# Mathematics Version 2.0

## Learning in Mathematics Version 2.0

Learning in Mathematics emphasises the importance of providing opportunities for students to develop proficiency in mathematics. This development of proficiency is achieved in how content is explored or developed, that is, how students experience the thinking and doing of mathematics.

#### Proficiency in Mathematics

The proficiencies of Understanding, Fluency, Reasoning and Problem-solving are embedded in all 6 strands and further the development of increasingly sophisticated knowledge and understanding of mathematical concepts, fluency in representations and procedures, and sound mathematical reasoning and problem-solving skills. Proficiency in mathematics enables students to respond to familiar and unfamiliar situations by employing mathematical processes to solve problems efficiently and to make informed decisions. Proficiency in mathematics also enables students to reflect on and evaluate approaches, and verify that answers and results are reasonable in the context.

##### Understanding

Mathematics provides opportunities for students to build and refine a robust knowledge of adaptable and transferable mathematical concepts, structures and procedures. Students make connections between related ideas, progressively draw on their reasoning skills to adapt and transfer understanding of familiar applications to unfamiliar contexts, and cultivate new ideas. They develop an understanding of the relationship between the ‘why’ and the ‘how’ of mathematics. Students build conceptual understanding and procedural fluency 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.

##### Fluency

Mathematics provides opportunities for students to develop, practise and consolidate skills; choose appropriate procedures; carry out procedures flexibly, accurately, efficiently and appropriately; and apply their recall of factual knowledge and understanding of concepts readily. Students are fluent when they connect their conceptual understanding to learned strategies and procedures, make reasonable estimates and calculate answers efficiently, and choose and use computational strategies efficiently; when they recognise robust or multiple ways of answering questions; when they choose appropriate representations and approximations; when they understand and regularly apply definitions, facts and theorems; and when they can manipulate mathematical objects, expressions, relations and equations to find solutions to problems.

##### Reasoning

Mathematics emphasises mathematical reasoning as central to thinking and working mathematically and as a critical component of proficiency in mathematics. Mathematical reasoning guides students in developing an increasingly sophisticated capacity for logical thought and actions, such as conjecturing, hypothesising, analysing, proving, experimenting, modelling, evaluating, explaining, inferring, justifying, refuting, abstracting and generalising. Students 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, make inferences about data or the likelihood of events, and prove that something is true or false. They are reasoning when they compare and contrast related ideas, and reflect on and explain their choices.

##### Problem-solving

Mathematics recognises the importance of providing students with meaningful opportunities to use mathematics to solve problems from both abstract mathematical and real-world contexts. Students engage in mathematical problem-solving when they are presented with a problem situation for which they do not immediately know the answer, and they work through a process of planning, choosing and applying strategies and heuristics to find a solution to the problem, reviewing and analysing their solution. Problems can be simple, where there is only one possible solution, or complicated, where the problem may have many valid approaches to develop solutions. Problem-solving is the ability of students to make choices, interpret, formulate, model and investigate problem situations mathematically, select and use technological functions and communicate solutions effectively. Students pose and solve problems when they use mathematics to represent unfamiliar or meaningful situations, design investigations and plan their approaches, make mathematical decisions as they draw on previously learnt concepts, skills, procedures and processes to solve problems, verify that their answers are reasonable, communicate solutions clearly and justify the reasonableness of their approaches.

#### Mathematical processes

Mathematical processes refer to the thinking, reasoning, communicating, problem-solving and investigation skills involved in working mathematically. Opportunities to learn process skills have been embedded across the strands, building in sophistication across the levels. Mathematical problem-solving and investigation draws on the processes of mathematical modelling, computational and algorithmic thinking, statistical investigation, probability experiments and simulations.

#### Mathematical modelling

Mathematical models are used to gain insight into and make predictions about real-world phenomena, to inform judgements and make decisions in personal, civic and work life. In the modelling process students formulate a real-world problem mathematically by making assumptions; recognise, connect and apply mathematical structures; analyse and solve the mathematical model; and interpret, generalise and communicate their results in response to the real-world situation. Mathematical modelling is an essential dimension of the contemporary discipline of mathematics and is key to informed and participatory citizenship.

#### Computational thinking and simulations

Students develop computational thinking through the application of its various components: decomposition, abstraction, pattern recognition, use of models and simulations, algorithms and generalisation. Computational thinking approaches involve experimental and logical analysis, empirical reasoning and computer-based simulations. The simulations can then be used to generate and test hypotheses and conjectures, identify patterns and key features (or counterexamples), and dynamically explore variation in the behaviour of structures, systems and scenarios.

#### Statistical investigation

Students develop the ability to conduct statistical investigations through informal exploration in the early levels. Later they use guided processes, which progressively lead them to conduct and review their own statistical investigations and to critique others’ processes and conclusions. Statistical investigation deals with uncertainty and variability in categorical (nominal or ordinal) or numerical (discrete or continuous) data arising from observations, surveys or experiments and can be initiated by a specific question, a situation or an issue.

#### Probability experiments and simulations

Students develop an understanding of experimentation through exploration and play-based learning in the early levels. They progress to conducting chance experiments and probability simulations from Level 3 onwards. Experimentation and simulation in mathematics can involve the use of digital and other tools, often to generate large sets of data for consideration, drawing on the interconnections between Statistics and Probability. Experimenting in mathematics requires students to plan what to do and evaluate what they find out using mathematical reasoning.

#### Computation, algorithms and the use of digital tools in mathematics

The capacity to purposefully select and effectively use the functionality of a digital device, platform, software or digital resource is a key aspect of engaging with computational thinking in the Mathematics curriculum. Digital tools can be used effectively to learn and apply mathematics in and across all of the strands. The use of digital tools addresses elements of the Digital Literacy capability. The functionalities may be accessed through hand-held devices such as calculators (arithmetic four operation, scientific, graphics, financial, CAS) and measurement tools (digital scales and other digital measuring devices), software on a computer or tablet (spreadsheet, dynamic geometry, statistical, financial, graphing, computer-algebra), an application on a personal device, virtual and augmented reality technologies or tools accessed from the internet or cloud. Different digital tools or platforms can carry out computations and implement algorithms using numerical, textual, statistical, probabilistic, financial, measurement, geometrical, graphical, logical and symbolic functionalities.

The term ‘computation’ is used in mathematics to refer to arithmetic and non-arithmetic calculations, operations, transformations, procedures and processes that are applied to mathematical objects to produce an output or result. A computation may be an arithmetic calculation; running an algorithm; applying transformations to the graph of a relation, function, network or set of data; developing a set, list, sequence or table of values from a rule; developing a diagram or shape; or the evaluation of an algebraic equation.

The objects of computations may be sets of numbers, text, data, points, shapes and objects in space, images, diagrams, networks, or symbolic and logical expressions, including equations.

Some computations may be dynamic; that is, they enable parameters, conditions and constraints to be varied and the corresponding results to be progressively shown. Examples include the effect of varying an outlier on the mean of a data set, the behaviour of an algorithm under a different set of inputs, sorting or ordering the elements of a set, observing the relative frequency of an event as the number of experiments increases, manipulating a shape in 2 dimensions or an object in 3 dimensions and observing any symmetries, or transforming the graph of a function by varying defining parameters, such as changing the gradient of a linear function.

An algorithm is a precise description of efficient steps and decisions needed to carry out a computation, or a set of rules to follow in order to accomplish a task. Algorithms often involve iterative (repetitive) and recursive (repeatedly applied) processes and can be represented as text, in diagrams or symbolically as flow charts or pseudocode. As students develop a conceptual understanding of how an algorithm works and fluency with using algorithms appropriately, they can reason and solve problems using algorithms as part of a computational thinking process.

#### Meeting the needs of diverse learners

The Victorian Curriculum F–10 values diversity by providing for multiple means of representation, action, expression and engagement, and allows schools the flexibility to respond to the diversity of learners within their community. All schools have a responsibility when implementing the Victorian Curriculum to ensure that students’ learning is inclusive, and relevant to their experiences, abilities and talents. For some students with diverse languages, cultures, abilities and talents, it may be necessary to provide a range of curriculum adjustments so they can access age-equivalent content in the Victorian Curriculum and participate in learning on the same basis as their peers.

Mathematics responds to the diversity of students in the mathematics classroom by connecting familiar experiences and objects in students’ lives. Familiar objects and situations add meaning to any mathematics exploration and help all students understand and use what they have learnt. Responding to student diversity also provides opportunities to deepen students’ understanding of mathematics and its applications. Strategies that could support the diverse needs of students in mathematics include providing:

• exposure to mathematical tasks to engage the intellectual curiosity and interest of students
• classroom discourse that promotes the investigation and growth of mathematical ideas
• technology and other tools to access and pursue mathematical investigations and other problem-solving tasks
• experience with mathematical concepts using multisensory methods to stimulate thinking skills
• access to familiar objects to represent and solve mathematical problems; coins, blocks, counters, buttons or other small objects can be used to demonstrate concepts such as greater than, less than, equal to, counting, adding, subtracting, sharing, grouping and fractions
• scaffolding procedures and processes using step-by-step instruction, demonstrating how to solve mathematical problems.