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Mathematics

Introduction

Mathematics is a human endeavour that has developed by practice and theory from the dawn of civilisation to the present day. Many societies and cultures have contributed to the growth of mathematics, often in times of scientific, technological, artistic and philosophical change and development. Complementary to this broad perspective of mathematics are the various mathematical practices that take place day to day in communities around the world.

While the usefulness of mathematics for modelling and problem solving is well known, mathematics also has a fundamental role in enabling cultural, social and technological advances, and empowering individuals as critical citizens in contemporary society and for the future. Number, space and measurement, chance and data are common aspects of most people’s mathematical experience in everyday personal, study and work situations. Equally important are the essential roles that mathematical structure and working mathematically play in people’s understanding of the natural and human worlds.

Mathematics can be described in terms of its objects, what they are and how they came to be; its established body of knowledge and why this is held to be true; its effective application in science, technology and other domains; and the practice and activities of mathematicians past and present. Aims for essential learning in school mathematics are for students to:

Mathematical knowledge includes knowledge of concepts, objects, definitions and structures. A small collection of mathematical ideas, objects, structures, and relationships between these, is taken as undefined and given in a context. New mathematical objects, structures and relationships are developed from these moving from simple to more complex and sophisticated ideas and practices. The motivation for accepting certain things as given building blocks for mathematical knowledge may be initially related to intuitive understanding of particular ideas and objects experienced with respect to the natural or human worlds. These and their subsequent developments are not empirical knowledge, but abstract mathematical entities.

Whether mathematical knowledge is viewed as being essentially mind dependent or mind independent, discovered or constructed, its abstract nature gives rise to the applicability of mathematics in a wide range of contexts, as mathematical objects, structures and relationships do not depend on a particular context for their existence, but are interpreted to model key features of these contexts. This abstraction poses a challenge to the teacher and student alike, and both will need to draw on knowledge of the world and link this to mathematical knowledge and its application in various situations.

Mathematical reasoning and thinking underpins all aspects of school mathematics, including problem posing, problem solving, investigation and modelling. It encompasses the development of algorithms for computation, formulation of problems, making and testing conjectures, and the development of abstractions for further investigation.

Computation and proof are essential and complementary aspects of mathematics that enable students to develop thinking skills directed toward explaining, understanding and using mathematical concepts, structures and objects. They provide a framework for the development of mathematical skills and techniques exemplified in the use of algorithms for computation and for the development of general case arguments.


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Structure of the domain

The Mathematics domain is organised into six sections, one for each level of achievement from Levels 1 to 6. Each level includes a learning focus statement and a set of standards organised by dimension. A glossary is included which provides definitions of or additional information about underlined terms.

Learning focus

Learning focus statements are written for each level. These outline the learning that students need to focus on if they are to progress in the domain and achieve the standards at the levels where they apply. They suggest appropriate learning experiences from which teachers can draw to develop relevant teaching and learning activities.

Standards

Standards define what students should know and be able to do at different levels and are written for each dimension. In Mathematics, standards for assessing and reporting on student achievement apply from Level 1. Standards for Structure are introduced from Level 3.

Dimensions

Standards in the Mathematics domain are organised in five dimensions:

Number

The Number dimension focuses on developing students’ understanding of counting, magnitude and order. The natural (counting) numbers with zero extend to positive and negative signed whole numbers (integers) and through part-whole relations and proportions of whole numbers to the rational numbers (fractions and finite decimals or infinite recurring decimals).

Proportions of lengths involving sides and/or diagonals of right-angled triangles and rectangles and arcs of a circle lead to the introduction of certain irrational real numbers such as the square root of 2, the golden ratio phi and fractions or multiples of pi.

Principal operations for computation with number include various algorithms for addition (aggregation), subtraction (disaggregation) and the related operations of multiplication, division and exponentiation carried out mentally, by hand using written algorithms, and using calculators, spreadsheets or other numeric processors for calculation.

Space

The Space dimension focuses on developing students’ understanding of shape and location. These are connected through forms of representation of two- and three-dimensional objects and the ways in which the shapes of these objects and their ideal representations can be moved or combined through transformations. Students learn about key spatial concepts including continuity, edge, surface, region, boundary, connectedness, symmetry, invariance, congruence and similarity.

Principal operations for computation with space include identification and representation, construction and transformation by hand using drawing instruments, and also by using dynamic geometry technology.

Measurement, chance and data

The Measurement, chance and data dimension focuses on developing students’ understanding of unit, measure and error, chance and likelihood and inference. Measure is based on the notion of unit (informal, formal and standard) and relates number and natural language to measuring characteristics or attributes of objects and/or events. Various technologies are used to measure, and all measurement involves error.

Students learn important common measures relating to money, length, mass, time and temperature, and probability – the measure of the chance or likelihood of an event. Other measures include area, volume and capacity, weight, angle, and derived rates such as density, concentration and speed.

Principal operations for computation with measurement include the use of formulas for evaluating measures, the use of technology such as dataloggers for direct and indirect measurement and related technologies for the subsequent analysis of data, and estimation of measures using comparison with prior knowledge and experience, and spatial and numerical manipulations.

Structure

The Structure dimension focuses on developing students’ understanding of set, logic, function and algebra. It is fundamental to the concise and precise nature of mathematics and the generality of its results. Key elements of mathematical structure found in each of the dimensions of Mathematics are membership, operation, closure, identity, inverse, and the commutative, associative and distributive properties as well as other notions such as recursion and periodic behaviour.

While each of these can be considered in its own right, it is in their natural combination as applied to elements of number, space, function, algebra and logic with their characteristic operations that they give rise to the mathematical systems and structures that are embodied in each of these dimensions.

Principal operations for computation with structure include mental, by hand and technology-assisted calculation and symbolic manipulation by calculators, spreadsheets or computer algebra systems, with sets, logic, functions and algebra.

Working mathematically

Working mathematically focuses on developing students’ sense of mathematical inquiry: problem posing and problem solving, modelling and investigation. It involves students in the application of principled reasoning in mathematics, in natural and symbolic language, through the mathematical processes of conjecture, formulation, solution and communication; and also engages them in the aesthetic aspects of mathematics.

In this dimension the nature, purpose and scope of individual work is connected to that of the broader mathematical community, and the historical heritage of mathematics through the discourse of working mathematically. Mental, by hand and technology-assisted methods provide complementary approaches to working mathematically.

Relationships between the dimensions

Number is related to the other dimensions through the aspects of counting, magnitude and order. It has logical and natural connections with Measurement, chance and data, and Space. Number systems provide the basis for the development of algebraic relationships in Structure and the contexts and explorations used in Working mathematically.

Space is related to the Number and Measurement, chance and data dimensions through the aspects of shape and location. The properties of patterns, transformations, and symmetry provide links to Structure and Working mathematically.

Measurement, chance and data is related to the Number and Space dimensions through the aspects of units, error, approximation, likelihood, angle, and the properties of two- and three-dimensional shapes. The application of measurement formulas and functions provide a link to Structure. A varied collection of practical contexts for generating and testing conjectures provides links to Working mathematically.

Structure is related to the Number, Space and Measurement, chance and data dimensions through the use of algorithms, patterns and functions. It is linked to Working mathematically through the key elements of mathematical language, concepts and relationships used in modelling and investigations.

Working mathematically is related to the Number, Space and Measurement, chance and data dimensions through the exploration of algorithms, patterns and functions, shapes and dimensions. It provides the processes for the development of inferences and deductions and for the exploration and proof of conjectures related to the Structure dimension.


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Stages of learning

The VELS take account of the developmental stages of learning young people experience at school. While student learning is a continuum and different students develop at different rates, they broadly progress through three stages of learning. General statements about characteristics of learners in these three stages are available at Stages of learning.

The following statements describe ways in which these characteristics relate to learning experiences and standards in each of the three stages of learning in the Mathematics domain.

Years Prep to 4 – Laying the foundations

During these years students develop fundamental knowledge of number, space, measurement and the foundations of the development and use of logical and systematic mathematical processes.

Early in this stage, mathematical activities centre on play and the manipulation of physical objects in settings that support engagement and behavioural and social development. Cognitive development of strong mathematical concepts is supported by the use of social and environmental contexts – students are encouraged to describe and discuss their immediate environment and daily activities using the terms and constructs of elementary mathematics. By sharing and interacting with others, students’ existing knowledge and concepts are further developed; and opportunities arise for challenging false notions such as that a six is harder to roll on a die than another number.

Early in this stage, students sort, count and compare concrete objects, and draw, arrange and manipulate simple shapes and objects. They use and describe basic measurement concepts related to themselves or familiar objects.

Later in this stage, students begin to recognise the structure of number and develop cognitive understanding of number as an object in its own right, and extend their number knowledge and representation of mathematical processes beyond their immediate environment. They can recognise and work with simple patterns in number and space and recognise the use of mathematics in daily life.

Years 5 to 8 – Building breadth and depth

During this stage, students develop many of the abstract and conceptual understandings of mathematics required for later success. Students become increasingly complex thinkers and can apply logical reasoning and related mathematical processes to both concrete and abstract ideas. However, the rate of cognitive, emotional and behavioural development that enables students to begin working with abstract ideas varies significantly between students and is dependent on both environmental and social factors. It remains important that students can recognise and appreciate contextual and personally relevant applications of the mathematics being studied.

With an increasingly outward focus on mathematical work, students expand the use of conjectures and hypotheses, and develop greater sophistication in the use of mathematical language.

Using conceptual understandings of the structure of number, students refine their mental and by-hand algorithms for computations, for example, developing and using criteria for deciding if an estimate is reasonable, and use them in investigations They become aware of ‘general case’ arguments for propositions and explore the role of counter-examples.

Towards the end of this stage, students use abstractions in number, space, measurement, chance and data, and structure as objects for further manipulation, for example, drawing graphs for functions specified by rules. Students are supported in their development of independence and creative and critical thinking processes by the use of a range of technology to explore mathematical ideas and processes.

Years 9 to 10 – Developing pathways

In this stage, students begin to consider their futures as adults. Their learning is strongly motivated by what they consider to be relevant and important to potential life and work pathways, so although they are increasingly using more abstract and conceptual ideas in mathematics, connections and relevance to real-life situations remain important.

Students develop a broad understanding of mathematical concepts and processes. They increasingly recognise connections between concepts and can apply them to investigations, problem solving and modelling in both familiar and unfamiliar contexts. Teachers need to provide students with opportunities for open-ended and extended investigations with peers. These not only support students’ increasing independence and need for intellectual challenge, but also cater for differences in conceptual development and perceived purposes of mathematics.

Competent learners can build new abstractions based on existing ones and create mathematical understandings that can be applied to both contextual situations and in purely mathematical terms. As students recognise the place of specialised learning in possible futures, they develop a wider sense of the purpose of their studies and at the same time appreciate links to other spheres of knowledge; they can perform computations with abstract irrational numbers, but also make connections with the world around them.

At this stage, students regularly use formal representations for logical, spatial and algebraic variables, and mathematical expressions. They develop their appreciation and knowledge of the role of deductive reasoning, general argument and proof in mathematics. They can apply metacognitive strategies to identify and reflect on assumptions about possible limitations to results of investigations. They use a wide range of technologies to carry out computations and analysis of abstract representations in all aspects of working mathematically.


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National Statements of Learning

The Victorian Essential Learning Standards (VELS) incorporate the opportunities to learn covered in the national Statements of Learning (www.mceetya.edu.au/mceetya/statements_of_learning,22835.html). The Statements of Learning describe essential skills, knowledge, understandings and capacities that all young Australians should have the opportunity to learn by the end of Years 3, 5, 7 and 9 in English, Mathematics, Science, Civics and Citizenship and Information and Communication Technologies (ICT).

The Statements of Learning were developed as a means of achieving greater national consistency in curriculum outcomes across the eight Australian states and territories. It was proposed that they be used by state and territory departments or curriculum authorities (their primary audience) to guide the future development of relevant curriculum documents. They were agreed to by all states and territories in August 2006.

During 2007, the VCAA prepared a detailed map to show how the Statements of Learning are addressed and incorporated in the VELS. In the majority of cases, the VELS learning focus statements incorporate the Statements of Learning. Some Statements of Learning are covered in more than one domain. In some cases, VELS learning focus statements have been elaborated to address elements of the Statements of Learning not previously specified. These elaborations are noted at the end of each learning focus statement.


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Numeracy in the Mathematics VELS

The Mathematics aims clearly specify that students are to:

Together these aims articulate capacities and dispositions that underpin numeracy and numerate behaviour. Specific numeracy capabilities are developed through student achievement of the content and performance standards in the five VELS dimensions: Number, Space, Measurement, chance and data, Structure and Working mathematically. As students engage in learning mathematics and demonstrate achievement of the standards at each level, they will acquire numeracy skills and related dispositions that enable them to use mathematics to pose and solve problems sensibly and confidently in a variety of different situations, and to communicate effectively about this with others.

The Mathematics standards, in conjunction with the National Numeracy Benchmarks and National Numeracy Testing (www.vcaa.vic.edu.au/prep10/aim/teachers/nationaltesting.html), provide teachers with a framework for monitoring student numeracy. Numeracy skills and dispositions are important in supporting studies in other domains in and across the three VELS strands: Physical, Personal and Social Learning, Discipline-based Learning and Interdisciplinary Learning. In turn, there are many contexts in VELS domains across the strands that require the effective application of concepts, skills and processes from Number, Space, Measurement, chance and data, Structure and Working mathematically and contribute to strong numeracy development as part of student learning.


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National Numeracy Benchmarks

National Numeracy Benchmarks are used for reporting achievement in three aspects of numeracy – ‘Number sense’, ‘Spatial sense’ and ‘Measurement and data sense’ – at Years 3, 5 and 7. The benchmarks define nationally agreed minimum acceptable standards for numeracy at these years.

Full details of the National Numeracy Benchmarks are available in Numeracy Benchmarks Years 3, 5 and 7, Curriculum Corporation (http://cms.curriculum.edu.au/numbench/index.htm), 2000.

The benchmarks describe minimum standards. For this reason, the Year 3 benchmarks relate to Level 2 Mathematics standards, the Year 5 benchmarks relate to Level 3 Mathematics standards and the Year 7 benchmarks relate to Level 4 Mathematics standards. Links to the numeracy benchmarks are located in the Mathematics standards.


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Pathways to VCE, VCE VET and VCAL

As students approach the end of the compulsory years of schooling they begin to make choices about their preferred areas of and pathways for learning. Students choose studies from the Victorian Certificate of Education (VCE) or recognised vocational training through either a Vocational Education Training (VET) program or the Victorian Certificate of Applied Learning (VCAL).

The Mathematics domain provides students with pathways to a wide range of VCE studies, VET programs, and VCAL units, as well as to the world of work. The standards ensure that students have both sufficient knowledge and skills, and appropriate learning strategies, to pursue all possible options.

The following diagram outlines some of the possible pathways available to students moving into the post-compulsory years. There are other possible sequences or combinations of options not shown here, such as VCE VET programs.

Victorian Essential Learning Standards – Mathematics Domain

VCE

Foundation Mathematics Units 1 & 2
General Mathematics Units 1 & 2
Mathematical Methods Units 1 & 2
Mathematical Methods (CAS) Units 1 & 2


Further Mathematics Units 3 & 4
Mathematical Methods Units 3 & 4
Mathematical Methods (CAS) Units 3 & 4
Specialist Mathematics Units 3 & 4

VET

Possible VET pathways with mathematics links include...
Agriculture
Automotive
Building and Construction
Electronics
Engineering
Laboratory Skills
Financial Services

VCAL

Any of the VCE Mathematics studies will meet the eligibility requirement for the numeracy component of the VCAL Literacy and Numeracy Skills strand.

Victorian Certificate of Education (VCE) Mathematics studies

The five VELS dimensions: Number, Space, Measurement, chance and data, Structure, and Working mathematically provide a comprehensive preparation for all VCE Mathematics studies. This applies to both the content of the VCE studies, in terms of suitable preparatory domain knowledge and skills, and also approaches to working mathematically involving problem solving, modelling, investigation and the effective use of technology.

Pathway 1 (2 units of VCE Mathematics)

Students choosing this pathway do not intend to study Mathematics beyond Units 1 and 2 level. Typically, they wish to study Mathematics at the Units 1 and 2 level to provide support for other VCE, VCAL, or VET studies, or in preparing for general employment. These students would generally study either Foundation Mathematics Units 1 and 2 or an implementation of General Mathematics Units 1 and 2 designed to meet these needs.

Pathway 2 (4 units of VCE Mathematics)

Students choosing this pathway intend to study General Mathematics Units 1 and 2 as preparation for Further Mathematics Units 3 and 4. This pathway would provide a flexible preparation for further study or work, with a significant study of statistics, but without study of calculus and related content on algebra and functions.

Pathway 3 (4, 6 or 8 units of mathematics)

Students choosing this pathway intend to study Mathematical Methods or Mathematical Methods (CAS) Units 1 and 2 (possibly in conjunction with General Mathematics Units 1 and 2), and subsequently study Mathematical Methods or Mathematical Methods (CAS) Units 3 and 4, possibly in conjunction with Further Mathematics Units 3 and 4. This pathway would provide a suitable preparation for further study in fields such as science, medicine or economics where a strong background in functions, algebra, calculus and probability is required.

Pathway 4 (8 or 10 units of mathematics)

Students choosing this pathway intend to study Mathematical Methods Units 1 and 2 or Mathematical Methods (CAS) Units 1 and 2 in conjunction with an implementation of General Mathematics Units 1 and 2 which emphasises mathematical structure and proof; leading to subsequent study of Mathematical Methods or Mathematical Methods (CAS) Units 3 and 4 and Specialist Mathematics Units 3 and 4, possibly in conjunction with Further Mathematics Units 3 and 4 This pathway provides a suitable preparation for further study in fields which require a strong mathematical background, such as computer science, engineering, actuarial studies and mathematics.

More information about VCE (www.vcaa.vic.edu.au/vce)

More information about VET (www.vcaa.vic.edu.au/vet)

More information about VCAL (www.vcaa.vic.edu.au/vcal)


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Downloads

Mathematics: Level 1 2 3 4 5 6


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