The Five Big Ideas at Secondary – Fluency

In the first in our series on The Five Big Ideas at Secondary, we speak to Helen Granger, Assistant Director for Secondary, about the importance of mathematical fluency and what it really means in the secondary maths classroom.

What do we mean by fluency?

Fluency is related to how well we know something and can easily access it, but there are different ways of ‘knowing’. We could say, ‘I know’ meaning, ‘I'm aware of’. For example, I know there's a place called Australia, but I've never been, so I can't picture it in detail or imagine what it would really be like to be Australian. We could also say ‘I know’ to mean that it's been part of my experience. For example, a visitor to Australia would know much more about it because they've experienced life there and could probably engage in a meaningful conversation about it.

Fluency refers to the sort of ‘knowing’ that means you are so familiar with it that you can access the information without making a conscious decision to do so. It is a knowledge or skill that's at your fingertips and is readily accessible without much effort.

Applying this to a mathematical context, fluency means being so familiar with mathematical concepts and patterns that you can apply them to unfamiliar situations. It means being able to work efficiently and accurately, with a sense of confidence, to manipulate the terms you’re working with. Part of this is having rapid recall of known facts, such as multiplication tables, but more than that, it is a sense of familiarity and of ‘living within’ mathematical structures.

Why is fluency in maths important?

Being mathematically fluent is an essential element in thinking mathematically. It means that each element of maths is not in isolation; it's not put into different disciplines. People can appreciate the interconnectedness of mathematical structures. Fluency, in terms of rapid recall, reduces the burden on working memory and allows thinking capacity to be channeled towards the new, challenging part of a concept.

But more than that, it brings a sense of confidence within the person. It's a sense of knowing what to do when faced with a problem or new concept. It allows people to feel empowered to handle maths, and not just reproduce set systems and set procedures. Having mathematical fluency is key to being able to think in a mathematical way.

How does it fit into the Five Big Ideas in Teaching for Mastery?

Fluency draws on the interconnected nature of maths and supports students in being able to think mathematically. It encourages students to express their strategies and approaches, empowering them to articulate their thinking. When we teach in a coherent fashion, we allow those connections to be made apparent and that's where our students can work in that flexible and fluent way.

Within the Five Big Ideas, fluency enables students to more easily navigate connected mathematical structures and work with confidence in those different areas. Fluency, the capacity to be able to move confidently between different scenarios, contexts and structures, is fundamental.

Click/tap image to enlarge

For example, in this task students must use known facts flexibly to deduce currently unknown facts. There is a focus on using the mathematical structure of multiplication fluently, in a new context, to solve each question, rather than simply applying an algorithm.

How can teachers incorporate strategies to develop students’ fluency into their maths lessons, and what barriers might teachers face?

As teachers of mathematics, we can often find ourselves under pressure, due to the amount of curriculum content we need to cover. This can sometimes be a barrier to taking the next step in tasks and activities with our students. However, there are some quite simple ways that we can help promote fluency within our classroom.

We can apply strategies such as focusing on and valuing the approaches a student takes, rather than the answer they get to. We can use questions like, ‘Can you show me on a diagram?’, ‘Can you show me on a different diagram?’. In the classroom, we can ensure that we take in a range of solution pathways to find an answer and explore the range of alternatives offered. Looking at multiple approaches broadens the mathematical experiences our students have.

Click/tap image to enlarge

This example encourages students to explore a problem in three different ways. They can develop an understanding that each of these pathways is a route to successfully deducing the number of counters.

As teachers, we should seek out ways to link up different areas of the curriculum. For example, we can include fractions in work on area and algebra in work on probability. This pushes students to use and apply knowledge and skills that haven't been an explicit part of the lesson.

Click/tap image to enlarge

Here students are given the opportunity to make connections between calculating perimeter with work on fractions. To deepen the challenge, this problem asks them to apply these skills within an unfamiliar context.

What are your top tips for teachers who want to develop their students’ fluency?

Look for opportunities to pause and ask, ‘How else could we do this?’, ‘When might we choose an alternative?’, ‘How else could this look?’. Exposing students to a range of representations and structures, linking them together and being explicit about the power of each of them, helps our students to make connections between those different mathematical concepts.

Encourage your students to view themselves as confident mathematicians. Give them opportunities to see the beauty of the full discipline of maths so that they can access a wide range of mathematical problems, look at wider perspectives and draw on a range of skills. Developing fluency will ensure that your students are able to grow into self-sufficient mathematicians.

Now watch the explainer video, What do we mean by fluency?

To find out more about the Five Big Ideas, and to discover how your school could harness fluency to enhance maths teaching, contact your local Maths Hub or explore some of the professional development available.

The materials referred to in this feature are taken from a NEW strand available for schools involved in the Sustaining phase of the Teaching for Mastery Programme: Developing fluency with multiplicative reasoning in Key Stage 3. Contact your local Maths Hub to find out how your school can get involved.