Working Mathematically relates to the syllabus objective:
Students develop understanding and fluency in mathematics through inquiry, exploring and connecting mathematical concepts, choosing and applying problem-solving skills and mathematical techniques, communication and reasoning
As an essential part of the learning process, Working Mathematically provides students with the opportunity to engage in genuine mathematical activity and develop the skills to become flexible and creative users of mathematics.
In this syllabus, Working Mathematically encompasses five interrelated components:
Students develop the ability to use a variety of representations, in written, oral or graphical form, to formulate and express mathematical ideas. They are communicating mathematically when they describe, represent and explain mathematical situations, concepts, methods and solutions to problems, using appropriate language, terminology, tables, diagrams, graphs, symbols, notation and conventions.
Students develop the ability to make choices, interpret, formulate, model and investigate problem situations, and communicate solutions effectively. They formulate and solve problems when they use mathematics to represent unfamiliar or meaningful situations, design investigations and plan their approaches, apply strategies to seek solutions, and verify that their answers are reasonable.
Students develop an increasingly sophisticated capacity for logical thought and actions, such as analysing, proving, evaluating, explaining, inferring, justifying and generalising. They are reasoning mathematically when they explain their thinking, deduce and justify strategies used and conclusions reached, adapt the known to the unknown, transfer learning from one context to another, prove that something is true or false, and compare and contrast related ideas and explain their choices.
Students build a strong foundation that enables them to adapt and transfer mathematical concepts. They make connections between related concepts and progressively apply the familiar to develop new ideas. Students develop an understanding of the relationship between the 'why' and the 'how' of mathematics. They build understanding when they connect related ideas, represent concepts in different ways, identify commonalities and differences between aspects of content, describe their thinking mathematically, and interpret mathematical information.
Students develop skills in choosing appropriate procedures, carrying out procedures flexibly, accurately, efficiently and appropriately, and recalling factual knowledge and concepts readily. They are fluent when they calculate answers efficiently, recognise robust ways of answering questions, choose appropriate methods and approximations, recall definitions and regularly use facts, and manipulate expressions and equations to find solutions.
The five components of Working Mathematically describe how content is explored or developed − that is, the thinking and doing of mathematics. They provide the language to build in the developmental aspects of the learning of mathematics. The components come into play when students are developing new skills and concepts, and also when they are applying their existing knowledge to solve routine and non-routine problems both within and beyond mathematics. At times the focus may be on a particular component of Working Mathematically or a group of the components, but often the components overlap. While not all of the Working Mathematically components apply to all content, they indicate the breadth of mathematical actions that teachers need to emphasise.
In addition to its explicit link to one syllabus objective, Working Mathematically has a separate set of outcomes for the components Communicating, Problem Solving and Reasoning. This approach has been adopted to ensure students' level of proficiency in relation to these components becomes increasingly sophisticated over the years of schooling.
Separate syllabus outcomes have not been developed for the Working Mathematically components Understanding and Fluency. These components are encompassed in the development of knowledge, skills and understanding across the range of syllabus strands, objectives and outcomes.
Teachers are able to extend students' level of proficiency in relation to the components of Working Mathematically by creating opportunities for their development through the learning experiences that they design.
Strand overview: Number and Algebra
The knowledge, skills and understanding developed in the Number and Algebra strand are fundamental to the other strands of this syllabus and are developed across the stages from Early Stage 1 to Stage 5.3.
Numbers, in their various forms, are used to quantify and describe the world. From Early Stage 1 there is an emphasis on the development of number sense, and confidence and competence in using concrete materials and mental, written and calculator techniques for solving appropriate problems. Algorithms are introduced after students have gained a firm understanding of basic concepts, including place value, and have developed mental strategies for computing with two- and three-digit numbers. Approximation is important and the systematic use of estimation is to be encouraged at all times. Students should always check that their answers 'make sense' in the contexts of the problems that they are solving.
In the early stages, students explore number and pre-algebra concepts by pattern making, and by discussing, generalising and recording their observations. This demonstrates the importance of early number learning in the development of algebraic thinking and the algebra concepts that follow.
The use of mental-computation strategies should be developed in all stages. Information and communication technology (ICT) can be used to investigate number patterns and relationships, and facilitate the solution of real problems involving measurements and quantities not easy to handle with mental or written techniques.
In Stage 2 to Stage 5, students apply their number skills to a variety of situations, including financial situations and practical problems, developing a range of life skills important for numeracy. Ratios and rates underpin proportional reasoning needed for problem solving and the development of concepts and skills in other aspects of mathematics, such as trigonometry, similarity and gradient.
Following the development of foundational number skills and pre-algebra concepts through patterning, a concrete approach to algebra is continued when students generalise their understanding of the number system to the algebraic symbol system. They develop an understanding of the notion of a variable, establish the equivalence of expressions, apply algebraic conventions, and graph relationships on the number plane.
Students recognise that graphing is a powerful tool that enables algebraic relationships to be visualised. The use of ICT for graphing provides an opportunity for students to compare and investigate these relationships dynamically. By the end of Stage 5.3, students have the opportunity to develop knowledge and understanding of the shapes of graphs of different relationships and the effects of performing transformations on these graphs.
Algebra has strong links with the other strands in the syllabus, particularly when situations are to be generalised symbolically.
Strand overview: Measurement and Geometry
Measurement enables the identification and quantification of attributes of objects so that they can be compared and ordered, while geometry is the study of spatial forms and involves representation of the shape, size, pattern, position and movement of objects in the three-dimensional world or in the mind of the learner. The study of geometry enables the investigation of three-dimensional objects and two-dimensional shapes, as well as the concepts of position, location and movement.
The presentation of Measurement and Geometry as a single strand recognises and emphasises their interrelationship.
The term 'geometry', derived from the Greek geo, meaning 'earth', and metria, meaning 'measure', traditionally has included relationships between the magnitude of the sides and angles of geometrical figures. While the units used to measure the magnitude of the sides may change, the relationships between the sides remain constant. A focus on the development of units of measure is pivotal in distinguishing the key ideas studied in measurement from the key ideas studied in geometry.
Important and critical skills for students to acquire are those of recognising, visualising and drawing shapes and describing the features and properties of three-dimensional objects and two-dimensional shapes in static and dynamic situations. Manipulation of a variety of real objects and shapes is crucial to the development of appropriate levels of imagery, language and representation. ICT, and dynamic geometry software in particular, can be used to facilitate the exploration and manipulation of shapes, geometric relationships and two-dimensional representations of three-dimensional objects.
Geometry uses systematic classification of angles, triangles, regular polygons and polyhedra. The ability to classify is a trait of human cultural development and an important aspect of education. Class inclusivity is a powerful tool in reasoning and determining properties. Justification and reasoning in both an informal and, later, a formal way are fundamental to geometry in Stage 4 and Stage 5.
When classifying quadrilaterals, for example, students need to begin to develop an understanding of inclusivity within the classification system. Quadrilaterals are inclusive of the parallelograms, trapeziums and kites, while parallelograms are inclusive of the rectangles and rhombuses, which are inclusive of the squares.
Measurement involves the application of number and geometry knowledge, skills and understanding when quantifying and solving problems in practical situations. Students need to make reasonable estimates for quantities, be familiar with commonly used units for length, area, volume and capacity, and be able to convert between these units. They should develop an idea of the levels of accuracy that are appropriate to particular situations. Competence in applying Pythagoras' theorem to solve practical problems is developed in Stage 4 and applied throughout the topics involving measurement.
Strand overview: Statistics and Probability
In the Statistics and Probability strand, statistics and probability are developed initially in parallel, with the links between them then built progressively across the stages.
The study of statistics within the strand includes the collection, organisation, display and analysis of data. Early experiences are based on real-life contexts using concrete materials. This leads to data collection methods and the display of data in a variety of ways. Students are encouraged to ask questions relevant to their experiences and interests, and to design ways of investigating their questions. They should be aware of the extensive use of statistics in society and be encouraged to critically evaluate claims based on statistical evidence. Data from a variety of sources, including print-based materials and the internet, can be analysed and evaluated. Electronic tools, such as spreadsheets and other software packages, may be used where appropriate to organise, display and analyse data.
The study of chance is introduced from Stage 1 to enable the development of understanding of chance concepts from an early age. Early emphasis is on understanding the notion of chance and the use of informal language associated with chance. The understanding of chance situations is further developed through the use of simple experiments that produce data, so that students can make comparisons of the likelihood of events occurring, and begin to order chance expressions on a scale from zero to one. In later stages, students link chance concepts to numerical probabilities, and explore increasingly sophisticated methods of determining the likelihoods of events, using experimental and theoretical approaches. Emphasis should be placed on developing skills in representing outcomes of events in ways that facilitate the calculation of probabilities.
Students need to develop the ability to use the language of statistics and probability, distinguishing between such concepts as simple and compound events, mutually exclusive and non-mutually exclusive events, discrete and categorical data, and independent and dependent variables. Appropriate analysis of data and the solution of associated problems depend on sound knowledge and understanding of such terms.
Skills in evaluation, and the ability to produce reasoned judgements, lead to students building further skills in critical evaluation of statistical information. In our contemporary society, there is a constant need to understand, interpret and analyse information displayed in tabular or graphical forms. Students need to recognise ways in which information can be displayed in a misleading manner, resulting in false conclusions.