Fluency in Maths - What is it and how do we teach it?

 

Peter Wheeldon - Head of Primary Maths
& Year 6 Teacher 

‘Mathematics is a creative and highly inter-connected discipline that has been developed over centuries, providing the solution to some of history’s most intriguing problems. It is essential to everyday life, critical to science, technology, and engineering, and necessary for financial literacy and most forms of employment. A high-quality mathematics education, therefore, provides a foundation for understanding the world, the ability to reason mathematically, an appreciation of the beauty and power of mathematics, and a sense of enjoyment and curiosity about the subject.’

Mathematics programmes of study: National curriculum in England (2013)[1]

 

The opening paragraph of the updated National Curriculum for Maths gives a clear and applied direction for Primary children and their learning. All the key words are present: creativity, critical thinking, and links to technology are all clear areas in which proficient mathematicians thrive. However, what about those who are still at their early stages of learning? How do we actually start teaching maths? Where do these amazing learners begin their journey?

What is Fluency?
 

Mathematics in Primary Education is built upon 3 core skills; fluency, reasoning, and problem solving. The key words in the previous paragraph are products of mastering the latter two skills, but (as found in any construction project) a weak foundation can lead to disaster! However, a solid foundation is the perfect base for developing a deep understanding of concepts and a launch-pad for further learning.
 
The National Curriculum defines fluency as being able to access ‘increasingly complex problems over time, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately.’[2] But what does this really mean and, most importantly, how can we teach this?
 
Imagine a mechanic fixing a car - the engine isn’t turning on. She can easily identify the overall problem, as it is a fundamental element of the car which needs to work, but finding the specific cause is another issue completely. Being able to draw upon fluency skills is similar to the initial training needed by the mechanic to find the part which is broken, and understanding what to do next.
 
How do children use fluency?
 

‘The first thing to say is that fluency is not only about number.’[3]

 
In the National Curriculum, number is the base upon which all other units are built. However, it is by far not the end of a child’s journey into mathematics - it is a key skill utilised in all other applications. Fluency in number basically means three things: that children have -

 
Efficiency - this implies that children do not get bogged down in too many steps or lose track of the logic of the strategy. An efficient strategy is one that the student can carry out easily.
 
Accuracy depends on careful recording, knowledge of number facts and other important number relationships, and double-checking results.
 
Flexibility (commonly known as Varied Fluency) requires the knowledge of more than one approach to solving a particular kind of problem, such as two-digit multiplication. Students need to be flexible in order to choose an appropriate strategy for the numbers involved, and also be able to use one method to solve a problem and another method to check the results.[4]
 
So, armed with these three skills, children should be able to confidently utilise their knowledge to find the best way to solve a calculation or problem and, most importantly, be able to do so in a variety of contexts - we call this ‘varied fluency’. In the classroom, we assess understanding of fluency by presenting the same idea in several different situations, models and problems. If children are able to effectively use their skills to solve these, then they have developed a firm and long-lasting understanding of fluency.

Let’s look at an example of this. If asking a child the answer to 3 × 4, they may choose to work it out mentally or use repeated doubling. They could also use their existing skills to know that 3 × 2 = 6
so they’d just need to double that. As fluency is also about choosing the most efficient and accurate method, it would be significantly more long-winded to choose to answer this through column multiplication.

 
However, if you were to ask them the answer to 17 × 18, it’s unlikely that they’d be able to tackle it in their heads straight away. They could use their knowledge of place value to break it down into smaller chunks or they could use a column-based method. For example, they’re more likely to be able to recall the answer to 17 × 10 first. From here, they might choose to double their answer before subtracting 17 × 2. What ultimately matters is that children know which method works most efficiently and accurately for them.[5]
 
Another way we can illustrate fluency is by choosing the most efficient method. Efficiency is the shortest path between the problem and the solution, and its benefits are twofold. First, it helps with recall speed, and second, it allows students to stay focused and not lose their train of logic while answering a problem.
 

Here’s a less efficient pathway: Counting by ones to solve an addition problem (solving 6 + 9 by starting at 6 and counting on 9 more to get 15)
 
Compared to a more efficient pathway: Start at 9 and count up 6 more to 15 ; an even more efficient way of solving this problem would be to make a group of ten by taking 1 away from 6 which makes 5 + 10 which equals 15.[6]

 
All of this uses children’s understanding as the base from which calculations can be adapted, applied and changed. Varied fluency is, fundamentally, children having a deep and secure understanding of what they have learned and, like the mechanic, can draw upon their knowledge when required to solve a problem or fix a car.
  
 
How can I help my child improve their fluency?
 
‘As with much of mathematics, the key to fluency is in making connections, and making them at the right time in a child's learning.’[7]
 
As with most learning, there are many different and contrasting ways to develop understanding - there is no one-fits-all model, and it is important to recognise that what may work brilliantly for one child may equally leave another in the dark.
 
However, there are several broad strategies used commonly in schools to help children understand new or complex ideas in maths: getting hands-on with their learning, giving them chance to discuss and question what they have learned, and finding ways in real-life to apply this new skill.
 

Using objects and pictures
By getting hands-on and using concrete objects to represent number, children are cementing their visual understanding of what numerals actually represent. Sometimes, using objects can unlock barriers to understanding fluency, especially in topics such as number skills, times tables and fractions.
 
 

Explaining understanding
In my class, we acknowledge that the highest level of understanding is not ‘doing’, it is ‘explaining’. Discussion is not simply children sharing how they did a particular calculation, but describing why and how it worked, and how their method is the same or different to those of others.[8]
 
Real-life links
Probably the most impactful way of helping children with their fluency is helping them to make links between the types of situations that a particular strategy might suit. Russell calls this mathematical memory, which is different from just memorising. She says that important mathematical procedures cannot be "forgotten over the summer" because they are based in a web of connected ideas about fundamental mathematical and realistic relationships.[9]
 
 
Final Thoughts
 
The past decade has been a revolution in our understanding of how children learn mathematics, and the introduction of the new National Curriculum in 2014 reflects this. However, what it does not take into account is the disparity between what educators now know, and how parents were taught and understand the same concepts. Although the way we teach maths may have changed, many of the concepts remain the same and any opportunity to push learning further at home is a welcome one. 
 
Fluency is the first step into understanding - however, this does not mean that it is the most important. Understanding in maths relies on a plethora of strategies; fluency may be the first port of call when learning a mathematical idea, but it is one stop on a multi-city cruise of problem solving, reasoning and explaining. 

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[1] Mathematics programmes of study: key stages 1 and 2 National curriculum in England (DFE, 2013)
[2] Mathematics programmes of study (DFE, 2013)
[3] Developing Number Fluency - What, Why and How (NCETM, 2019)
[4] Developing Computational Fluency with Whole Numbers in the Elementary Grades (Russell, 2020)
[5] What is Maths Fluency? (Twinkl, 2022)
[6] Mathematics Practice and Fluency (3P Learning, 2019)
[7] Developing Number Fluency (NCETM, 2019)
[8] Developing Number Fluency (NCETM, 2019)
[9] Computational Fluency (Russell, 2020)