# Mathematics

## Middle School

Learning mathematics creates opportunities for and enriches the lives of all Australians. The Australian Curriculum: Mathematics provides middle school students with essential mathematical skills and knowledge in number and algebra, measurement and geometry, and statistics and probability. It develops the numeracy capabilities that all students need in their personal, work and civic life, and provides the fundamentals on which mathematical specialties and professional applications of mathematics are built.

Mathematics has its own value and beauty and our middle school mathematics program aims to instill in students an appreciation of the elegance and power of mathematical reasoning. Mathematical ideas have evolved across all cultures over thousands of years, and are constantly developing. Digital technologies are facilitating this expansion of ideas and providing access to new tools for continuing mathematical exploration and invention. The middle school curriculum focuses on developing increasingly sophisticated and refined mathematical understanding, fluency, reasoning, and problem-solving skills. These proficiencies enable students to respond to familiar and unfamiliar situations by employing mathematical strategies to make informed decisions and solve problems efficiently.

At Murray Bridge High School we aim to help students understand the links between the various components of mathematics, as well as the relationship between mathematics and other disciplines. Mathematics is composed of multiple but interrelated and interdependent concepts and systems which students apply beyond the mathematics classroom. In science, for example, understanding sources of error and their impact on the confidence of conclusions is vital, as is the use of mathematical models in other disciplines. In geography, interpretation of data underpins the study of human populations and their physical environments; in history, students need to be able to imagine timelines and time frames to reconcile related events as well as analyze and interpret data; and in English, deriving quantitative and spatial information is an important aspect of making meaning of texts.

We aim to help all students benefit from access to the power of mathematical reasoning and learn to apply their mathematical understanding creatively and efficiently. The Mathematics middle school curriculum provides students with carefully paced, in-depth study of critical skills and concepts. We encourages students to become self-motivated, confident learners through inquiry and active participation in challenging and engaging experiences.

## Senior School

Senior school students study mathematics from the Senior Secondary Australian Curriculum: Mathematics, which consists of four SACE Board accredited subjects in mathematics, with each subject organized into four units. The subjects are differentiated, each focusing on a pathway that will meet the learning needs of a particular group of senior secondary students. In order for students to achieve their SACE it is necessary that students successfully complete (at a C grade or better) at least one semester of a mathematical subject. The subjects available are described below.

**Essential Mathematics** focuses on using mathematics effectively, efficiently and critically to make informed decisions. It provides students with the mathematical knowledge, skills and understanding to solve problems in real contexts for a range of workplace, personal, further learning and community settings. This subject provides the opportunity for students to prepare for post-school options of employment and further training.

**General Mathematics **focuses on using the techniques of discrete mathematics to solve problems in contexts that include financial modelling, network analysis, route and project planning, decision making, and discrete growth and decay. It provides an opportunity to analyze and solve a wide range of geometrical problems in areas such as measurement, scaling, triangulation and navigation. It also provides opportunities to develop systematic strategies based on the statistical investigation process for answering statistical questions that involve comparing groups, investigating associations and analyzing time series.

**Mathematical Methods **focuses on the development of the use of calculus and statistical analysis. The study of calculus in Mathematical Methods provides a basis for an understanding of the physical world involving rates of change, and includes the use of functions, their derivatives and integrals, in modelling physical processes. The study of statistics in Mathematical Methods develops the ability to describe and analyze phenomena involving uncertainty and variation.

**Specialist Mathematics **provides opportunities, beyond those presented in Mathematical Methods, to develop rigorous mathematical arguments and proofs, and to use mathematical models more extensively. Specialist Mathematics contains topics in functions and calculus that build on and deepen the ideas presented in Mathematical Methods as well as demonstrate their application in many areas. Specialist Mathematics also extends understanding and knowledge and introduces the topics of vectors, complex numbers and matrices. Specialist Mathematics is the only mathematics subject that has been designed to not be taken as a stand-alone subject.

## Numeracy Improvement

### Scaffolding Numeracy in the Middle Years

Along with NAPLaN students in the middle years also undergo a diagnostic testing process as a part of the implementation of the Scaffolding Numeracy in the Middle Years Program.

The Scaffolding Numeracy in the Middle Years (SNMY) was an Australian Research Council Project awarded to RMIT University, the Victorian Department of Education and Training and the Tasmanian Education Department from July 2003 to June 2006. The project investigated the effectiveness of a new assessment-guided approach to improving student numeracy outcomes in Years 4 to 8. In particular, it was aimed at identifying and refining a learning and assessment framework for the development of multiplicative thinking at this level using rich assessment tasks. The program has since been implemented in many secondary schools across Australia with great results in improving students with difficulties in mathematics.

Unlike NAPLaN, which has a very broad focus, the SNMY diagnostic test is designed to get a very finely detailed picture of a where a student sits within the Learning and Assessment Framework for Multiplicative Thinking (LAF), a hierarchy of key ideas and strategies related to the development of multiplicative thinking and the “Big Ideas” of mathematics. It is organised into eight zones, from initial explorations with concrete materials through to complex multiplicative structures. It is common for classes to contain students across a broad range of the LAF.

Multiplicative thinking is the term used to describe a student’s ability to work flexibly with the concepts, strategies and representations of multiplication (and division) as they occur in a wide range of contexts. Multiplicative thinking is characterised by:

• a capacity to work flexibly and efficiently with an extended range of numbers (for example, larger whole numbers, decimals, common fractions, ratio and percent)

• an ability to recognise and solve a range of problems involving multiplication or division including direct and indirect proportion

• the means to communicate this effectively in a variety of ways (for example, words, diagrams, symbolic expressions and written algorithms).

Once identified, teachers can then begin to implement highly focussed learning plans, which were developed by teams of teachers working collaboratively across schools and clusters. They were designed to scaffold student learning from one zone of the Learning and Assessment Framework to the next, improving a student’s ability to work and manipulate numbers with various strategies to better equip them to solve unfamiliar problems and give them a much better chance of success in mathematics as they progress through the Big Ideas.

These big ideas include:

**Trusting the Count**: students believe that if they count the same collection again they will get the same amount; they can draw on mental objects for each of the numbers to ten based on visual imagery that allow them to “see” these numbers in terms of their parts and as they relate to numbers of which they are a part (e.g. 8 is 6 and 2, double 4, 2 less than 10).

**Place Value **(a multiplicative structure): students see 10 ones as 1 ten and are able to work fluently with counts of tens and counts of ones independently; they understand and can use the relationship that 10 these is 1 of those to extend the whole number system to hundreds and beyond.

**Multiplicative Thinking** (initial ideas): students move beyond an understanding of multiplication and division as repeated addition; they have access to efficient strategies for multiplication and division based on the number of groups rather than the number in each group (e.g. 3 of anything is double it and one more group).

**Partitioning **(another aspect of multiplicative thinking): students extend their ideas about multiplication and division to make connections to fractions, decimals and per cent; they understand and use the “for each” idea to solve simple proportional reasoning problems.

**Proportional Reasoning** (a key defining aspect of multiplicative thinking): students recognise and work with relationships between numerical quantities; they represent these in multiple ways (e.g. graphs, tables, expressions) and solve problems involving fractions, decimals, per cent, rate, ratio and proportion.

The diagnostic tests, as well as informing teaching at whole class level, will also be used to identify and assist those students who are most need of specific intervention, with small groups being given highly focused support in areas of identified weakness by mathematics teachers. It is hoped that with improved access to higher level thought processes and curriculum, students will become more engaged and successful in their studies of mathematics, leading to greater success, both at school and as members of a 21st century society.