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Mathematical Fluency: What Is It and Why Does It Matter?

• Judy Hickman

Let’s address fluency in math by defining what fluency is, why it matters, and how the three stages of fluency are defined by Florida in the B.E.S.T. Standards for Mathematics.

What is mathematical fluency.

“When we are fluent in a language, we can respond and converse without having to think too hard. The language comes naturally, and we do not use up space in our brain thinking about what word to use. Fluency comes from using the language in multiple settings, from trying things out, and failing and trying again.” – Dr. Nic, Creative Maths  

This approach to fluency in any language applies to the language of mathematics, too.

In mathematics, fluency builds on a foundation of conceptual understanding, strategic reasoning, and problem-solving to achieve automaticity. Students connect conceptual understanding (Stage 1) with strategies and methods (Stage 2) and use the methods in a way that makes sense to them (Stage 3) .

When students go through these stages to build fluency, they gain an understanding of the operations and the strategies and methods in their toolbox for solving them, and they become strategic thinkers who can efficiently compute arithmetic.

Fluency is often misunderstood as being able to quickly compute basic math facts, regardless of conceptual understanding, otherwise known as memorization. But being fluent in mathematics is more than memorization, accuracy, and speed.

Accuracy goes beyond memorizing a procedure to get the right answer; it involves understanding the meaning of the procedure, applying it carefully, and checking to see if the answer makes sense. Emphasizing speed can discourage flexible thinking. True fluency is built when students are permitted to stop, think, and use strategies that make sense to efficiently solve a problem.

Why is mathematical fluency important?

By building fluency in math, students can efficiently use foundational skills to solve deeper, more meaningful problems that they encounter in the world around them. Fluency contributes to success in the math classroom and in everyday life.

For example, math fluency is useful for:

• adding scores while playing a game
• using mental math to decide the best buy while shopping at a grocery store
• estimating a percent when determining a tip for a delivery driver
• and so much more!

Throughout everyday life, fluent math thinkers use strategies and methods that they understand to efficiently compute operations and check that their answers are reasonable.

“While being fluent with math facts doesn’t make word problems easy, it does reduce the number of cognitive resources needed to tackle the computation portion of the process, allowing those resources to be allocated to other components of the process.” – Differentiated Teaching

3 Stages of Fluency Defined by Florida’s B.E.S.T. Standards for Mathematics

Let’s examine the three stages of fluency as defined by Florida’s B.E.S.T Standards for Mathematics .

Stage 1: Exploration

• Students investigate arithmetic operations to increase understanding by using manipulatives, visual models, and engaging in rich discussion.
• Models help build on prior learning and make connections between concepts.
• Exercises classified as Stage 1 will prompt students to use a model to solve.

Stage 2: Procedural Reliability

• Students utilize skills from the exploration stage to develop an accurate, reliable method that aligns with the student’s understanding and learning style.
• Students may need the teacher’s help to choose a method, and they are learning how to use a method without help.
• Students choose any method to solve problems independently. Then students are asked to describe their method to ensure that they understand the method and why it works.

Stage 3: Procedural Fluency

• Students build on their conceptual understanding from Stages 1 and 2 and use an efficient and accurate procedure to compute an operation, including the standard algorithms.
• Students are no longer asked to describe their method because they are proving that they can solve accurately and without assistance.

Note: E mbedded within Stages 1-3 is Automaticity . Automaticity is the ability to act according to an automatic response which is easily retrieved from long-term memory. It usually results from repetition and practice.

How do math programs and curriculum incorporate fluency?

When looking for a new math curriculum, districts should consider math programs that use a variety of models (Stage 1) and strategies (Stage 2) as well as standard algorithms (Stage 3) to teach math.

Practice problems should encourage the use of various methods to solve problems as well as student explanations of the methods they choose to use (Stage 2). Student exploration, collaboration, and peer discussion will also aid students in the development of their mathematical thinking.

Programs that integrate foundational mathematical thinking and reasoning skills will help students become mathematical thinkers who can strategically choose efficient methods to solve problems.

By acquiring mathematical fluency, students will have a greater cognitive capacity to solve more complex problems in the real world.

Related Articles

Understanding florida's mathematical thinking and reasoning (mtr) standards.

Topics: Florida , MTR

• Mathematics proficiencies

Introduction

The Australian Curriculum: Mathematics aims to be relevant and applicable to the 21st century. The inclusion of the proficiencies of understanding, fluency, problem-solving and reasoning in the curriculum is to ensure that student learning and student independence are at the centre of the curriculum. The curriculum focuses on developing increasingly sophisticated and refined mathematical understanding, fluency, reasoning, and problem-solving skills. These proficiencies enable students to respond to familiar and unfamiliar situations by employing mathematical strategies to make informed decisions and solve problems efficiently.

The proficiency strands describe the actions in which students can engage when learning and using the content of the Australian Curriculum: Mathematics.

Understanding

Students build a robust knowledge of adaptable and transferable mathematical concepts. They make connections between related concepts and progressively apply the familiar to develop new ideas. They develop an understanding of the relationship between the ‘why’ and the ‘how’ of mathematics. Students build understanding when they connect related ideas, when they represent concepts in different ways, when they identify commonalities and differences between aspects of content, when they describe their thinking mathematically and when they interpret mathematical information

Students develop skills in choosing appropriate procedures; carrying out procedures flexibly, accurately, efficiently and appropriately; and recalling factual knowledge and concepts readily. Students are fluent when they calculate answers efficiently, when they recognise robust ways of answering questions, when they choose appropriate methods and approximations, when they recall definitions and regularly use facts, and when they can manipulate expressions and equations to find solutions.

Problem-Solving

Students develop the ability to make choices, interpret, formulate, model and investigate problem situations, and communicate solutions effectively. Students formulate and solve problems when they use mathematics to represent unfamiliar or meaningful situations, when they design investigations and plan their approaches, when they apply their existing strategies to seek solutions, and when they verify that their answers are reasonable.

Students develop an increasingly sophisticated capacity for logical thought and actions, such as analysing, proving, evaluating, explaining, inferring, justifying and generalising. Students are reasoning mathematically when they explain their thinking, when they deduce and justify strategies used and conclusions reached, when they adapt the known to the unknown, when they transfer learning from one context to another, when they prove that something is true or false, and when they compare and contrast related ideas and explain their choices.

Useful Links

• Australian Curriculum: Mathematics F–10
• Review by Kaye Stacey of 'Adding it up: helping children learn mathematics' report
• Peter Sullivan presentation: Designing learning experiences to exemplify the proficiencies
• Peter Sullivan presentation: Create your own lessons
• Peter Sullivan paper: Using the proficiencies to enrich mathematics teaching and assessment

Explore Mathematics proficiencies portfolios and illustrations

Building fluency through problem solving

Editor’s Note:

This is an updated version of a blog post published on January 13, 2020.

Problem solving builds fluency and fluency builds problem solving. How can you help learners make the most of this virtuous cycle and achieve mastery?

Fluency. It’s so important that I have written not one , not two , but three blog posts on the subject. It’s also one of the three key aims for the national curriculum.

It’s a common dilemma. Learners need opportunities to apply their knowledge in solving problems and reasoning (the other two NC aims), but can’t reason or solve problems until they’ve achieved a certain level of fluency.

Instead of seeing this as a catch-22, think of fluency and problem solving as a virtuous cycle — working together to help learners achieve true mastery.

Supporting fluency when solving problems

Fluency helps children spot patterns, make conjectures, test them out, create generalisations, and make connections between different areas of their learning — the true skills of working mathematically. When learners can work mathematically, they’re better equipped to solve problems.

But what if learners are not totally fluent? Can they still solve problems? With the right support, problem solving helps learners develop their fluency, which makes them better at problem solving, which develops fluency…

Here are ways you can support your learners’ fluency journey.

Don’t worry about rapid recall

What does it mean to be fluent? Fluency means that learners are able to recall and use facts in a way that is accurate, efficient, reliable, flexible and fluid. But that doesn’t mean that good mathematicians need to have super-speedy recall of facts either.

Putting pressure on learners to recall facts in timed tests can negatively affect their ability to solve problems. Research shows that for about one-third of students, the onset of timed testing is the beginning of maths anxiety . Not only is maths anxiety upsetting for learners, it robs them of working memory and makes maths even harder.

Just because it takes a learner a little longer to recall or work out a fact, doesn’t mean the way they’re working isn’t becoming accurate, efficient, reliable, flexible and fluid. Fluent doesn’t always mean fast, and every time a learner gets to the answer (even if it takes a while), they embed the learning a little more.

Give learners time to think and reason

Psychologist Daniel Willingham describes memory as “the residue of thought”. If you want your learners to become fluent, you need to give them opportunities to think and reason. You can do this by looking for ways to extend problems so that learners have more to think about.

Here’s an example: what is 6 × 7 ? You could ask your learners for the answer and move on, but why stop there? If learners know that 6 × 7 = 42 , how many other related facts can they work out from this? Or if they don’t know 6 × 7 , ask them to work it out using facts they do know, like (5 × 7) + (1 × 7) , or (6 × 6) + (1 × 6) ?

Spending time exploring problems helps learners to build fluency in number sense, recognise patterns and see connections, and visualise — the three key components of problem solving.

Developing problem solving when building fluency

Learners with strong problem-solving skills can move flexibly between different representations, recognising and showing the links between them. They identify the merits of different strategies, and choose from a range of different approaches to find the one most appropriate for the maths problem at hand.

So, what type of problems should you give learners when they are still building their fluency? The best problem-solving questions exist in a Goldilocks Zone; the problems are hard enough to make learners think, but not so hard that they fail to learn anything.

Here’s how to give them opportunities to develop problem solving.

Centre problems around familiar topics

Learners can develop their problem-solving skills if they’re actively taught them and are given opportunities to put them into practice. When our aim is to develop problem-solving skills, it’s important that the mathematical content isn’t too challenging.

Asking learners to activate their problem-solving skills while applying new learning makes the level of difficulty too high. Keep problems centred around familiar topics (this can even be content taught as long ago as two years previously).

Not only does choosing familiar topics help learners practice their problem-solving skills, revisiting topics will also improve their fluency.

Keep the focus on problem solving, not calculation

What do you want learners to notice when solving a problem? If the focus is developing problem-solving skills, then the takeaway should be the method used to answer the question.

If the numbers involved in a problem are ‘nasty’, learners might spend their limited working memory on calculating and lose sight of the problem. Chances are they’ll have issues recalling the way they solved the problem. On top of that, they’ll learn nothing about problem-solving strategies.

It’s important to make sure that learners have a fluent recall of the facts needed to solve the problem. This way, they can focus on actually solving it rather than struggling to recall facts. To understand the underlying problem-solving strategies, learners need to have the processing capacity to spot patterns and make connections.

The ultimate goal of teaching mathematics is to create thinkers. Making the most of the fluency virtuous cycle helps learners to do so much more than just recall facts and memorise procedures. In time, your learners will be able to work fluently, make connections, solve problems, and become true mathematical thinkers.

Jo Boaler (2014). Research Suggests that Timed Tests Cause Math Anxiety. Teaching Children Mathematics , 20(8), p.469.

Willingham, D. (2009). Why don’t students like school?: A Cognitive Scientist Answers Questions About How the Mind Works and What It Means for Your Classroom. San Francisco: Jossey-Bass.

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The latest news and views on education from oxford university press., the role of reasoning in supporting problem solving and fluency.

A recent webinar with Mike Askew explored the connection between reasoning, problem solving and fluency. This blog post summaries the key takeaways from this webinar.

Using reasoning to support fluency and problem solving 

You’ll probably be very familiar with the aims of the National Curriculum for mathematics in England: fluency, problem-solving and reasoning. An accepted logic of progression for these is for children to become fluent in the basics, apply this to problem-solving, and then reason about what they have done. However, this sequence tends towards treating reasoning as the icing on the cake, suggesting that it might be a final step that not all children in the class will reach. So let’s turn this logic on its head and consider the possibility that much mathematical reasoning is in actual fact independent of arithmetical fluency.

What does progress in mathematical reasoning look like?

Since we cannot actually ‘see’ children’s progression in learning, in the way we can see a journey’s progression on a SatNav, we often use metaphors to talk about progression in learning. One popular metaphor is to liken learning to ‘being on track’, with the implication that we can check if children going in the right direction, reaching ‘stations’ of fluency along the way. Or we talk about progression in learning as though it were similar to building up blocks, where some ideas provide the ‘foundations’ that can be ‘built upon’. 

Instead of thinking about reasoning as a series of stations along a train track or a pile of building blocks, we can instead take a gardening metaphor, and think about reasoning as an ‘unfolding’ of things. With this metaphor, just as the sunflower ‘emerges’ from the seed, so our mathematical reasoning is contained within our early experiences. A five-year-old may not be able to solve 3 divided by 4, but they will be able to share three chocolate bars between four friends – that early experience of ‘sharing chocolate’ contains the seeds of formal division leading to fractions. 1  

Of course, the five-year-old is not interested in how much chocolate each friend gets, but whether everyone gets the same amount – it’s the child’s interest in relationships between quantities, rather than the actual quantities that holds the seeds of thinking mathematically.  

The role of relationships in thinking mathematically

Quantitative relationships.

Quantitative relationships refer to how quantities relate to each other. Consider this example:

I have some friends round on Saturday for a tea party and buy a packet of biscuits, which we share equally. On Sunday, I have another tea party, we share a second, equivalent packet of the biscuits. We share out the same number of biscuits as yesterday, but there are more people at the table. Does each person get more or less biscuits? 2

Once people are reassured that this is not a trick question 3 then it is clear that if there are more people and the same quantity of biscuits, everyone must get a smaller amount to eat on Sunday than the Saturday crowd did. Note, importantly, we can reason this conclusion without knowing exact quantities, either of people or biscuits. 

This example had the change from Saturday to Sunday being that the number of biscuits stayed the same, while the number of people went up. As each of these quantities can do three things between Saturday and Sunday – go down, stay the same, go up – there are nine variations to the problem, summarised in this table, with the solution shown to the particular version above. 

Before reading on, you might like to take a moment to think about which of the other cells in the table can be filled in. (The solution is at the end of this blog).

It turns out that in 7 out of 9 cases, we can reason what will happen without doing any arithmetic. 4 We can then use this reasoning to help us understand what happens when we do put numbers in. For example, what we essentially have here is a division – quantity of biscuits divided between number of friends – and we can record the changes in the quantities of biscuits and/or people as fractions:

So, the two fractions represent 5 biscuits shared between 6 friends (5/6) and 5 biscuits shared between 8 (5/8). To reason through which of these fractions is bigger we can apply our quantitative reasoning here to see that everyone must get fewer biscuits on Sunday – there are more friends, but the same quantity of biscuits to go around. We do not need to generate images of each fraction to ‘see’ which is larger, and we certainly do not need to put them both over a common denominator of 48.  We can reason about these fractions, not as being static parts of an object, but as a result of a familiar action on the world and in doing so developing our understanding of fractions. This is exactly what MathsBeat does, using this idea of reasoning in context to help children understand what the abstract mathematics might look like.

Structural relationships : 

By   structural relationships,   I mean   how we can break up and deal with a quantity in structural ways. Try this:

Jot down a two-digit number (say, 32) Add the two digits (3 + 2 = 5) Subtract that sum from your original number (32 – 5 = 27) Do at least three more Do you notice anything about your answers?

If you’ve done this, then you’ll probably notice that all of your answers are multiples of nine (and, if like most folks, you just read on, then do check this is the case with a couple of numbers now).

This result might look like a bit of mathematical magic, but there must be a reason.

We might model this using three base tens, and two units, decomposing one of our tens into units in order to take away five units. But this probably gives us no sense of the underlying structure or any physical sensation of why we always end up with a multiple of nine.

If we approach this differently, thinking about where our five came from –three tens and two units – rather than decomposing one of the tens into units, we could start by taking away two, which cancels out.

And then rather than subtracting three from one of our tens, we could take away one from each ten, leaving us with three nines. And a moment’s reflection may reveal that this will work for any starting number: 45 – (4 + 5), well the, five within the nine being subtracted clears the five ones in 45, and the 4 matches the number of tens, and that will always be the case. Through the concrete, we begin to get the sense that this will always be true.

If we take this into more formal recording, we are ensuring that children have a real sense of what the structure is: a  structural sense , which complements their number sense. 

Decomposing and recomposing is one way of doing subtraction, but we’re going beyond this by really unpacking and laying bare the underlying structure: a really powerful way of helping children understand what’s going on.

So in summary, much mathematical reasoning is independent of arithmetical fluency.

This is a bold statement, but as you can see from the examples above, our reasoning doesn’t necessarily depend upon or change with different numbers. In fact, it stays exactly the same. We can even say something is true and have absolutely no idea how to do the calculation. (Is it true that 37.5 x 13.57 = 40 x 13.57 – 2.5 x 13.37?)

Maybe it’s time to reverse the logic and start to think about mathematics emerging from reasoning to problem-solving to fluency.

Mike Askew:  Before moving into teacher education, Professor Mike Askew began his career as a primary school teacher. He now researches, speaks and writes on teaching and learning mathematics. Mike believes that all children can find mathematical activity engaging and enjoyable, and therefore develop the confidence in their ability to do maths. 

Mike is also the Series Editor of  MathsBeat , a new digitally-led maths mastery programme that has been designed and written to bring a consistent and coherent approach to the National Curriculum, covering all of the aims – fluency, problem solving and reasoning – thoroughly and comprehensively. MathsBeat’s clear progression and easy-to-follow sequence of tasks develops children’s knowledge, fluency and understanding with suggested prompts, actions and questions to give all children opportunities for deep learning. Find out more here .

You can watch Mike’s full webinar,  The role of reasoning in supporting problem solving and fluency , here . (Note: you will be taken to a sign-up page and asked to enter your details; this is so that we can email you a CPD certificate on competition of the webinar). 

Solution to  Changes from Saturday to Sunday and the result

 1 If you would like to read more about this, I recommend Lakoff, G., & Núñez, R. E. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. Basic Books.

2 Adapted from a problem in: Lamon, S. (2005). Teaching Fractions and Ratios for Understanding. Essential Content Knowledge and Instructional Strategies for Teachers, 2nd Edition. Routledge.

3 Because, of course in this mathematical world of friends, no one is on a diet or gluten intolerant!

4 The more/more and less/less solutions are determined by the actual quantities: biscuits going up by, say, 20 , but only one more friend turning up on Sunday is going to be very different by only having 1 more biscuit on Sunday but 20 more friends arriving. 

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One thought on “ the role of reasoning in supporting problem solving and fluency ”.

Hi Mike, I enjoyed reading your post, it has definitely given me a lot of insight into teaching and learning about mathematics, as I have struggled to understand generalisations and concepts when dealing solely with numbers, as a mathematics learner. I agree with you in that students’ ability to reason and develop an understanding of mathematical concepts, and retain a focus on mathematical ideas and why these ideas are important, especially when real-world connections are made, because this is relevant to students’ daily lives and it is something they are able to better understand rather than being presented with solely arithmetic problems and not being exposed to understanding the mathematics behind it. Henceforth, the ideas you have presented are ones I will take on when teaching: ensuring that students understand the importance of understanding mathematical ideas and use this to justify their responses, which I believe will help students develop confidence and strengthen their skills and ability to extend their thinking when learning about mathematics.

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Here’s Why Mathematical Fluency is Critical For Problem-Solving and Reasoning

In summary: Mathematical fluency skills help students think faster and more clearly, giving them the energy, attention and focus to tackle complex problem-solving and reasoning questions.

The future needs problem-solvers with reasoning skills. But as education shifts its focus to the critical and creative angle of mathematics problems, we can’t lose sight of the abilities and skills that make this thinking possible:  mathematical fluency .

We’ve covered mathematical fluency in another article ( What is mathematical fluency? ), but here’s the TL;DR version:

Mathematical fluency is the ability to quickly and accurately recall mathematical facts and concepts. It’s made up of 5 key parts – accuracy, flexibility and appropriate response, efficiency, automaticity, and number sense. 

Fluency builds the foundations students use to tackle more complex, multi-step questions in problem-solving and reasoning activities, and it’s crucial to their success. Here’s why:

Mathematical fluency saves energy

Students have only so much energy. You’ll have noticed this before and after lunch breaks. The same principles apply when it comes to problem-solving and reasoning activities.

Let’s say your PSR activity has five steps, and each one of them has four or five problems to solve. The more energy students spend on figuring out those smaller questions, the less they’ll have when it comes to critically and creatively tackling the whole question.

Further, when students aren’t succeeding at one part of a larger problem, it can make the entire activity seem like an overwhelming exercise.

If we’re to look at a student’s brain, those with high fluency skills would have efficient neural pathways, meaning there’s less energy spent and less time is taken for the question to be received and for the answer to be found.

The good news is that these neural pathways are strengthened with repeated exercise, like with any learned behaviour.

By getting students to practice fluency, you’re strengthening the mind muscles they need to do heavier lifting and for longer.

Fluency saves time

Hand-in-hand with saving energy, fluency saves time for students, and this has two distinct benefits: it helps students stay focused on the logical progression of problems and perform better on tests.

Focus  – In a multi-step problem that asks students to use several approaches (like a mix of geometry, algebra, fractions and so on), being able to recall or solve the minutiae with little or no effort keeps them from losing focus on their logical progression.

It’s like being on a hike where you’re expected to find your food and water, camp, and mountain climb – if you’re stuck focusing on each step and breathing in and out, you probably won’t feel much like setting up a tent or getting your ropes and climbing gear in order.

Better test-taking  – Tests have time limits and they’re stressful. Fluency alleviates these pressures; first, by enabling students to attempt to complete more questions, and by getting around the roadblocks of basic computations (like counting on fingers, writing down, working out or reaching for the calculator).

Math fluency builds confidence and reduces mathematics anxiety

Motivation, engagement and progress all rely on students’ confidence that they can complete tasks. For students with mathematics anxiety (the feeling of being overwhelmed or paralysed by mathematics), this is especially important.

Strong fluency allows students to work and see success independently, growing their sense of autonomy and confidence, and helping them see whole problems as small, achievable steps.

It’s like any kind of sporting competition or arts performance; the drilled basics allow the athlete or artist to work on more complicated movements and strategies and prepares them mentally for big events.

In this case, our events are tests, problem-solving, or being introduced to new concepts and material.

Download printable worksheets for math fluency Explore resources

Early mathematical fluency is an indicator of later success

Students who have better fluency in their early education are likely to perform better as they enter secondary school. But it goes further than that  – mathematically fluent early learners see significant gains in their mathematics achievements later on .

We can make educated guesses for why this is – the pace of education and the progressive complexity of mathematics means that those who don’t develop strong fluency early will have a harder time keeping up.

This is especially true when it comes to problem-solving and reasoning.

Preparing students early with fact fluency gives them the tools they need to take on the harder problems they’ll inevitably face in their secondary schooling. If we don’t, it’s like throwing an entry-level karate student into the ring with a black belt master – they won’t have the strategy, reflex or thinking to take them on.

Mathematical fluency prepares students for the problem-solving future

It’s hard to imagine what the future careers of our students will look like. But judging by the push into an automatic world, we can almost guarantee they’ll need three key things to be successful:

• The ability to understand and manipulate data
• Critical thinking skills that will allow them to act strategically and tactically
• Creative thinking skills that enable them to approach problems in a variety of ways

How to reinforce your students’ mathematical fluency

We recommend three things:

Playing mathematics games

Practice requires repetition, and repetition is fun when it’s gamified. But games have a few further benefits:

• They encourage thinking about mathematics on a strategic level
• They need less teacher input and encourage autonomous learning
• They build students computational fluency
• They connect the classroom environment to the home learning environment

Daily mathematics fluency activities

Mathematics skills become strong when they’re done regularly. After a concept has been introduced, you should look to have activities planned to cement students’ knowledge until you’re confident they can work on it or use it independently.

Give students time to discover

Plan lessons that allow students time to discover number patterns, structures, and concepts and test them out in different situations to see if what they discovered works. This builds autonomy and gives students the chance to reflect on their learning.

Develop mathematical fluency, problem-solving, and reasoning with our award-winning programs

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What Is Fluency In Maths & How Do Schools Develop It?

Rebecca Jakes

Here experienced maths lead and Third Space Learning user, Rebecca Jakes gives a clear definition of what fluency is and means, what fluency looks like within a mastery curriculum, and, crucially for other primary teachers, four proven techniques and resources that she has seen develop fluency at Key Stage 2. 

When the new maths curriculum was introduced a few years ago, with its triple emphasis on fluency, reasoning and problem solving , there was a sharp intake of breath amongst teachers and leaders across the country.

The higher expectation overall alongside the introduction of a new curriculum for mastery and depth was quite daunting at first. How would the children cope? How would teachers cope?

Central to this new form of mathematics education all was this concept of maths fluency. What did it mean, how should we teach it, and would children be able to achieve the necessary level by their KS2 SATs?

A few years down the line I have recently found myself reflecting on the changes from the vantage point of being a maths leader and maths adviser across several schools and assessing which interventions and initiatives we’ve used that have really worked to develop pupils maths fluency at KS2.

Fluency Definition: What is Fluency in Maths?

Fluency in maths is about developing number sense and being able to choose the most appropriate method for the task at hand; to be able to apply a skill to multiple contexts.

The National Curriculum states that pupils should become fluent in the fundamentals of mathematics through varied and frequent practice. While a part of this is about knowing key mathematical facts and recalling them efficiently, fluency means so much more than this as it allows pupils to delve much deeper.

How Does Mathematical Fluency Fit with Reasoning and Problem Solving?

The mastery curriculum for primary schools places problem solving at the heart of mathematics with the main aim that every child can learn to solve sophisticated multi-step word problems in an unfamiliar context.

To enable them to achieve this, pupils must develop their conceptual understanding, mathematical thinking and use of mathematical language. This is where fluency and reasoning come in.

How Fluency And Reasoning Work In Primary Maths

Fluency in maths works through intelligent practice (rather than just mechanical repetition). Once a child has grasped a mathematical concept, the idea is that they are exposed to varied fluency activities which develop their understanding.

These activities also require them to use verbal reasoning to justify and explain their thinking in order to solve word problems in an unfamiliar context.

Making this happen in the classroom is quite straightforward when you focus on the unit you are teaching.

However we’ve all felt that frustration when children are given problems that include applying skills that have been previously taught. They look at you with a blank face, completely at a loss of what to do.

This article on the science of learning and memory goes into more detail on what’s happening here.

So how are schools ensuring children don’t forget the basics as they move from one unit of Maths to another? Simple – they are adding in some extra time every day or week to enable them to simply practise those Maths skills that they have already been taught.

What is the Best Way to Teach KS2 Maths Fluency?

The mastery curriculum has meant that finally children are allowed to just practise until they are confidently calculating. And, over the last couple of years, I’ve witnessed how truly beneficial this can be for pupils.

Children are becoming fluent in calculations and times tables and they’re loving it!

Their fluency in calculations such as their multiplication facts has led to pupils tackling more complex problems with greater confidence and resilience.

Because they are no longer having to tackle remembering how to do the calculations needed or the most appropriate strategy to choose each time, they are able to put all their energy into how to solve their mathematical problem.

As a primary teacher and consultant I am fortunate to see the many different ways in which schools are giving children time to hone the skills needed through daily or weekly practice.

There is a lot of variation and different methods for tackling this, but the one thing that is constant between them is that they are producing good or excellent outcomes in Maths.

So what’s working?

The following are the four different approaches to improving fluency in Maths at KS2 that are used by schools in my area .

Approaches to Fluency 1: Mastery Time

As Maths leader at my previous school, Woodcot Primary, I introduced a set time each day of around 30 minutes following a carousel approach to fluency practice.

During this time the class teacher has a guided group where they either:

• Work with pupils who have not fully grasped a concept during that day’s lesson.
• Pre-teach target children.
• Work with target children on areas of arithmetic they are not yet fluent in.

The rest of the class work on whatever the focus is for that day. This is usually practising fluency in calculations or nailing the times table and related division facts they are working on.

In KS1 there is a heavy focus on number bonds and memorising number facts . Most year groups have one day of the week where children practise using online or offline KS1 maths games to embed number relationships.

Using this approach has seen a big impact on pupil confidence in maths and resilience. As pupils gain fluency in calculations, they are no longer worrying about making mistakes, leaving them to focus on unpicking sophisticated problems with enthusiasm.

For Year 6 pupils it has had a great impact on arithmetic scores, leaving children plenty of time to learn the reasoning and problem solving skills needed for the KS2 SATs.

• 8 Reasons Great Arithmetic Skills are Key to SATs Success
• How Woodcot Primary Increased Their KS2 Maths Results from 45% to 88% here.

Approaches to Fluency 2: Maths Meetings

Haselworth Primary School hold daily ‘Maths Meetings’ in all of their year groups.

In KS1 these focus on counting (forwards, backwards, 2s, 5s, 10s etc.) in a fun context such as ‘counting tennis’, where pupils bat numbers back and forwards with a partner. Number bonds and quick recalls of doubles and halves are also a focus in Years 1 and 2.

Read more: 10 ways to memorise number facts at KS2

KS2 Maths Meetings have more of a focus on times table facts with daily practice and a weekly test. They also have a daily fluency activity based on agreed key fluency skills for each year group.

In addition to this, Year 6 do daily Maths SATs Facts from Vocabulary Ninja and Fluent in Five arithmetic practice from Third Space Learning .

Approaches to Fluency 3: Weekly Repetition

‘Number of the Week’ sheets are becoming a common addition to fluency practice. The example here is one taken from a Year 4 class at a school I moderated with recently.

Teachers reported that the biggest impact from doing this was that pupils were able to visualise a number and its properties much more quickly in other contexts.

Fluent in Five Arithmetic Pack: Years 3 to 6 [Weeks 1 to 6]

Use on a whiteboard or print out our SATs style questions, complete with progression document & help for pupils to choose mental or written method

The sheet can be varied for different abilities or year group focus; concrete resources can be used to support learning when necessary and the pupils reportedly enjoy the familiarity of the activity.

Approaches to Fluency 4: Targeted Fluency Focus

At my new school Brockhurst Primary, each year group has a set fluency focus per half term.

During each half term, teachers provide fluency activities on a daily or weekly basis and ensure there are visual reminders around the classroom to bring it to the forefront of the children’s minds.

Every half term children take home a ‘Fluency in Maths: Key Facts’ sheet with one area of Maths to focus on, enabling parents to become involved in learning and have a greater understanding of the expectations in Maths for their child.

By the end of the half term, children should know these facts and the aim is for them to achieve true automaticity so they can recall them instantly.

Many classes also hold a ‘daily Maths mile’ where children walk or jog a mile (or however far they can get in fifteen minutes) while practising their area of focus as a class.

The Best Free Resources to Develop Fluency in Maths for Year 3, Year 4, Year 5 & Year 6

Finally, here are my recommendations for the my top three engaging (and mostly free) resources to develop fluency in maths at KS2. They’re great for developing fluency in short bursts throughout the term.

1. Fluent In Five Daily Arithmetic Practice – Third Space Learning

You can download weeks 1-6 for free here.

This free arithmetic resource is designed around the teaching for mastery approach, and the daily activities really do only take five minutes.

In Year 4, I use them as at the beginning of a lesson and find them brilliant for sharing ideas for mental calculations. They are particularly good for children who don’t make obvious links like number bonds and near doubles and are overly reliant on written methods.

For Years 5 and 6 they provide an opportunity to practise arithmetic fluency on a daily basis, and offer the perfect chance for teachers to address misconceptions or set arithmetic targets. The other advantage of Fluent in Five is the emphasis on finding the most efficient strategy to use when solving the arithmetic questions with the ‘try written method’ or ‘try mental method’

Schools who use Third Space Learning’s 1-to-1 KS2 Maths intervention programme have access to all 36 weeks of Fluent in Five, alongside arithmetic tests, SATs papers and diagnostic quizzes.

They also have an amazing video CPD library in their online Maths Hub which is my ‘go to’ when I’m introducing a new area of Maths or if I have a pupil or group of pupils who can’t grasp a concept.

2. Mathsframe

Many of the games and activities on Mathsframe are free and come with a range of support and challenge.

I particularly like the ones shown above for children who are still building conceptual understanding (but also for those who have learnt a method but don’t actually understand what they are doing).

As a paid member, you also have access to assessments and variations of problem solving questions.

3. Prodigy Game

My class would play prodigy all day long if I let them. It’s free and takes minutes to set up. As a teacher, all you need to do is set tasks regularly to match what the children are learning in class or to revisit prior learning.

Each pupil has a login and they have to solve maths problems to move through the game.

They can meet other people from their class but not anyone from the outside world so it’s also pretty safe. It’s great for home learning; the only drawback is getting them to stop playing at the end of a session.

Other resources to build Maths fluency at KS2

For those without internet access to interactive online games, we’re always adding to our collection of maths games blogs. Choose any of these for maths games KS2 will get the most out of, or if you’re looking for support specifically to develop fluency in times tables then these times tables games are ideal.

Another favourite for building fluency in maths is Times Tables Rockstars which will give your pupils the confidence, speed and accuracy they need to fly through the Year 4 Multiplication Tables Check .

How I Teach Mathematical Fluency to Year 3 and Year 4

So, after seeing so many approaches to developing fluency and number sense, what have I done in my own practice? The answer is simple – I’ve shared it and used it.

In Year 4 where I am based, the teaching team were reflecting on the ‘use it or lose it’ issue that arises when an area of Maths is mastered but not returned to.

So after sharing the ideas I have seen we decided to incorporate our favourite bits into one model:

Sticking with the carousel model, each day pupils practise the mathematical concept or fluency they are focusing on in addition to the fluency focus from previous terms until they’re giving accurate answers every time.

On one day they have a guided group with an adult (either class teacher or LSA) and if need be the teacher pre-teaches or works with small groups of pupils who haven’t grasped concepts in that day’s Maths lesson.

One session is on laptops and the ‘Maths facts’ session is using the weekly repetition example shown above.

Having just read Clare Sealy’s amazing blog on teaching telling the time , we are now thinking about swapping one of the activities to one based on fluency in reading time.

As Clare quite rightly said, we all tear our hair out when trying to teach time but only do so once a year. Maths Express is accompanied by using ‘ Fluent in Five ’ at the beginning of each Maths lesson or as a morning challenge when children arrive at school.

Why have we called it Maths Express? The Year 6 teacher and Maths Lead at Brockhurst had given this name to her sessions and we liked the sound of it. Some of the boys in our year group are mad on trains so we decided to use it too.

The children love the sessions and one child commented, ‘I know why it’s called Maths Express. We are all on the train together heading in the same direction and we want to get to the same place.’ I couldn’t have put it any better myself.

• What is Maths Mastery and Teaching for Mastery?
• Maths mastery resources to support your maths teaching
• Ultimate KS2 Checklist Of 33 Mental Maths Strategies

How Third Space Learning’s online tuition supports the development of maths fluency

Each of the lessons in our interventions provide a journey through each concept. They begin with a recap of previously learnt topics that link to the current lesson, using real world examples and deep questioning to introduce and develop the pupil’s knowledge of the topic. Pupil and tutor then work together on procedural fluency for the topic of the week, building towards more complex problems as a pupil masters the concept. This process gives us a great opportunity to work on misconceptions as they arise, and the tutors can assess and use this knowledge moving forward. 

DO YOU HAVE STUDENTS WHO NEED MORE SUPPORT IN MATHS?

Every week Third Space Learning’s specialist online maths tutors support thousands of students across hundreds of schools with weekly online 1 to 1 maths lessons designed to plug gaps and boost progress.

Since 2013 these personalised one to 1 lessons have helped over 150,000 primary and secondary students become more confident, able mathematicians.

Learn how the programmes are aligned to maths mastery teaching or request a personalised quote for your school to speak to us about your school’s needs and how we can help.

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Developing mathematical fluency: comparing exercises and rich tasks

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• Published: 26 September 2017
• Volume 97 , pages 121–141, ( 2018 )

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Achieving fluency in important mathematical procedures is fundamental to students’ mathematical development. The usual way to develop procedural fluency is to practise repetitive exercises, but is this the only effective way? This paper reports three quasi-experimental studies carried out in a total of 11 secondary schools involving altogether 528 students aged 12–15. In each study, parallel classes were taught the same mathematical procedure before one class undertook traditional exercises while the other worked on a “mathematical etude” (Foster International Journal of Mathematical Education in Science and Technology , 44 (5), 765–774, 2013b ), designed to be a richer task involving extensive opportunities for practice of the relevant procedure. Bayesian t tests on the gain scores between pre- and post-tests in each study provided evidence of no difference between the two conditions. A Bayesian meta-analysis of the three studies gave a combined Bayes factor of 5.83, constituting “substantial” evidence (Jeffreys, 1961 ) in favour of the null hypothesis that etudes and exercises were equally effective, relative to the alternative hypothesis that they were not. These data support the conclusion that the mathematical etudes trialled are comparable to traditional exercises in their effects on procedural fluency. This could make etudes a viable alternative to exercises, since they offer the possibility of richer, more creative problem-solving activity, with comparable effectiveness in developing procedural fluency.

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1 Introduction

Attaining fluency in key mathematical procedures is essential to students’ mathematical development (Department for Education [DfE], 2013 ; National Council of Teachers of Mathematics [NCTM], 2014 ; Truss, 2013 ). Being secure with important mathematical procedures gives students increased power to tackle more complicated mathematics at a more conceptual level (Codding, Burns, & Lukito, 2011 ; Foster, 2013b , 2016 ), since automating skills frees up mental capacity for being creative (Lemov, Woolway, & Yezzi, 2012 , p. 36). Devising ways to support the development of robust fluency with mathematical procedures is currently a focus of attention. For example, in England the national curriculum for mathematics emphasises procedural fluency as the first stated aim (DfE, 2013 ), and the current “mastery” agenda stresses “intelligent practice” as a route to simultaneously developing procedural fluency and conceptual understanding (Hodgen, 2015 ; National Association of Mathematics Advisors [NAMA], 2016 ; National Centre for Excellence in the Teaching of Mathematics [NCETM], 2016 ).

However, a focus on procedural fluency is sometimes seen as a threat to reform approaches to the learning of mathematics, which emphasise sense making through engagement with rich problem-solving tasks (Advisory Committee on Mathematics Education [ACME], 2012 ; Office for Standards in Education [Ofsted], 2012 ). In a technological age, in which calculators and computers can perform mathematical procedures quickly and accurately, it may be argued that teaching problem solving should be prioritised over practising procedures. It may also be that an excessive focus on basic procedures fails to kindle students’ interest in mathematics and could be linked to students, especially girls, not choosing to pursue mathematics beyond a compulsory phase (Boaler, 2002 ). Nevertheless, in a high-stakes assessment culture, where procedural skills are perceived to be the most straightforward ones to assess, the backwash effect of examinations is likely to lead to schools and teachers feeling constrained to prioritise the development of procedural fluency over these other aspects of learning mathematics (Foster, 2013c ; Ofsted, 2012 ; Taleporos, 2005 ).

In this context, it has been suggested that a mathematics task genre of etudes might be capable of addressing procedural fluency at the same time as offering a richer experience of learning mathematics (Foster, 2013b , 2014 ). Etudes are mathematics tasks in which extensive practice of a well-defined mathematical procedure is embedded within a rich mathematical problem (Foster, 2013b ). Such tasks aim to generate plentiful practice incidentally as learners tackle a rich, open-ended problem. East Asian countries which perform well in large-scale international assessments such as the Programme for International Student Assessment (PISA) and the Trends in International Mathematics and Science Study (TIMSS) are thought to succeed in emphasising mathematical fluency without resorting to low-level rote learning of procedures (Askew et al., 2010 ; Fan & Bokhove, 2014 ; Leung, 2014 ).

There have been many attempts to design tasks that incorporate meaningful practice (Kling & Bay-Williams, 2015 ) or exploit systematic variation (NAMA, 2016 ) to address fluency goals within deeper and more thought-provoking contexts. Not only might this lead to greater interest and motivation for students (Li, 1999 ), it is conceivable that it could assist in the development of procedural fluency by to some extent shifting students’ focus away from the details of the procedure, perhaps thereby aiding automation. From the point of view of being economical with students’ learning time, Hewitt ( 1996 ) described the generation of purposeful practice by subordinating the role of practice to a component of a larger mathematical problem. In this way, attention is placed not on the procedure being performed but instead on the effect of its use on a desired goal (Hewitt, 2015 ).

Mathematical etudes draw on these intentions to situate procedural practice within rich, problem-solving tasks. Although anecdotally etudes have been very favourably received by mathematics teachers, and appear to be popular with students, it is not known whether or not they are as effective as traditional exercises at developing procedural fluency. While etudes might be expected to offer other advantages, such as greater engagement and opportunities for creative problem solving and exploration, it is not known whether this comes at a cost of effectiveness in narrow terms of developing procedural fluency. It seems possible that diverting students’ attention away from the details of carrying out a procedure and onto some wider mathematics problem could hinder their immediate progress in procedural fluency. However, on the contrary, problem-solving aspects of an etude could potentially focus students on the details of a procedure in a way that supports development of fluency. So this paper investigates whether or not etudes are as effective as exercises for developing students’ procedural fluency.

2 Mathematical etudes

2.1 background.

Procedural fluency involves knowing when and how to apply a procedure and being able to perform it “accurately, efficiently, and flexibly” (NCTM, 2014 , p. 1). The Mathematical Etudes Project Footnote 1 aims to devise creative ways to help learners of mathematics develop their fluency in important mathematical procedures. It might be supposed that any varied diet of rich problem-solving tasks would automatically generate plentiful opportunities for students to gain practice in a multitude of important mathematical procedures, and that this would be a natural way for procedural fluency to be addressed in the curriculum. However, a rich, open-ended task may be approached in a variety of ways (Yeo, 2017 ), and, where a choice of approaches is possible, students may be drawn to those which utilise skills with which they are already familiar and comfortable. In this way, areas of weakness may remain unaddressed. For example, a student lacking confidence with algebra may be able to solve a mathematical problem successfully using numerical trial and improvement approaches, or perhaps by drawing an accurate graph. From the point of view of problem solving, selecting to use tools with which one is already competent is an entirely appropriate strategy, but if algebraic objectives were central to why the teacher selected the task, then the task has failed pedagogically. In this way, an open-ended task cannot necessarily be relied on to focus students’ attention onto a specific mathematical procedure. Even if it does succeed in doing this, it may not generate sufficient practice of the particular technique to develop the desired fluency, since a broader problem is likely to contain other aspects which also demand the student’s time and attention.

For this reason, an etude cannot simply be a problem which provides an opportunity for students to use the desired procedure; it must place that procedure at the centre of the students’ activity and force its repeated use. Success with the task must be contingent on repeated accurate application of the desired procedure. The Mathematical Etudes Project has developed numerous practical classroom tasks which embed extensive practice of single specified mathematical procedures within rich problem-solving contexts (Foster, 2011 , 2012a , b , 2013a , b , d , 2014 , 2015a ). It is whether such tasks are as effective as traditional exercises in developing fluency or not that is the subject of this research.

The term “etude” is borrowed from music, where an etude is “originally a study or technical exercise, later a complete and musically intelligible composition exploring a particular technical problem in an esthetically satisfying manner” (Encyclopaedia Britannica, 2007 ). Originally, etudes were intended for private practice, rather than performance, but later ones sought to achieve the twin objectives of satisfying an audience in concert as well as working as an effective tool for the development of the performer’s fluency. This latter sense inspires the idea of a mathematical etude, which is defined as a mathematics task that embeds “extensive practice of a well-defined mathematical technique within a richer, more aesthetically pleasing mathematical context” (Foster, 2013b , p. 766). In musical etudes, such as those by Chopin, the self-imposed constraint of focusing on (normally) a single specific technique may contribute to the beauty of the music.

The idea of practising a basic skill in the context of more advanced skills is common in areas such as sport (Willingham, 2009 , p. 125), and has been used within mathematics education. For example, Andrews ( 2002 ) outlined “a means by which practice could be embedded within a more meaningful and mathematically coherent activity” (p. 16). Boaler advised that it is best to “learn number facts and number sense through engaging activities that focus on mathematical understanding rather than rote memorization” (Boaler, 2015 , p. 6), and many have argued that algorithms do not necessarily have to be learned in a rote fashion (Fan & Bokhove, 2014 ). Watson and De Geest ( 2014 ) described systematic variation of tasks for the development of fluency, and it is known that, to be effective, practice must be purposeful, and systematically focused on small elements, and that feedback is essential (Ericsson & Pool, 2016 ). The challenge is to devise mathematics tasks which do this within a rich context.

The three etudes trialled in the studies described in this paper will now be discussed. Two of these etudes address solving linear equations in which the unknown quantity appears on both sides (studies 1 and 2), and the third etude concerns performing an enlargement of a given shape on a squared grid with a specified positive integer scale factor (study 3).

2.2 Linear equations etudes

The first two etudes described focus on solving linear equations. Both are intended to generate practice at solving linear equations in which the unknown quantity appears on both sides.

2.2.1 Expression polygons etude

In this etude, students are presented with the diagram shown in Fig. 1 , called an expression polygon (Foster, 2012a , 2013a , 2014 , 2015a ). Each line joining two expressions indicates that they are equated, and the initial task for students is to solve the six equations produced, writing each solution next to the appropriate line. For example, the top horizontal line joining x  + 5 to 2 x  + 2 generates the equation x  + 5 = 2 x  + 2, the solution to which is x  = 3, so students write 3 next to this line. In addition to recording their solutions on the expression polygon, a student could write out their step-by-step methods on a separate piece of paper.

Expression polygons etude. (Taken from Foster, 2015a )

Having completed this, the students will obtain the solutions 1, 2, 3, 4, 5 and 6. The pattern is provocative, and students typically comment on it (Foster, 2012a , 2015a ). This leads naturally to a challenge: “Can you make up an expression polygon of your own that has a nice, neat set of solutions?” Students make choices over what they regard as “nice” and “neat”. They might choose to aim for the first six even numbers, first six prime numbers, first six squares or some other significant set of six numbers. Regardless of the specific target numbers chosen, the experimentation involved in producing their expression polygon is intended to generate extensive practice in solving linear equations. Working backwards from the desired solution to a possible equation, and modifying the numbers to make it work, necessitates unpicking the equation-solving process, which could contribute to understanding of and facility with the procedure. Students are expected to attend more to the solutions obtained than they would when working through traditional exercises, where the answers typically form no pattern and are of no wider significance than that individual question. As students gain facility in solving equations, they focus their attention increasingly on strategic decisions about which expressions to choose. They might even go on to explore what sets of six numbers may be the solutions of an expression polygon, or experiment with having five expressions rather than four, for instance. In this way, the task is intended to self-differentiate through being naturally extendable (Foster, 2015a , b ).

2.2.2 Devising equations etude

2.3 enlargements etude.

In this third etude, which addresses the topic of performing an enlargement of a given shape, students are presented with the diagram shown in Fig. 2 , containing a right-angled isosceles triangle on a squared grid. The task is to find the locus of all possible positions for a centre of enlargement such that, for a scale factor of 3, the image produced lies completely on the grid. Students can generally find, without too much difficulty, one centre of enlargement that will work, but finding all possible points is demanding and may entail reverse reasoning from the possible image vertex positions to those of the original triangle. Further extensions are possible by considering different starting shapes, different positions of the starting shape on the grid and different scale factors. In all of this work, the enlargement procedure is being practised extensively within a wider investigative context.

Enlargement etude grid. (Taken from Foster, 2013d )

2.4 Summary

Each of the three etudes described above is intended to generate extensive opportunities for practising a single specified procedure within a rich problem-solving context. However, although etudes might be anticipated to have benefits for students in terms of greater engagement and creative problem solving, it is not known how effective they are in comparison with the standard approach of traditional exercises in the narrow objective of developing students’ procedural fluency. It might be thought that incorporating other aspects beyond repetition of the desired procedure might to some extent diminish the effectiveness of a task for developing students’ procedural fluency. However, the opposite could be the case if the problem-solving context to some extent directs students’ attention away from the performance of the procedure and onto conceptual aspects, leading to greater automation. Consequently, the research question for these studies is: Are etudes as effective as traditional exercises at developing students’ procedural fluency or not?

In these exploratory studies, it is important to emphasise that a choice was made to compare etudes and exercises only in very narrow terms of procedural fluency. While it is likely that etudes offer other, harder-to-measure benefits for students, such as providing opportunities for creative, open-ended, inquiry-based exploration and problem solving, unless they are at least about as good as traditional exercises at developing students’ procedural fluency, it is unlikely, in a high-stakes assessment culture, that schools and teachers will feel able to use them regularly as an alternative. Traditional exercises are widely used by teachers not because they are perceived to be imaginative and creative sets of tasks but because they are believed to work in the narrow sense of developing fluency at necessary procedures. If there were some other way to achieve this, that did not entail the tedium of repetitive drill, it would presumably be preferred—provided that it were equally effective at the main job. For this reason, in these studies the focus was entirely on the effect of etudes on procedural fluency. Rather than trying to measure the plausible but more nebulous ways in which etudes might be superior, in this first exploratory set of studies it was decided to focus solely on the question of the effectiveness of etudes for the purpose of developing procedural fluency.

3 Study 1: Expression polygons

The aim of this study was to investigate whether a particular etude (“Expression polygons”, see Section 2.2.1 ) is as effective as traditional exercises at developing students’ procedural fluency in solving linear equations, relative to the alternative hypothesis that the etude and the exercises are not equally effective.

A quasi-experimental design was used, with pairs of classes at the same school assigned to either the intervention (the etude) or control (traditional exercises) condition. Data was collected across one or two lessons, in which students in the intervention group tackled an etude while those in the control group worked through as many traditional exercises as possible in the same amount of time. Pre- and post-tests were administered at the beginning and end of the lesson(s).

A classical t test on the gain scores (post-test − pre-test) would be suitable for detecting a statistically significant difference between the two groups; however, within the paradigm of null hypothesis significance testing, failure to find such a difference would not constitute evidence for the null hypothesis of no difference—it would simply be inconclusive (Dienes, 2014 ). No evidence of a difference is not evidence of no difference. This is because failure to detect a difference might be a consequence of an underpowered study, which might have been able to detect a difference had a larger sample size (or more sensitive test) been used. For this reason, it is necessary to use a Bayesian approach for these studies, in order to establish how likely is the hypothesis of no difference between the two treatments (etude and traditional exercises) in terms of gain in procedural fluency, relative to the alternative hypothesis that there is a difference. Thus, Bayesian t tests were carried out on the gain scores obtained in each study, as described below.

3.1.1 Instrument

The “Expression polygons” etude (Foster, 2012a , 2013a , 2015a ) discussed in Section 2.2.1 was used for the intervention groups, and the control groups were provided with traditional exercises and asked to complete as many as possible in the same amount of time (see Fig. 3 for both), normally around 20 min. The exercises consisted of linear equations in which the unknown appears on both sides, leading to small integer solutions. Pre- and post-tests were designed (Fig. 4 ), each consisting of four equations of the same kind as those used in the exercises. In this way, it was hoped that any bias in the focus of the tests would be toward the control group (exercises), since the matching between the exercises and the post-test was intended to be as close as possible. Each test was scored out of 4, with one mark given for the correct solution to each equation. The post-test included a space at the end for open comments, asking students to write down “what you think about the work you have done on solving equations”. This question was intended to capture students’ perceptions of the two different tasks.

Study 1 materials: expression polygons etude (intervention) and traditional exercises (control)

Study 1 pre-test and post-test

3.1.2 Participants

Schools were recruited through a Twitter request for help, and schools A, B and C (Table 1 ) took part in this study. These schools were a convenience sample, spanning a range of sizes and composition. In most schools in England, mathematics classes are set by attainment, and this was the case for schools A and B, while school C used mixed attainment classes. In all of the schools, teachers were asked to:

choose two similar classes (e.g. Year 8 or 9 parallel sets) who you are teaching to solve linear equations with the unknown on both sides (e.g. equations like 7 x  − 1 = 5 x  + 3). In these materials, all the solutions are whole numbers, but some may be negative.

A total of 241 mathematics students from Years 8 and 9 (age 12–14) participated. Forty-eight students’ pre- and post-tests could not be matched, either because they were not present for one of them or (for the vast majority) because they did not put their name clearly on the test. These students’ tests were excluded from the analysis, leaving N  = 193. The large number of tests which could not be matched here was mainly due to the fact that in one particular class (20 students) none of the students wrote their names on either of their tests, and so none of the data from this class could be used.

3.1.3 Administration

Teachers were asked to use the materials with a pair of “parallel” classes across one or preferably two of the students’ normal mathematics lessons. Allocation was at class level, and schools were responsible for choosing pairs of classes that they regarded as similar, which were normally a pair from the same Year group which were setted classes at the same level (e.g., both set 3 out of 6). In most cases, the same teacher taught both classes, so as to minimise teacher effects.

Pre- and post-tests were administered individually in the same amount of time and until most students had finished (normally about 10 min. for each). Both classes were then taught by the teacher how to solve linear equations with the unknown on both sides. Teachers were asked to teach both classes “as you would normally, in the same way, and for approximately the same amount of time”. Following this, the control class received traditional exercises (Fig. 3 ), with the expectation that the number of questions would be more than enough for the time available (normally about 20 min.) and that students would not complete all of them, which generally proved to be the case. The intervention group received the “Expression polygons” etude (Fig. 3 ). Teachers were advised that “It is important that [the students] go beyond solving the six equations and spend some time generating their own expression polygons (or trying to).” Teachers were asked to allow the two classes the same amount of time to work on these tasks: “however much time you have available and feel is appropriate; ideally at least a whole lesson and perhaps more”. It is estimated that this was generally about 20–30 min. During this phase, teachers were asked to help both classes as they would normally, using their professional judgement as to what was appropriate, so that the students would benefit from the time that they spent on these tasks. Then the post-test was administered in the same way as the pre-test.

3.2 Results

The mean and standard deviation of the scores for both conditions at pre- and post-test, along with mean gain scores calculated as the mean of (post-test − pre-test) for each student, are shown in Table 2 and Fig. 5 . The similarity of the mean scores on the pre-test is reassuring regarding the matching of the parallel classes. A Bayesian t test was carried out on the gain scores, using the BayesFactor Footnote 2 package in R , comparing the fit of the data under the null hypothesis (the etude is as effective as the traditional exercises) and the alternative hypothesis (the etude and the exercises are not equally effective). A Bayes factor B indicates the relative strength of evidence for two hypotheses (Dienes, 2014 ; Rouder, Speckman, Sun, Morey, & Iverson, 2009 ), and means that the data are B times as likely under the null hypothesis as under the alternative. With a Cauchy prior width of .707, an estimated Bayes factor (null/alternative) of 1.03 was obtained, indicating no reason to conclude in favour of either hypothesis. (Conventionally, a Bayes factor between 3 and 10 represents “substantial” evidence [Jeffreys, 1961 ].) Prior robustness graphs for all of the Bayesian analyses described in this paper are included in the Appendix . In this case, calculation indicates that an exceptionally wide Cauchy prior width of more than 2.39 would be needed in order to obtain a “substantial” (Jeffreys, 1961 ) Bayes factor. The 95% credible interval Footnote 3 for the standardised effect size was [− .545, .005].

Study 1 results. (Error bars indicate ± 1 standard error)

Students’ comments on the study were few and generally related to the teaching episode rather than the etude or exercises. Insufficient responses meant that analysis of students’ perceptions of the two tasks was not possible.

3.3 Discussion

Such an inconclusive result does not allow us to say that either the exercises or the etude is superior in terms of developing procedural fluency, and neither does it allow us to say that there is evidence of no difference. Scrutiny of the students’ work suggested that in the time available many had engaged only superficially with the etude, whereas students in the control group had generally completed many exercises. It is possible that the style of the etude task was unfamiliar and/or that students were unclear regarding what they were supposed to do. For this reason, it was decided to devise a new etude to address the same topic of linear equations, one that it was hoped would be easier for students to understand and more similar in style to tasks that they might be familiar with. This etude formed the basis of study 2.

4 Study 2: Devising equations

The aim of this study was to investigate whether a different etude (“Devising equations”, see Section 2.2.2 ) is as effective as traditional exercises at developing students’ procedural fluency in solving linear equations, relative to the alternative hypothesis that the etude and the exercises are not equally effective.

The same quasi-experimental design was used as in study 1, with pairs of classes at the same school assigned to either the intervention (the “Devising equations” etude, see Section 2.2.2 ) or control (traditional exercises).

4.1.1 Instrument and administration

This time the intervention group received the “Devising equations” etude, as described in Section 2.2.2 (see Fig. 6 ). The control group received the same set of traditional exercises as used in study 1 (see Fig. 3 ) and were asked to complete as many as possible in the same amount of time as given to the etudes group. The same pre- and post-tests were used as in study 1 (see Fig. 4 ). Administration was exactly as for study 1, except that this time the only advice given to teachers regarding the etude was that students should “generate and solve their own equations”.

Study 2 “Devising equations” task

4.1.2 Participants

Schools were again recruited through a Twitter request. Schools D, E, F, G and H (Table 1 ) took part, all of which used attainment setting for mathematics. Teachers were again asked to choose parallel classes, and a total of 213 mathematics students from Years 8 and 9 (age 12–14) participated. This time, 19 students’ pre- and post-tests could not be matched, because students did not always put their names on their tests, leaving N  = 194.

4.2 Results

Results are shown in Table 3 and Fig. 7 . As in study 1, a Bayesian t test was carried out on the gain scores (Dienes, 2014 ; Rouder et al., 2009 ), with a Cauchy prior width of .707, this time giving a Bayes factor (null/alternative) of 5.92. This means that the data are nearly six times as likely under the null hypothesis (the etude is as effective as the traditional exercises) as under the alternative hypothesis (the etude and the exercises are not equally effective). Conventionally, a Bayes factor between 3 and 10 represents “substantial” evidence (Jeffreys, 1961 ). The prior robustness graph (see Appendix ) indicates that any Cauchy prior width of more than .317 would have led to a Bayes factor of at least 3, which suggests that this finding is robust. The 95% credible interval for the standardised effect size was [−.326, .233].

Study 2 results. (Error bars indicate ± 1 standard error)

Again, students’ comments were insufficiently plentiful or focused on the task to enable an analysis.

4.3 Discussion

Study 2 provides substantial evidence that there is little difference across one or two lessons between the effect on students’ procedural fluency of using traditional exercises or the “Devising equations” etude. Examination of students’ work showed a much greater engagement with this etude than with the “Expression polygons” one used in study 1, as evidenced by far more written work, so it is plausible that the effect of this etude might consequently have been stronger and, in this case, was closely matched to that of the exercises.

In an attempt to extend the bounds of generalisability of this finding, a third study was conducted, using the enlargements etude discussed in Section 2.3 , in order to see whether a similar result would be obtained in a different topic area.

5 Study 3: Enlargements

The aim of this study was to investigate whether a third etude (“Enlargements”, see Section 2.3 ) is as effective as traditional exercises at developing students’ procedural fluency in a different (geometric) topic area: performing an enlargement of a given shape on a squared grid with a specified positive integer scale factor. As before, the alternative hypothesis was that the etude and the exercises are not equally effective.

The same quasi-experimental design was used as in studies 1 and 2, with pairs of parallel classes in each school assigned to either the intervention (this time the “Enlargements” etude) or control (traditional exercises) condition.

5.1.1 Instrument and administration

The “Enlargements” etude (Foster, 2013d ) discussed in Section 2.3 was used for the intervention groups, and the control groups were provided with traditional exercises and asked to complete as many as possible in the same amount of time (see Fig. 8 for both). The exercises consisted of a squared grid containing five right-angled triangles and four given points. Each question asked students to enlarge one of the given shapes by a scale factor of 2, 3 or 5, using as centre of enlargement one of the given points. Pre- and post-tests were administered (Fig. 9 ), in which students were asked to enlarge a given triangle with a scale factor of 4 on a squared grid about a centre of enlargement marked with a dot. The pre- and post-tests were intended to be as similar as possible in presentation to the traditional exercises, again in the hope that any bias in the focus of the post-test would be in favour of the control group. As before, the post-test included a space at the end for open comments, asking students to write down “what you think about the work you have done on enlargements”. Each test was scored out of 4, with one mark for each correctly positioned vertex and one for an enlarged triangle of the correct shape, size and orientation (not necessarily position). Administration was exactly as for studies 1 and 2, except that this time teachers were asked to ensure

that the students understand that they are meant to try to find as many possible positions for the centre of enlargement as they can—perhaps even the whole region where these centres can be. Students could also go on to explore what happens if the starting triangle is in a different position, or is a different shape, or if a different scale factor is used (original emphasis).

The purpose of this was to try to ensure that the students would engage extensively with the etude and not assume that finding one viable centre of enlargement was all that was required.

Study 3 materials: enlargement etude (intervention) and traditional exercises (control)

Study 3 pre-test and post-test

5.1.2 Participants

As before, schools were recruited through a Twitter request. Schools I, J and K in Table 1 took part, all of which used attainment setting for mathematics lessons. Teachers were again asked to choose parallel classes, and a total of 151 mathematics students from Years 9 and 10 (age 13–15) participated. Year 9–10 classes were used this time, rather than Year 8–9 classes, due to teachers’ choices about suitability for this different topic. This time, only 10 students’ pre- and post-tests could not be matched, again because of missing names on some of the tests, leaving N  = 141.

5.2 Results

Analysis proceeded as before, and the results are shown in Table 4 and Fig. 10 . Again, a Bayesian t test was carried out on the gain scores (Dienes, 2014 ; Rouder et al., 2009 ), with a Cauchy prior width of .707, this time giving a Bayes factor (null/alternative) of 5.20, meaning that the data are about five times as likely under the null hypothesis (the etude is as effective as the traditional exercises) as under the alternative hypothesis (the etude and the exercises are not equally effective). Conventionally, a Bayes factor between 3 and 10 represents “substantial” evidence (Jeffreys, 1961 ). The prior robustness graph (see Appendix ) indicates that any Cauchy prior width of more than .365 would have led to a Bayes factor of at least 3, which suggests that the finding is robust. The 95% credible interval for the standardised effect size was [−.384, .257].

Study 3 results. (Error bars indicate ± 1 standard error)

Once again, student comments were too few to allow a reasonable analysis.

5.3 Discussion

Study 3 provides substantial evidence that there is no difference across one or two lessons between the effect on students’ procedural fluency of using traditional exercises or this enlargement etude. Examination of students’ work showed a lot of drawing on the sheets, with many students correctly finding the locus of all possible positions for the centre of enlargement. It may be that this greater degree of engagement (relative to study 1) could account for this etude being of comparable benefit to the exercises, as was the case in study 2.

6 General discussion

The Bayes factors obtained in these three studies were combined using the BayesFactor package in R and the meta.ttestBF Bayesian meta-analysis function. Again using a Cauchy prior width of .707, this time an estimated combined Bayes factor (null/alternative) of 5.83 was obtained (Table 5 ). This again falls within the conventionally accepted range of 3 to 10 for “substantial” evidence (Jeffreys, 1961 ). This means that, taken together, the three studies reported support the conclusion in favour of the null hypothesis that the etudes are as effective as the traditional exercises in developing students’ procedural fluency, relative to the alternative hypothesis that the etudes and the exercises are not equally effective.

The smaller Bayes factor for study 1 may have resulted from a less clearly articulated etude that was unfamiliar in style to the students, requiring a greater degree of initiative in constructing expressions than is normally expected in mathematics classrooms. If it is the case that students were less sure what was expected of them, this could explain why the etudes group carried out less equation solving here than the exercises group did. As reported above, in studies 2 and 3, a greater effort was made in the teacher instructions to explain the intentions of the task, and a greater engagement with the etudes was inferred from the quantity of written work produced.

7 Conclusion

These three exploratory studies suggest that the etudes trialled here are as effective as the traditional exercises in developing students’ procedural fluency. Consequently, for a hypothetical teacher whose sole objective was to develop students’ procedural fluency, it should be a matter of indifference whether to do this by means of exercises or etudes. Given the plausible benefits of etudes in terms of richness of experience and opportunity for open-ended problem solving and creative thinking, it may be that etudes might on balance be preferred (Foster, 2013b ).

It should be stressed that only three etudes were tested in these studies across only two mathematics topics and with students aged 12–15. Further studies using other etudes in other topic areas and with students outside this age range would be necessary to extend the generalisability of this finding. In addition, studies including delayed post-tests would be highly desirable, but were not practicable for this initial exploratory study. It would also be important to examine evidence for the hypothesised benefits of etudes beyond the narrow focus of these studies on procedural fluency. For example, it is plausible that etudes are more engaging for students, provide opportunities for students to operate more autonomously and solve problems, promote discussion and reasoning and support conceptual understanding of the mathematics. Classroom observation data, other kinds of assessments, as well as canvassing teacher and student perspectives, would be necessary to explore the extent to which this might be the case.

Caution must be exercised in interpreting these findings, since the constraints of the participating schools did not allow random allocation of students to condition (etude or exercises). Instead, schools selected pairs of “parallel” classes, generally based on level of class within the Year group (e.g., set 3 out of 6). It is reassuring that pre-test scores were generally close across the two conditions, but there remains the possibility that the parallel classes differed on some relevant factor. It should also be noted that some pre- and post-tests could not be matched, as students did not write their names on their tests, meaning that these tests had to be excluded from the data. In studies 2 and 3, the percentages of tests excluded were 9% and 7%, respectively, but in study 1 the percentage was much higher (20%). However, this was largely the result of one particular class, in which none of the students wrote their names on either test; ignoring this class, the percentage of missing data was a less severe 12%. However, these higher than desirable percentages of missing data are a reason to be cautious in interpreting these findings.

The extent of the guidance given to teachers about how to use the etudes was necessarily highly limited by the constraints of these studies. For practical reasons, the entire instructions on conducting the trials were restricted to one side of A4 paper. No professional development was involved, as these trials were carried out at a distance, and in most cases the participating schools and teachers were recruited via Twitter and contacted solely by email, and were not known personally to the researcher. It may be supposed that students would derive far greater benefit from etudes if they were deployed by teachers who had received professional development which involved prior opportunities to think about and discuss ways of working with these sorts of tasks. It remains for future work to explore this possibility.

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Appendix: Prior robustness graphs for the three Bayesian analyses

Horizontal dashed lines show the conventional cut-off Bayes Factor of 3 for “substantial” evidence (Jeffreys, 1961 ).

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Foster, C. Developing mathematical fluency: comparing exercises and rich tasks. Educ Stud Math 97 , 121–141 (2018). https://doi.org/10.1007/s10649-017-9788-x

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