Boys’ achievement of the outcomes in Working Mathematically, Number, Measurement, Chance and Data, Space, and Algebra will enable them to deal with quantitative and spatial ideas; to collect, display, analyse and interpret data; and to describe and reason about patterns and relationships.
The Mathematics learning area takes a major – although certainly not sole – responsibility for ensuring that boys:
- select, integrate and apply numerical and spatial concepts and techniques;
- recognise when and what information is needed, locate and obtain it from a range of sources, and evaluate, use and share it with others; and
- describe and reason about patterns, structures and relationships in order to understand, interpret, justify and make predictions.
The Mathematics learning area supports and reinforces the literacy work of the School by setting expectations and providing feedback and support to boys that is consistent with current thinking in literacy. It also takes a major responsibility for assisting boys to learn to read, write, listen to and talk about mathematics, and to develop the range of special symbols, vocabulary and diagrammatic representations that mathematics contributes to language. In these ways, the Mathematics learning area makes a direct contribution to enabling boys to use language to understand, develop and convey ideas and information and interact with others.
Mathematics plays a central role in the generation of technology generally and has, itself, changed significantly as a result of the impact of computing technologies. Through the achievement of a number of the Mathematics learning area outcomes, boys select, use and adapt technologies.
Through the Mathematics learning area, boys come to appreciate the way in which mathematics is embedded in the very fabric of our own and other societies. Their understanding of the cultural and intellectual significance of mathematical activity, and how it influences what we are and might be, enhances the extent to which they:
- understand their cultural, geographic and historical contexts, and have the knowledge, skills and values necessary for active participation in Australian life;
- interact with people and cultures other than their own and are equipped to contribute to the global community; and
- participate in creative activity of their own and understand and engage with the artistic, cultural and intellectual work of others.
The outcomes in all strands, such as number, space, measurement, algebra, and chance and data, address the mathematical attitudes, appreciations, and individual and collaborative work habits that enable boys to be critical, creative and confident users of mathematics. With the provision of a supportive environment for learning mathematics, appropriate mathematical challenge and teaching processes that foster autonomous learning in mathematics means boys:
- visualise consequences, think laterally, recognise opportunity and potential, and are prepared to test options;
- are self-motivated and confident in their approach to learning and are able to work individually and collaboratively; and
- recognise that each person has the right to feel valued and be safe and, in this regard, understand their rights and obligations and behave responsibly.
Finally, the Mathematics learning area makes an indirect contribution to several Overarching Outcomes – and hence to other learning areas – as a result of its responsibilities for numeracy. Numeracy generally benefits learning in most parts of the school curriculum, while being innumerate can inhibit and even prevent boy’s learning elsewhere. In particular, being numerate can significantly enhance boys’ capacity to:
- understand and appreciate the physical, biological and technological world and have the knowledge and skills to make decisions in relation to it;
- understand their cultural, geographic and historical contexts and have the knowledge, skills and values necessary for active participation in Australian life;
- value and implement practices that promote personal growth and wellbeing; and
- participate in creative activity of their own and understand and engage with the artistic, cultural and intellectual work of others.
In the early adolescent years, boys continue to align strongly with their peer groups. They are concerned with understanding their physical and social world, but tend to focus on how it affects them personally and how they will find their place within it. As a result, for many adolescents success with Mathematics is not its own reward, and neither is pleasing their parents or teachers.
Boys want to know how the mathematics they are learning will help them outside the Mathematics classroom: at home or in the workplace. It is also closely connected to how good they think they are at mathematics and whether they believe that they are capable of making sense of mathematics. Consequently, the curriculum should seek to challenge and extend all boys in mathematics, but within a supportive environment for learning.
Boys should be encouraged to persevere and to produce work to a quality and standard appropriate for the purpose and audience. Having worked on problems individually or collaboratively, boys should reflect upon and discuss successful and unsuccessful mathematical strategies and ideas. They should also prepare some oral or written summaries, explanations and reports in which they describe and justify their mathematical processes and conclusions. While an overly pedantic attention to the technical language of mathematics is unlikely to be productive for most boys during these years, reading, writing and talking mathematics can support learning and boys should be assisted to develop their mathematical literacy.
In the early adolescent years, boys are increasingly able to use mathematics to help make sense of the world, as distinct from using their understanding of the world to make sense of mathematics. Their widening experience of the community and other learning areas increases the range of situations to which they can apply mathematics, for example, they may draw on their understanding of the effect of scaling on area and volume to understand why babies dehydrate more quickly than adults. Various mathematical ideas are coming together, so that, in dealing with scale, boys should be encouraged to draw upon and integrate ideas from number, space, measurement and algebra.
Boys should also investigate purely mathematical ideas and relationships, gaining some experience with the cycles of conjecture, explanation and justification which typify pattern finding and problem solving.
The curriculum should enable boys to improve their capacity to represent numbers in a variety of ways and move flexibly between representations. The understanding of decimal place value should continue to be a focus of learning experiences for many boys in these years. This is necessary for such everyday purposes as ordering numbers, reading scales and measuring small quantities, the important ideas being those of order and relative magnitude. Both of these become more difficult when dealing with very small and very large numbers for which concrete or visual referents are not available.
Boys should be learning to apply number operations to a wide range of problem situations, developing the skills necessary to select operations and procedures, and judge the reasonableness of results. They need to maintain and consolidate their techniques for mental arithmetic, estimation, calculator use and paper-and-pencil work so that they become confident of their capacity to deal with everyday computational situations correctly and efficiently. Some will also extend the types of numbers on which they can operate to include addition and subtraction of negative numbers which arise in realistic settings. Boys in the early adolescent years are approaching the time when they will begin to earn money of their own and take responsibility for managing their own finances. Consequently, social and commercial arithmetic become increasingly relevant. The use of calculator technology should be assumed.
Boys should be improving their measurement skills and their ability to estimate quantities. They should become proficient with commonly used measuring equipment, develop a good feel for the size of various standard units and become competent at estimating in standard units. They should be learning that all measurement is approximate and that efficient measurement requires a sensible choice of unit; in some situations, the most accurate unit possible may be chosen, but in others a rough measure may be the best choice. Boys should carry out practical tasks involving measurement. They should, individually and collaboratively, plan, make judgments about which measurements to make, organise and carry out the measurements, and decide whether the results are of the right magnitude.
Boys should be developing a range of sensible methods of indirect measurement. These include mensuration formulae, Pythagoras’ theorem, rates and differences, similarity and scale. An important idea which should develop from the investigation of right triangles is that similar right triangles have equal corresponding ratios – the basic principle of right triangle trigonometry.
Chance and data
Boys should learn to estimate probabilities experimentally and through the analysis of simple sample spaces. They should work collaboratively and with teacher support to set up and carry out simulations.
Practical investigations should be undertaken which involve all of the facets of data handling. Boys may plan and execute surveys about opinions on such matters as environmental issues, tastes in music or what sports they want the school to offer, or collect census type data from peers, investigating, for example, how many boys have computers at home. The activities should include careful consideration of procedures for choosing samples and designing and trialling questionnaires, the comparative advantages of different methods of organising and representing data (including tables, databases, plots and graphs, and summary statistics) and the difficulties which arise at the interpretation stage, which can be used to inform later projects.
Boys should be learning to interpret various representations of data including means, measures of variability and association, line plots, histograms, stem-and-leaf plots, box plots, scatter plots, and lines of best fit; understand the conditions under which their use is appropriate; and compare and select from different possible representations of the same data. Calculators are a necessary tool in this. Boys should be gaining experience which will, over time, enable them to distinguish between a population and a sample, informally draw inferences from data collected by themselves and others, construct convincing arguments based on such data, and evaluate arguments.
Experiences should be practical and exploratory although many boys will also benefit from a more analytic study of geometry. All boys should examine two-dimensional and three-dimensional geometric shapes, investigating and describing relationships between classes of shapes, for example, all squares are rhombuses and all squares are also rectangles, but not all rhombuses and rectangles are squares. They should analyse the properties of various shapes and apply this knowledge to problems, for example, they may develop and justify an approach to ensuring that bases for a softball game are always correctly placed.
Boys should visualise, demonstrate and describe the effect of reflections, rotations, translations and enlargements on shape, size, orientation and arrangement and recognise and produce associated symmetries. They should identify and explain the use of these transformations in designs such as floor tiles, Escher woodcuts drawing, devices such as wheels, and in natural things such as crystals and shells. Boys should improve their proficiency at drawing figures and constructing objects. They may build full-sized and scale models of a range of shapes which fit specifications. They should also use and compare several different conventions for representing three-dimensional objects in two-dimensions, for example, plans and elevations, isometric drawings, perspective drawings, orthogonal drawings, contour maps, and Mercator and other geographic maps.
Boys should learn to use conventional geometric language and techniques to show routes, paths and regions and represent locations and arrangements in networks and other diagrams.
Boys should have extensive experience in observing patterns and relationships among quantities and representing them symbolically and graphically. The study of functions should begin with a sampling of relationships familiar to boys. Boys should use sketch graphs to model relationships drawn from their own daily experiences, such as their mood at different times of the day, and from their understanding of other forms of variation, such as the speed of a car at various stages of a race. Their sketches should reflect the difference between discrete and continuous data and between situations which are essentially deterministic (e.g. the relationship between the radius and circumference of circles) and those which involve an element of chance (e.g. the relationship between body weight and age in children). A function grapher from late in Year 9 onward should enable boys to develop an intuitive grasp of the general shapes of particular kinds of functions (including linear, quadratic, exponential, reciprocal and periodic) fairly rapidly and to be able to visualise the effects of translation and reflection on these functions.
Boys should learn to manipulate simple algebraic expressions which occur in meaningful contexts, for example, they may produce different expressions for the general term in a matchstick sequence and then rearrange the expressions to show that they describe the same relationship, that is, they are equivalent. Boys should be developing facility with notational conventions and properties such as additive and multiplicative inverses and the distributive property of multiplication over addition in order to rearrange expressions and solve equations. From spatial or numerical investigations, they should establish and come to recognise common identities such as a2– b2 = (a – b)(a + b) and apply these in various contexts.
Boys should formulate equations and inequalities from a range of numerical, spatial and measurement contexts, and develop a repertoire of ways to solve them including ‘guess, check and improve’ and graphical methods, with some analytic methods for dealing with at least linear equations. Some boys may also learn to solve pairs of simultaneous equations and quadratic equations analytically. They should informally shade regions on graphs to represent various constraints, for example, given a constraint such as ‘the floor of the house will be a rectangle and have an area of between 100 and 144 square metres’, they shade a region on a graph to show all possible room dimensions.
Mathematics and values
As explicit acknowledgement of core shared values is one of the principles of the Curriculum Framework, the integration of these values will enhance the learning opportunities within school communities. Boys are assisted in developing these values through discussion and modelling as part of the learning and teaching processes within the school environment.