Stages of Learning in Mathematics
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.
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.