistics course could be constructed using the new Learning Matrix and Achievement Standards are provided here. They are indicative only and do not mandate any particular context or approach.

More detailed sample Teaching and Learning Programmes will be developed during piloting.

## Assessment Matrix

## Conditions of Assessment for internally assessed standards

This section provides guidelines for assessment against internally assessed Standards. Guidance is provided on:

- appropriate ways of, and conditions for, gathering evidence
- ensuring that evidence is authentic
- any other relevant advice specific to an Achievement Standard.

**NB**: Information on additional generic guidance on assessment practice in schools is published on the NZQA website. It would be useful to read in conjunction with these Conditions of Assessment.

The school's Assessment Policy and Conditions of Assessment must be consistent with the Assessment Rules for Schools With Consent to Assess. These rules will be updated during the NCEA review. This link includes guidance for managing internal moderation and the collection of evidence.

#### For all Achievement Standards

Internal assessment provides considerable flexibility in the collection of evidence. Evidence can be collected in different ways to suit a range of teaching and learning styles, and a range of contexts. Care needs to be taken to offer students opportunities to present their best evidence against the Standard(s) that are free from unnecessary constraints.

It is recommended that the design of assessment reflects and reinforces the ways students have been learning. Collection of evidence for the internally assessed Standards could include, but is not restricted to, an extended task, an investigation, digital evidence (such as recorded interviews, blogs, photographs or film), or a portfolio of evidence.

It is also recommended that the collection of evidence for internally assessed Standards should not use the same method that is used for any external Standards in a course, particularly if that method is using a time-bound written examination. This could unfairly disadvantage students who do not perform well under these conditions.

A separate assessment event is not needed for each Standard. Often assessment can be integrated into one activity that collects evidence towards two or three different Standards from a programme of learning. Evidence can also be collected over time from a range of linked activities (for example, in a portfolio). This approach can also ease the assessment workload for both students and teachers.

Effective assessment should suit the nature of the learning being assessed, provide opportunities to meet the diverse needs of all students, and be valid and fair.

Authenticity of student evidence needs to be assured regardless of the method of collecting evidence. This needs to be in line with school policy. For example: an investigation carried out over several sessions could include teacher observations or the use of milestones such as a meeting with the student, a journal, or photographic entries recording progress etc.

- Ākonga need to work independently on this task, although planning and collection of data (if required) can be done in groups.
- Ākonga will either be given a general purpose and may select an aspect to focus on, or they could come up with their own purpose.
- Ākonga must plan and collect data; however, additional data can be provided to them (eg collect data for a comparison, and then be provided with a time series dataset).
- Kaiako can provide feedback on a student's plan.
- Data collecting may involve physical collection (eg by taking measurements), creating a questionnaire, collecting data from the internet, or other valid collection methods.
- Data collected should be appropriate to ākonga and their environment.
- Kaiako will provide guidance to ākonga on the selection of appropriate data sets.
- Ākonga will have access to appropriate technology and resources.

- Kaiako can provide feedback on a student's plan for their exploration.
- Problems that relate to Aotearoa New Zealand or the Pacific region may include the values, beliefs, and customs of the people of Aotearoa New Zealand and the Pacific region, especially Māori and diverse Pacific Islands people.
- Ākonga will have access to appropriate technology and resources.

Students need to be familiar with methods, ie, procedures and reasoning related to the following:

Number:

- more complex rates, ratio, and proportion (including scale diagrams)
- negative and fractional powers
- percentage: increase, decrease, and inverse
- standard form.

Algebra:

- formulae
- graphing
- manipulating and simplifying expressions
- inequations
- quadratic and simple exponential equations
- simultaneous linear equations with two unknowns
- optimal solutions
- relating graphs, tables, equations, and patterns
- relating rate of change to the gradient of a graph
- forming, graphing, or manipulating linear models.

Geometry

- properties of similar shapes
- Pythagoras’ theorem in 3D situations
- trigonometric ratios in right-angled triangles
- transformations (reflection, rotation, translation, and enlargement): their key features and symmetry of patterns.

Measurement

- surface area of prisms, pyramids, cones, and spheres
- volume of composite prisms, pyramids, cones, and spheres
- conversions between more complex metric units such as area, volume, and derived measures, eg, km/h.