Bringing the Maths Proficiencies to Life

What does it mean to be a successful maths learner?

Ask 100 teachers and you'll get 100 different answers! But, across those answers, some fascinating themes will emerge.

Here's what 100 teachers (well, 91) had to say when I asked them this question. What do you notice about this word cloud? What catches your eye?

In 2001, a group of researchers were asked this same question. Rather than create a word cloud, they produced a 400-page book (researchers, right?!) and, at the heart, was a new concept that they believed summarised what is most important:

“We have chosen mathematical proficiency to capture what we believe is necessary for anyone to learn mathematics successfully.” – National Research Council, 2001

And so, the maths proficiencies were born.

What are the maths proficiencies?

Here's how I define all five proficiencies (yes, there are five! Productive Disposition often gets bumped off the list… why do you think that might be?):

The 5 Proficiencies
1. Understanding: To connect and represent mathematical ideas
2. Fluency: To use methods flexibly, accurately and appropriately
3. Problem Solving: To observe, form conjectures and use evidence to (dis)prove mathematical ideas
4. Reasoning: To logically think through, reflect on and justify ideas
5. Productive Disposition: To believe in diligence and self-efficacy

Together, the maths proficiencies form this bright entanglement:

Key idea: the proficiencies are intertwined and interdependent

Students don't just learn how to reason, or how to become fluent. Each proficiency informs and supports the others. You can't separate them.

How do you teach the proficiencies?

If the proficiencies are interdependent, then how do you teach them?

Let's go back to the original intention of the proficiencies: they are indicators of mathematical success.

So, for any new skill or concept, design task outcomes with the proficiency in mind (e.g. questions used, discussion focus, opportunities for reflection).

Let's look at an example: finding the perimeter of shapes:

A perimeter task:
Shows are rectangle with dimensions 7 metres by 15 metres

Here are ways that the problem might be tweaked to focus on each proficiency:

  • Fluency: Calculate the perimeter in at least 2 ways. Which method seems best to use?
  • Understanding: Look at different (pre-prepared) solutions to the problem. Identify which are correct/incorrect.
  • Reasoning: Justify your solution. Why is it correct?
  • Problem solving: Start with a perimeter (e.g. 44 m) and identify possible rectangles. How many can you find? How do you know when you've found them all?

If you respond to any of these four prompts, notice how you'll necessarily use a range of proficiencies in order to answer it – but your framing of the problem and how you wrap it up with students, will bring one proficiency to the forefront.

The shape example highlights another important point:

To draw out and emphasise each proficiency, you don't need to do massive amounts of work and preparation. Instead, small changes to an existing task are more than enough.

Where does the curriculum fit in?

Diagram shows the Maths Proficiencies sitting across curriculum strands (i.e. Number and Algebra, Measurement and Space, and Statistics and Probability)

The beauty of the proficiencies is in providing another way to look at maths teaching and learning.

The proficiencies aren't in place of or in competition with curriculum content – instead, they sit underneath and draw out the richness of what it means to truly learn maths.

Join the Conversation


  1. As always Michaela, you bring maths to life! We love using the proficiencies as part of our daily and weekly programs.

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