Proficiencies in the mathematics curriculum

Antje and David Leigh-Lancaster, Leigh-Lancaster Consulting 

Introduction 

In Australian Curriculum: Mathematics V9, ACARA has embedded the proficiencies into the content descriptions and achievement standards to make their role within mathematics more explicit. 

A key reason for this is to highlight the interconnected and integral role the proficiencies play in the teaching and learning of mathematics, rather than treating them as standalone components to be addressed separately.

This article outlines how the proficiencies fit together in Australian Curriculum: Mathematics V9, and how teachers can make these visible and practical through everyday classroom planning and instruction.

What are the four proficiencies in mathematics?

The four proficiencies in mathematics are: understanding, fluency, reasoning and problem-solving. 

A simple way to think about them is:

• understanding gives meaning
• fluency gives access
• reasoning gives coherence
• problem-solving gives purpose.

These proficiencies can be seen as the way by which students come to know and use mathematics. Together with the curriculum content, the proficiencies provide students with the skills, knowledge and dispositions to apply their mathematics knowledge across a range of contexts.

A detailed description of each of the four proficiencies can be found on ACARA’s website: Understand this learning area – Mathematics.  (opens external website in a new window)

Identifying the proficiencies in the curriculum

Each year level description begins with a common introductory paragraph about proficiency in mathematics. For example:

This opening paragraph offers a useful way to think about how mathematical proficiency is positioned in the teaching and learning of the curriculum. It highlights:

• how students engage: ... in a range of approaches to learning and doing mathematics ... 
• what this learning develops: … understanding of and fluency with concepts, procedures and processes ...
• how that development occurs: … by making connections, reasoning, problem-solving and practice ... 
• what proficiency enables: … to respond to familiar and unfamiliar situations by employing mathematical strategies to make informed decisions and solve problems efficiently.

Words that highlight a proficiency

In the Australian Curriculum: Mathematics V9, the proficiencies are often indicated by the actions students are expected to take. Some words can apply to more than one proficiency, depending on the context. 

This table presents a selection of words often associated with each proficiency, along with some examples of where they appear in the mathematics curriculum.

Understanding: classify, compare, connect, describe, explain, interpret, identify, order, recognise, represent

• Content description – Year 3, Number 
  recognise, represent and order natural numbers using naming and writing conventions for numerals beyond 10,000 AC9M3N01

Fluency: calculate, choose, count, demonstrate, efficient, estimate, find, manipulate, measure, perform, order, recall, use

• Content description – Year 6, Algebra
  find unknown values in numerical equations involving brackets and combinations of arithmetic operations, using the properties of numbers and operations AC9M6A02

Reasoning: analyse, compare, discuss, explain, infer, interpret, justify, conjecture, predict, reason

• Content description – Year 8, Space
  establish properties of quadrilaterals using congruent triangles and angle properties, and solve related problems explaining reasoning AC9M8SP02

Problem-solving: apply, choose, communicate, evaluate, formulate, interpret, investigate, model, represent, solve

• Content description – Year 2, Number
  use mathematical modelling to solve practical problems involving additive and multiplicative situations, including money transactions; represent situations and choose calculation strategies; interpret and communicate solutions in terms of the situation AC9M2N06

Planning with the proficiencies

A practical way to plan with the proficiencies in mind is to start with the content and then ask: Where are the opportunities to develop understanding, fluency, reasoning and problem-solving?

The following Year 4 content description provides a useful example.

Example

Content description: count by fractions including mixed numerals; locate and represent these fractions as numbers on number lines AC9M4N04

To develop understanding, plan for students to:

•  build the idea that fractions are numbers, not just parts of objects, and that they can be counted, for example: ²⁄₃, ⁴⁄₃ , ⁶⁄₃ = 2, ⁸⁄₃ …  
• recognise that the denominator tells us how many equal parts make up one whole, while the numerator represents the number of equal parts we have
• understand that fractions can be counted beyond one whole
• notice when different counts land at the same point on the number line, for example: ³⁄₂ , ⁶⁄₄ and ¹²⁄_8. 

To develop fluency, plan for students to:

• practise counting by fractions and mixed numerals flexibly and efficiently, including moving across whole numbers and counting from different starting points
• move flexibly between improper fractions and mixed numerals
  locate and represent fractions on number lines with increasing accuracy, by identifying the whole numbers first and then partitioning the intervals
• partition number lines by using the denominator to determine the number of equal intervals.

To develop reasoning, plan for students to:

• explain and justify why a fraction such as ⁴⁄₃ is located to the right of 1 on a number line
• discuss whether the placement of a fraction on a number line makes sense, and explain why
• compare and order fractions by reasoning about their size relative to each other or to whole numbers, for example: ⁷⁄₄ is less than 2 because 2 is ⁸⁄₄
• explain how different counts can land on the same point on the number line.

To develop problem-solving, plan for students to:

• investigate different fraction counting patterns that include a given number, such as 3
• investigate which fractions can be used to count between two given numbers, for example, from ³⁄₄ to 4¹⁄₂  
• apply their knowledge in new ways, for example, if a strip of paper represents the distance from 0 to 2 on a number line, ask students to locate the number ⁷⁄₄ on the paper strip by folding only
• model and solve scenarios on a number line, for example, a single frog jump covers ¹⁄₅ of a metre; how many jumps does it take to pass the 2 ¹⁄₂ metre mark?

Alignment to AERO

When developing proficiencies, consider AERO's model of learning and teaching, which identifies effective and efficient teaching practices aligned with how students learn. Some examples:

• Understanding is developed through explicit teaching and modelling that manage cognitive load and establish key concepts. 
• Fluency is strengthened through scaffolded practice and gradual release, allowing students to consolidate learning and build efficiency. 
• Reasoning is developed by prompting students to explain, justify and transfer their thinking, using logic to connect ideas and apply knowledge across varied contexts.
• Problem-solving is introduced as students demonstrate increasing mastery, with guided and independent opportunities to apply their learning in unfamiliar situations.

Broader planning considerations

There are some school-wide approaches that can strengthen the benefits that the proficiencies bring to the teaching and learning of mathematics:

• plan for understanding by identifying the underlying mathematical concepts and relationships in the content, and
• ensuring these are explored alongside procedures and facts
• recognise that becoming fluent takes time and practice, and plan for regular retrieval practice so students can revisit and consolidate previous learning
• develop common questions across and within teaching teams that can be used during lessons to prompt students to explain their thinking and justify their reasoning
• plan for problem-solving by explicitly teaching strategies and developing a schedule for introducing and revisiting them across year levels.

References and related resources