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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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• 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

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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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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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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.

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

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The Need For Speed: why fluency counts for maths learning

Home » Publications » The Need For Speed: why fluency counts for maths learning

Toni Hatten-Roberts

October 19, 2023 · AP57

1. Introduction

Australia has a problem in maths achievement which is shown in stark relief by the declining number of students taking the subject in senior years. A recent report by the Australian Mathematical Sciences Institute (AMSI) identified significant falls in the number of Year 11 and 12 students choosing to enrol in high-level mathematics subjects. This dropped from an average of 71-73% over the last 10 years, to a new low of 66%. This is likely to impact Australia’s future workforce. [1]

One of the most significant reasons students look to ‘drop’ maths (or any subject for that matter) is based on their ability to be successful at it. Generally speaking, a student is less likely to continue if he or she has trouble succeeding. According to the PISA, [2]  Australian 15-year-olds are falling behind in their mathematical skills. Compared to their Australian peers in 2003, they are at least one year behind, and three years behind those in Singapore, the top-performing country. The report found that 46% of 15-year-olds do not meet the national standard of proficiency in mathematics, indicating that almost half of the student population is struggling in this subject.

This decline in mathematics performance reflects the way the subject is being taught across Australia and other Western countries, with the prioritising of conceptual understanding of maths over procedural and factual fluency, with the latter often derided as damaging to students’ understanding. [3]

The current dominant thought in mathematics classrooms around Australia, is that students must build conceptual knowledge first to help them invent and understand the procedures. This is despite mathematical fluency being a foundational skill that underpins these higher-level skills. Without a strong foundation in basic mathematics, students will likely struggle to apply problem-solving and reasoning skills effectively.

Mathematical fluency refers to the ability to perform mathematical calculations using well-rehearsed procedures quickly and accurately and includes the ability to recall facts to the point of automaticity. It also involves a strong understanding of mathematical vocabulary and symbols, as well as the ability to read and interpret mathematical expressions and equations. Fluency provides a foundation for higher-level mathematics skills needed for problem-solving, reasoning, and critical thinking, as well as real-world problem-solving while promoting efficiency and confidence. When students are fluent in basic mathematical skills, mathematics anxiety is reduced and a positive attitude towards mathematics is fostered.

However, if fluency is the foundation of mathematical development, Australian students have yet to master it, as evidenced by their poor performance. Based on experience working with more than a hundred schools in Queensland, Northern Territory, NSW, Victoria, Western Australia, the ACT, and Tasmania, teachers consistently report that students struggle with recalling basic mathematical facts. If you were to ask a high school maths teacher what one maths skill they believe is most important for students to learn in primary school, they would likely say the ability to quickly recall multiplication tables and division facts. This foundational knowledge is crucial for future mathematical concepts such as geometry, fractions, factors, rates, ratios, and algebra, all taught beyond Year 4.

Students who have foundational maths fact knowledge easily recalled from memory, are more likely to develop the prerequisite skills for solving more complex problems and can interpret more abstract principles.

Studies on mathematics achievement [4] have shown that students who excel in mathematics at an early grade level, are likely to maintain their success in subsequent grades. Conversely, students who struggle in the earlier grades are more likely to face challenges in the future. [5] It is essential for schools to provide high-quality mathematics instruction that includes routines for fact and procedural fluency.

Alongside this, there is a need for a ‘point of time’ indicator, much like the check introduced in the UK to identify students who have gaps in automatic recall of facts, particularly multiplication. This will help identify early, the students who have not mastered procedural fluency for addition, subtraction, multiplication and division using the standard algorithm.

2. The importance of timed assessments

Timed assessments are a measure where students can recall the facts with automaticity, with little hesitation and provide the teacher with information on the mastery point of student learning. Students who have high rates in reading typically have low rates of error. The same can be found in research on timed maths fluency. [6] Monitoring whether students have attained fluency, determined by both accuracy and speed can only be done through regularly timed tests that track and measure how close to automaticity students are getting. [7]

Suggestions that timed maths facts tests to measure fluency are a cause for anxiety lacks research to support such claims with no causal evidence found. [8] Research by Gunderson, et al. (2018) [9] found the main cause of mathematics anxiety is based on whether students lack skills. Schools that follow a ‘science of reading’ approach consistently use timed fluency to measure reading, yet none report anxiety about reading. Why? Because the students are time-tested on what they have explicitly been practising. When monitoring reading fluency, teachers can measure current student performance and allow insight into future performance in reading based on how many words per minute they read with accuracy.

The same applies to mathematics.  Data collated can be used to support the development of factual fluency as a necessary prerequisite to higher mathematics acquisition. [10] Additionally, evidence suggests that students who are confident in this area of mathematics, confidence permeates to other areas of mathematical problem-solving. [11] From a cognitive science perspective, timed tests have the added benefit of an instructional approach used for retrieval practice, a strategy for learning. Effortless retrieval of declarative facts reduces the cognitive load when students work with higher levels of mathematical problems. Students who recall their basic facts accurately and quickly have greater cognitive resources available to learn more complex tasks or concepts. So, a daily timed test, after some paired verbal rehearsal of a set of facts with a classmate, becomes a daily learning experience just like reading fluency routines.

TEXT BOX The Maths Wars and misconceptions about fluency

The ‘Maths Wars’ describes a long-standing debate in the field of mathematics education regarding the best way to teach maths. The debate centres around varying beliefs regarding two key issues: what knowledge to prioritise when teaching new content to students and the relevant priority placed on mathematical fluency through timed assessments.

There are competing views as to the relative importance teachers place on developing either procedural knowledge to solve mathematical problems, or on constructing students’ conceptual knowledge. [12] Conceptual proponents argue that mathematics education should emphasise problem-solving, reasoning, and critical thinking skills, as these are essential for success in higher-level maths and real-world contexts. This is partly based on a belief that students only learn when they discover mathematical concepts for themselves, such as through independent inquiry or exploration activities. Moreover, it is also implied that students must be encouraged to explore mathematical concepts in-depth, well before exposing students to standard procedures or algorithms. It is also argued that this exploration will lead to a deeper understanding of mathematics and better problem-solving skills. However this dichotomy is false, as conceptual and procedural knowledge are deeply intertwined and iterative.   Both concepts and procedures reinforce each other. Research shows that there is no optimal ordering for teaching either concepts or procedures first, as outlined in a CIS analysis paper last year, Myths That Undermine Maths Teaching, by Sarah H. Powell, Elizabeth M. Hughes and Corey Peltier [AP38 August 2022, Page 2].

It is often feared that memorisation of key maths facts ­ like the multiplication tables ­ is insufficient for students to truly understand mathematics and that an approach prioritising mathematical fluency could lead to a narrow view of mathematics that neglects important conceptual and reasoning skills. Moreover, it is claimed that learning mathematics facts under timed conditions can create anxiety and a dislike of mathematics for some students, particularly those who may struggle with basic mathematics skills. Instead, it is argued that a more exploratory, project-based approach to mathematics education is more engaging and less anxiety-inducing for students. [13] However, while it is important to consider students’ emotional well-being and engagement in mathematics, the lack of mathematical fluency and skill can itself create anxiety and frustration for students. [14]

3. Why does fluency matter?

Automaticity in mathematics frees up working memory and allows for the instant recall of a body of knowledge that supports students to manipulate new information as they build more complex schemas in mathematics. [15]   A mathematically fluent student can easily recall basic facts such as multiplication tables, and addition and subtraction facts, and can mentally perform calculations without having to rely on calculative devices. It should be a primary learning objective for all students to have computational fluency, particularly in the younger grades.

Like all subjects delivered in the school context, much of what students need to learn in mathematics will take effort and is considered ‘biologically secondary knowledge’. [16] Biologically secondary knowledge, the knowledge not acquired from a biological predisposition, requires attention, practice, retrieval and overlearning of foundational procedures and facts to then draw on efficiently for more complex tasks. The evidence from cognitive science suggests that, as learners, we are more alike than different when it comes to the way the brain is thought to encode, store and retrieve information. [17]

Geary’s (2012) [18] cognitive architecture of primary and secondary knowledge aligns some mathematical knowledge to our primary architecture — innate knowledge that has evolved in humans for generations, such as quantifying a small number of objects (1 – 3) referred to as subitising.  Research has also shown that infants have the ability to recognise greater and smaller amounts, the magnitude of numbers between 1 to 3 and can add and subtract quantities of up to 3 and 4. [19] However, most mathematics is a domain-specific secondary knowledge that must be taught to students explicitly. Additionally, mathematics has its own set of vocabulary and metalanguage. This vocabulary (the language of maths) must become the ‘sight words’ for students, orthographically mapped and conceptually understood, much the same as any vocabulary learnt through the approaches supported by the science of reading.

Finding an answer to a basic calculation using a calculator or by using a mental calculation strategy, takes time.  Storage in working memory is limited in duration as well as capacity. If a calculation is needed, other associated knowledge being held in limited working memory runs the risk of ‘timing out’ and can be lost. Cognitive science tells us that children in grades K-3 have far less working memory capacity than adults. [20] When students can recall algorithms and facts with automaticity, working memory is freed and students have a better capacity to work with problems. [21] When we know better, we do better, and what we know now is that we must exert effort to fill long-term memory in subjects such as mathematics.

In upper primary, students need fluent procedures to solve operations as well as automatic recall of multiplication and division facts as the underpinning knowledge to develop the concepts taught, such as rate and ratio, fractions and beginning algebra. Often curriculum document standards ask students to experiment and learn each of the non-standard algorithms or invented procedures to understand the concept before they learn the standard algorithm. Cognitive experts recommend students automate recall of one standard algorithm so that, given a problem, students know the steps to follow to solve it rather than offering a suite which can confuse students. [22]

Early primary students are especially good at remembering, but not reasoning. They will usually become frustrated when asked to solve by reasoning before achieving recall of memorised facts and fluency in procedures. So, a problem such as 8 + __ = 10, is better solved in the early years by the recall of facts to 10, as opposed to knowing that the missing addend is solved by using the inverse strategy of subtraction, where students are expected to use their subtraction knowledge to solve the problem rather than recalling from memory the learnt fact of 8 + 2 = 10. [22]

The inverse relationship concept is strengthened, however, when students continually solve by their automatic recall of facts knowledge and recall of all the memorised facts of a fact family [i] recalling in regular verbal chants both addition and subtraction facts.

Teaching conceptual and procedural knowledge together helps strengthen each other over time. Many students are likely to decide early on in primary school that they are ‘just not good at mathematics’ if they lack mathematical fact knowledge that could support them to answer automatically.  When presented with mathematical problems to solve, they don’t have the reasoning capability to do this, nor the mathematical fact fluency to rely on. The question is whether a large group of students identified as having a learning difficulty in mathematics actually do, or whether it is really the result of poor whole class teaching due to a lack of basic skills development.

Dyscalculia is a learning difficulty where students experience delays in numeracy development and lack basic number sense, impacting every aspect of number processing and thus any mathematics learning. There is a strong possibility, however, of an over-identification of dyscalculia throughout our schools due to our current instructional approach that begins with our early years of mathematics instruction and methods of teaching. Not, unlike a period of whole language pedagogy, where a significant number of students are identified as having a reading difficulty, or even dyslexia, but is really the result of poor whole-class instruction usually bereft of explicit phonics in reading instruction offered by balanced literacy advocates.

Research indicates that conceptual understanding and procedural fluency develop concurrently, with a two-way relationship between building conceptual and procedural knowledge. Instead of prioritising the concept over procedure, it is important to teach mathematics explicitly and build upon a student’s prior knowledge.

[i] A fact family would include 8 + 2 = 10; its turnaround, 2 + 8 = 10 and its two subtraction facts, 10 – 2 = 8 and 10 – 8 = 2.

4. Which mathematical facts matter for developing fluency?

Mathematical facts and times tables need more than basic rote learning as practised in the past and must go beyond posters in bedrooms. Practice must include verbal rehearsal to support the rehearsal benefits offered by McDaniel et al., (2009) [23] . Previously mastered times-tables must be mixed with new multiplication facts, adding only one or two new facts amongst known facts, and then removing well-known facts for some time to be reviewed later. [24]

To benefit from the strength of memory that becomes a piece of knowledge, time and effort will need to be devoted in helping children automate. Achieving automaticity requires first rehearsal, then retrieval practice over days and weeks (overlearning). Repeated deliberate practice is needed for transfer, and by using the combination of visual representations, verbal rehearsal, and writing, the key facts become an easily retrievable element from long-term memory. Saying the facts as rhythmic phrases becomes just another oral phrase students can pull to their minds without thinking. Skip counting patterns, such as counting in 3s or 4s work well in building conceptual knowledge. However, merely skip counting the answers will not help the ‘phrase’ recall of  ‘4 times 3 equals 12’, which supports students in embedding the facts into long-term memory.

The addition and subtraction facts of at least to 10 should be taught to automaticity from Year 1 with multiplication following soon after in Year 2 and 3, and division from there. These facts become a set of more than 350 known facts that can be drawn upon for other mathematical problem-solving. Students benefit from hearing, seeing, verbally rehearsing and writing the facts of basic operations needed for mathematics beyond Year 3. Students need to learn these fundamentals gradually by reviewing mixed sets, always including a cumulative review of past known facts . Mixed sets require recalling new facts, such as the 4 times tables but mixing them amongst previously taught facts such as 2s 3s and 5s, not just one set of facts per week or several ‘sets’ per fortnight.

To keep them recallable, students must continue to retrieve them after some time away (spaced practice) and this is part of the retrieval routine of the maths review (PowerPoint daily review presentations) where students recall and apply the facts to build automaticity. [25] Other important facts needed to develop a knowledge base, are what are called declarative facts, such as measurement conversion, 1000 ml = 1 L; attributes of angles; definition of a fraction (showing a visual) with its parts — numerator and denominator using maths specific academic language are just a few.

 5. What are the most effective teaching practices?

Research on memory and learning is quite clear that the limitations in our working memory affect learning when acquiring new academic knowledge. [26]   New information must pass through working memory, and managing the load as students are introduced to it is important to protect against overload. Explicit instruction involves breaking down complex skills or concepts into smaller, more manageable steps and providing clear explanations, models and worked examples [27] , helping to mitigate the limits in working memory.  This has been proven to be highly effective when teaching mathematics.

If we provide novices with an open-ended mathematics investigation or problems where students are left to sort the information through what is often termed a ‘productive failure’ [28] approach, there is a risk that students become distracted, lose understanding through misconception or become frustrated due to a lack of knowledge to hook the new learning onto. Eventually, for many students, this becomes a ‘blow’ to their self-perceived ability to learn mathematics. Instructional approaches that do not consider the way human cognitive architecture and the limitations of working memory impact learning are likely to be ineffective. Cognitive Science tells us that students will only remember what they have extensively practised and continue to be retrieved over many years. [29]

For students to be successful and considered proficient in maths, requires an explicit model of teaching, where students gain knowledge and skills, through interleaved [ii] practice over time. Explicit instruction also includes modelling mathematical procedures, alongside the standard algorithm, step by step.  An algorithm supported by a teacher’s ‘think aloud’, and using a concrete visual representation assists students’ conceptual knowledge.

It also allows the teacher to manage the cognitive load for the students by breaking complex mathematical concepts into smaller pre-skills that will be needed for more complex tasks. Explicit instruction is built on high levels of active engagement by the student, not the ‘chalk and talk’ often associated with ‘traditionalist’ mathematics.  The teacher engages in frequent checking for understanding, which allows for the opportunity to receive timely feedback to limit misconceptions, build confidence and gain a deeper understanding of the concepts being taught.

Explicit instruction lessons should begin with a daily review or quiz of previously taught mathematical facts and procedures. There is often a gap between what teachers teach and what students learn, due primarily to the way the mathematics curriculum is currently delivered in Australia. Primary mathematics is typically taught in blocks, where two to three weeks are dedicated to each separate topic across the term and is ticked off as content covered, and is assessed at the end of the block with no checking for long-term understanding using delayed tests. Yet, without spaced, practice [iii] and repeated retrieval over time, much of what students have covered is often forgotten and will need re-teaching due to the lack of practice and the ‘tick, flick and move on’ delivery mode teachers feel compelled to use. Students need to be provided with mass practice in the initial encoding of each procedural skill where students have an opportunity to use them efficiently, independently and accurately. Once mastery has occurred, students need spaced and cumulative review over time. [30]

Schools that acknowledge the science of learning and the need for an explicit model of teaching, understand the need to build procedural and fact fluency. This can be practised through the daily use of review, where students retrieve and apply their facts and procedural knowledge in regular routines. The technique of using a ‘maths daily review’ is the perfect vehicle to support students in developing fluency in facts and procedures, as the frequent retrieval allows students the opportunity to practise the many repetitions of the skill needed to move the knowledge to long-term memory.

Teachers use the review to assess mastery and check for understanding of their students’ developing fluency. The 20 to 30-minute daily review routine has the bonus of cumulative review where students must retrieve previously taught topics to ensure mastery has been attained. Mathematical activities that make use of spaced practice, whether in the form of ‘daily reviews’, ‘retrieval grids’, ‘do nows’, or homework based on retrieving concepts from the previous week’s learning, last month, last term or unit, ensure students have fluency for later use.

[ii] Interleaving occurs when different topics in a course of study are jumbled up and learned concurrently. Take a mathematics lesson for example, instead of learning about a concept such as fractions for 2-3 weeks then moving on to a different topic, an interleaved approach would combine several other concepts such as measurement, time, algebra, statistics etc. into the daily lessons.  

[iii] Spaced practice involves spreading out learning into smaller chunks over a longer period of time rather than conducting the learning over longer sessions. It works by allowing information to be forgotten and then repeatedly re-learnt. This process helps to commit the information to long term memory.

6. How could we better systematically monitor mathematical fluency in Australia?

We know students who are proficient with mathematical facts, being able to recall facts with speed and accuracy, are able to work at higher levels of mathematics more easily and have the bonus of self-efficacy. It is surely time then to bring in some form of widespread monitoring that can be used diagnostically to identify those students who may struggle with later mathematics.  If the inability to know times tables to automaticity is an indicator of students who may struggle at later mathematics and has been identified as such an important sub-skill for mathematical thinking, adopting a widespread point-in-time assessment is required.

Being fluent in mathematics is no different from having basic fluency in literacy. Just as timed reading tests of fluency are used as a measure of reading proficiency, likewise a multiplication facts speed test can identify students’ ability to recall the facts with automaticity and identify those who may require remediation sooner, rather than later. Currently in the UK, students are tested on their fluency to recall multiplication facts in Year 4 as a benchmark of proficiency and precision. The newly-introduced Multiplication Tables Check is an annual statutory check on the times tables knowledge for all state-funded Year 4 students in England and Wales. The test is taken towards the end of the year and data is collected by both the UK Ministry of Education and schools, within a window of time much like the Australian NAPLAN assessment. Australian curriculum documents, similar to the UK, expect students to ‘use their proficiency with addition and multiplication facts to add and subtract, multiply and divide numbers efficiently’ identified in the Year 4 achievement standard; students in the UK by the end of Year 3, should be fluent in the 2, 3, 4, 5, 8, 10 times tables, and by the end of Year 4 should know all their times tables up to 12, i.e., the 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 times tables.

The Multiplication tests consist of an on-screen test consisting of 25 times table questions randomly selected (no more than 30% of test items is the same as any other test form) for each student with a higher ratio of 6s, 7s, 8s, 9s and 12s as these facts are considered more difficult. Students have six seconds to answer each question, with a three-second pause between questions. On average, the check should take no longer than five minutes to complete.

The purpose of the check is to determine whether students can fluently recall their times tables up to 12, which the UK Government cites as “essential for future success in mathematics” and goes on to report that the students’ school will also use the result to identify students who need additional support. While there is no official pass mark, fail, or expected standard threshold, schools are said to make their judgments as to the intervention based on the results. One such school setting using the test, Athena Learning Trust (UK) consisting of three primary schools, begins intervention in autumn for those students not comfortably achieving 23 out of a possible 25.

Such a test provides valuable information for schools and teachers to identify students who need additional support. Given the current state of mathematics in Australia, it would seem logical to invest in such a test here at a similar point of time during Term 3 of Year 4, with the opportunity for allowing intervention to begin in Term 4. This test has little cost, takes minimal time and delivers valuable information. The question should be ‘why wouldn’t we’ rather than ‘why would we’. Presently NAPLAN assessment in May for Year 3 and Year 5 students fails to test the speed and accuracy of facts, yet by the end of Year 4 students are expected to ‘use their proficiency with addition and multiplication to add and subtract, multiply and divide numbers efficiently’. [31] Yet as a system, how do we know? If this is such an important milestone indicator of future maths proficiency, as indicated by the research [32] then adding a timed test on fact fluency is essential in supporting schools in identifying students well before high school when it is too late.

7. Conclusion

Understanding the role and importance of mathematical fluency and addressing it, is a fundamental challenge facing education in Australia. The national decline in mathematical standards as measured by a multitude of metrics has been steady and consistent for at least the past 20 years. The current approach has not worked and if continued, will likely lead to worsening outcomes for students and the nation.

Just as with reading, mathematical fluency is equally vital for students’ success and confidence in the subject. To ensure it is attained, monitoring is necessary through daily formative timed assessment in addition and subtraction in the early years and multiplication beginning in Year 2 to Year 4. Students who haven’t mastered maths facts by Term 3 of Year 4 will need a daily intervention program.

The fluency of students could be better monitored by adopting a universal screening tool that tests the accuracy and speed with which students can solve multiplication tables questions. This Multiplication Tables Check for all students would be administered by state and territory education departments, similar to the UK, for all Year 4 students. These results could be used by schools ­ with the support of departments ­ to provide interventions to improve numeracy as well as monitoring improvements once such interventions have been provided.

Within schools, research suggests a range of approaches could build mathematical fluency; among the best evidenced include the following:

• Developing an understanding of the role of automaticity, working memory and cognitive load theory can provide a basis for a successful new approach to teaching mathematics, particularly in the early years of schooling. An explicit instruction approach based on Rosenshine’s Principles24 where students review foundational information including multiplication facts through sharp-paced daily and cumulative reviews 20 minutes a day is a successful way of operationalising this and achieving mathematical fluency of basic facts and procedures by Australian students
• Daily practice by primary school students of oral facts, as well as practice in standard procedures, maths vocabulary, counting patterns and place value to underpin number understanding for students to work confidently. Students who have not mastered multiplication fluency by secondary school risk future failure and disengagement in higher-level maths courses
• The proper teaching and regular review of number facts, maths vocabulary and procedural fluency. This is essential to high-quality instruction and requires overlearning.
• The sharing of these results with families, which ultimately will build confidence in maths teaching in our primary schools in the wider community.

[1] Australian Mathematical Sciences Institute, (2022, April 27) https://amsi.org.au/2022/04/27/

maths-crisis-year-12-maths-enrolments-reach-all-time-low/

[2] OECD. (2018a). PISA 2018 results . Oecd.org. https://www.oecd.org/pisa/publications/pisa-2018-results.htm

[3] VanDerHeyden, A. M., & Codding, R. S. (2020). Belief-Based versus Evidence-Based Math Assessment

and Instruction. Communiqué (National Association of School Psychologists), 48(5), 1–20–25.

[4] Geary, D. C. (2011). Cognitive Predictors of Achievement Growth in Mathematics: A 5-Year

Longitudinal Study. Developmental Psychology, 47(6), 1539–1552.         https://doi.org/10.1037/a0025510

[5] Price, G. R., Mazzocco, M. M. M., & Ansari, D. (2013). Why mental arithmetic counts: Brain

activation during single digit arithmetic predicts high school math scores. The Journal of Neuroscience, 33(1), 156–163. https://doi.org/10.1523/JNEUROSCI.2936-12.2013

[6] VanDerHeyden, A., McLaughlin, T., Algina, J., & Snyder, P. (2012). Randomized Evaluation of a

Supplemental Grade-Wide Mathematics Intervention. American Educational Research Journal, 49(6), 1251–1284. https://doi.org/10.3102/0002831212462736

[7] Riley-Tillman, T. C., VanDerHeyden, A. M., & Burns, M. K. (2012).  RTI applications,

Volume 1: Academic and behavioral interventions  (Vol. 1). Guilford Press.

[8] Grays, S., Rhymer, K., & Swartzmiller, M. (2017). Moderating effects of mathematics anxiety on

the effectiveness of explicit timing. Journal of Behavioral Education, 26(2)

[9] Gunderson, E. A., Park, D., Maloney, E. A.,Beilock, S. L. & Levine, S. C. (2018) Reciprocal

relations among motivational frameworks, math anxiety, and math achievement

in early elementary school. Journal of Cognition and Development, 19, 21–46. doi:10.1080

/15248372.2017.14215

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Learn Mathematics.

[11]   Foundations for Success: The Final Report of the National Mathematics Advisory Panel. (2008).

In US Department of Education. US Department of Education.

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https://www.proquest.com/magazines/no-more-math-wars/docview/1761255371/se-2

[13] Boaler, J. (2014). Research Suggests that Timed Tests Cause Math Anxiety. Teaching

Children Mathematics, 20(8), 469–474. https://doi.org/10.5951/teacchilmath.20.8.046

[14] Gunderson, E. A., Park, D., Maloney, E. A.,Beilock, S. L. & Levine, S. C. (2018) Reciprocal

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[15]   Tricot, A., & Sweller, J. (2014). Domain-Specific Knowledge and Why Teaching Generic Skills

Does Not Work. Educational Psychology Review, 26(2), 265–283. https://doi.org/10.1007/s10648- 013-9243-1

[16] Geary, D. (2012). Evolutionary educational psychology. In K. Harris, S. Graham, & T. Urdan (Eds.),

APA educational psychology handbook (Vol. 1, pp. 597–621). Washington, DC: American Psychological

Association.

[17] Willingham, D., & Daniel, D. (2012). Teaching to what students have in common.

Educational leadership ,  69 (5), 16-21.

[18]    Sweller, J. (2012). Human cognitive architecture: Why some instructional procedures work and others

do not. In APA educational psychology handbook, Vol 1: Theories, constructs, and critical issues (pp. 295–325). American Psychological Association. https://doi.org/10.1037/13273-011

[19]    Lin, D. C. G. J. (1998). Numerical Cognition: Age-Related Differences in the Speed of Executing

Biologically Primary and Biologically Secondary Processes. Experimental Aging Research, 24(2),

101–137. https://doi.org/10.1080/036107398244274

[20]   Gathercole, S. E., Lamont, E., & Alloway, T. P. (2006). Working Memory in the Classroom. In

Working Memory and Education (pp. 219–240). https://doi.org/10.1016/B978-012554465-8/50010-7

[21] Sweller, J., Clark, R., & Kirschner, P. (2010). Teaching general problem-solving skills is not a substitute for,

or a viable addition to, teaching mathematics. Notices of the American Mathematical Society, 57 (10), 1303-1304.

[22] Hartman, J. R., Hart, S., Nelson, E. A., & Kirschner, P. A. (2023). Designing mathematics standards in

agreement with science. International Electronic Journal of Mathematics Education, 18(3),             em0739. https://doi.org/10.29333/iejme/13179

[23] McDaniel, M. A., Brown, P. C., & Roediger, H. L. (2014). Make it stick: The science of successful learning .

Harvard University Press.

[24] Burns, M. K., Aguilar, L. N., Young, H., Preast, J. L., Taylor, C. N., & Walsh, A. D. (2019). Comparing

the Effects of Incremental Rehearsal and Traditional Drill on Retention of Mathematics Facts

and Predicting the Effects With Memory. School Psychology, 34(5), 521–530. https://doi.org/10.1037/spq0000312

[25] Dehaene, S. (2020). How we learn: Why brains learn better than any machine… for now. Viking.

[26] Kirschner, P. A., Sweller, J., & Clark, R. E. (2006). Why Minimal Guidance During Instruction Does Not Work: An Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential, and Inquiry-Based Teaching. Educational Psychologist , 41 (2), 75–86.

[27] Rosenshine, B. (2012). Principles of Instruction Research-Based Strategies That All Teachers Should         Know. American Educator, 36(1), 12–19. https://files.eric.ed.gov/fulltext/EJ971753.pdf

[28]   Ashman, G., Kalyuga, S., & Sweller, J. (2020). Problem-solving or Explicit Instruction: Which

Should Go First When Element Interactivity Is High? Educational Psychology Review, 32(1), 229–247. https://doi.org/10.1007/s10648-019-09500-5

[29] Willingham, D. T. (2004). Practice makes perfect, but only if you practice beyond the point of

perfection. American Educator, 28 (1), 31-33.

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Reasoning Skills

Developing opportunities and ensuring progression in the development of reasoning skills

Achieving the aims of the new National Curriculum:

Developing opportunities and ensuring progression in the development of reasoning skills.

The aims of the National Curriculum are to develop fluency and the ability to reason mathematically and solve problems. Reasoning is not only important in its own right but impacts on the other two aims. Reasoning about what is already known in order to work out what is unknown will improve fluency; for example if I know what 12 × 12 is, I can apply reasoning to work out 12 × 13. The ability to reason also supports the application of mathematics and an ability to solve problems set in unfamiliar contexts.

Research by Nunes (2009) identified the ability to reason mathematically as the most important factor in a pupil’s success in mathematics. It is therefore crucial that opportunities to develop mathematical reasoning skills are integrated fully into the curriculum. Such skills support deep and sustainable learning and enable pupils to make connections in mathematics.

This resource is designed to highlight opportunities and strategies that develop aspects of reasoning throughout the National Curriculum programmes of study. The intention is to offer suggestions of how to enable pupils to become more proficient at reasoning throughout all of their mathematics learning rather than just at the end of a particular unit or topic.

We take the Progression Map for each of the National Curriculum topics, and augment it with a variety of reasoning activities (shaded sections) underneath the relevant programme of study statements for each year group. The overall aim is to support progression in reasoning skills. The activities also offer the opportunity for children to demonstrate depth of understanding, and you might choose to use them for assessment purposes as well as regular classroom activities.

Place Value Reasoning

Addition and subtraction reasoning, multiplication and division reasoning, fractions reasoning, ratio and proportion reasoning, measurement reasoning, geometry - properties of shapes reasoning, geometry - position direction and movement reasoning, statistics reasoning, algebra reasoning.

The strategies embedded in the activities are easily adaptable and can be integrated into your classroom routines. They have been gathered from a range of sources including real lessons, past questions, children’s work and other classroom practice.

Strategies include:

• Spot the mistake / Which is correct?
• True or false?
• What comes next?
• Do, then explain
• Make up an example / Write more statements / Create a question / Another and another
• Possible answers / Other possibilities
• What do you notice?
• Continue the pattern
• Missing numbers / Missing symbols / Missing information/Connected calculations
• Working backwards / Use the inverse / Undoing / Unpicking
• Hard and easy questions
• What else do you know? / Use a fact
• Fact families
• Convince me / Prove it / Generalising / Explain thinking
• Make an estimate / Size of an answer
• Always, sometimes, never
• Making links / Application
• Can you find?
• What’s the same, what’s different?
• Odd one out
• Complete the pattern / Continue the pattern
• Another and another
• Testing conditions
• The answer is…
• Visualising

These strategies are a very powerful way of developing pupils’ reasoning skills and can be used flexibly. Many are transferable to different areas of mathematics and can be differentiated through the choice of different numbers and examples.

Nunes, T. (2009) Development of maths capabilities and confidence in primary school, Research Report DCSF-RR118 (PDF)

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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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• Colin Foster   ORCID: orcid.org/0000-0003-1648-7485 1  

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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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Exploring Students’ Procedural Fluency and Written Adaptive Reasoning Skills In Solving Open-Ended Problems

Stephanie gayle b. andal & rose r. andrade, volume 2, issue 1, march 2022.

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Developing students’ mathematical skills requires both procedure and reasoning. However, the declination of possessing these skills is still evident today. Hence, this study aimed to describe the students’ procedural fluency in terms of accuracy, flexibility, and efficiency and written adaptive reasoning in terms of explanation and justification in solving open-ended problems. The study employed descriptive-correlational design through purposive sampling of thirty students from a National High School in Laguna, Philippines. The quantitative data revealed that in procedural fluency, students can quickly submit a complete solution leading to correct answer. However, they fail to provide two or more solutions in solving open-ended problems. The results also showed that students can clearly explain the problem but struggle to justify their solution. Moreover, procedural fluency is positively correlated to their adaptive reasoning. Consequently, students with an average level of mathematical achievement scored significantly higher than those at a low mathematical level in terms of flexibility. Pedagogical implications suggest that problem-solving activities for students should not solely focus on getting the correct procedures and answers. Further, it is recommended that teachers should expose students in open-ended problems and allow them to try and justify their own unique solutions irrespective of their mathematical achievement.

Keywords: mathematical achievement, open-ended problems, procedural fluency, problem-solving, written adaptive reasoning

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Cite this article:

Andal, S.B. & Andrale, R.R. (2022). Exploring Students’ Procedural Fluency and Written Adaptive Reasoning Skills in Solving Open-Ended Problems. International Journal of Science, Technology, Engineering and Mathematics, Volume 2 Issue 1, pp. 1 - 25. DOI: https://doi.org/10.53378/352872

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Developing Math Reasoning In Elementary School And Beyond: The Mathematical Skills Required And How To Teach Them

Developing math reasoning skills in elementary school is crucial to succeed in developing a math mastery approach to learning which will support development through to middle school and high school. Students need strong applied reasoning alongside their math skills to be able to succeed – there’s no point in memorizing a theorem if you don’t know when to use it!

The Ultimate Guide to Problem Solving Techniques

Help your students to develop their problem solving skills with this free worksheet.

My approach to elementary school level math teaching and learning is that it should be about exploring, reasoning and challenging thinking, rather than learning rote/abstract rules for calculations and facts.

Though I recognize that fluency in math and memorizing key number facts is essential in elementary school mathematics to acquire the basics – these are the prerequisite skills that ought to be used and applied in real life contexts.

To succeed on standardized tests, it is clear that children require deep knowledge of facts and mathematical concepts. Moreover, they need to be able to use and apply these facts to a range of contexts, and different types of word problems , including the more complex multi-step and two-step word problems

What is reasoning in math?

Let’s start with the definition of mathematical reasoning. Reasoning in math is the process of applying logical and critical thinking to a mathematical problem to determine the truth in given mathematical statements. Mathematical reasoning helps students make connections and decide on the correct strategy to reach a solution.

Math reasoning is sometimes seen as the glue that bonds students’ mathematical skills together; it’s also seen as bridging the gap between fluency and problem solving. It allows students to use their fluency to carry out  math strategies for problem solving accurately .

In my opinion, it is only when we teach children to reason and give them the freedom to look for different strategies when faced with an unfamiliar context that we are really teaching mathematics in elementary school.

What are the types of mathematical reasoning?

There are two different types of reasoning:

Inductive reasoning

Deductive reasoning.

Inductive reasoning is also called bottom-up logic. When using inductive reasoning, people come to a conclusion based on observations. However, their conclusion may or may not be factual. For example, a student may observe that the following numbers are divisible by 4: 12, 36, 40, 48. They also notice that each of the numbers in the set is even. Therefore, they conclude that all even numbers are divisible by 4. This, however, is false. 

Deductive reasoning is called top-down logic and works in the opposite way to inductive reasoning. When using deductive reasoning, people use known facts to reach a conclusion. For example, a student may be trying to determine if all even numbers are divisible by 4. They may use the examples 22 ÷ 4 and 30 ÷ 4 to prove that not all even numbers are divisible by 4. This makes deductive reasoning more reliable.

Why focus teaching and learning on mathematical reasoning?

Logical reasoning requires metacognition (thinking about thinking) . It influences behavior and attitudes through greater engagement, requesting appropriate help (self-regulation) and seeking conceptual understanding.

Reasoning promotes these traits because it requires children to use their mathematical vocabulary . In short, reasoning requires a lot of active talk.

It is worth mentioning that with reasoning, active listening is equally important and if done right can also ensure increased learning autonomy for students.

The theory behind mathematical reasoning in elementary school

The infographic (below) from Helen Drury cleverly details what should underpin a mathematics teaching and learning syllabus. It’s a good starting point when you’re thinking about your mathematics curriculum in the context of fluency reasoning and problem solving .

I’ve also been very influenced by the Five Principles of Extraordinary Math Teaching by Dan Finkel

These are as follows, and are a great starting point to developing math reasoning at the elementary school level

1. Start math lessons with a question

2. Students need to wonder and struggle

3. You are not the answer key

4. Say yes to your students’ original ideas (but not yes to methodical answers)

See also this free guide to elementary math problem solving and reasoning techniques .

How to make reasoning central to math lessons in elementary school

Pose lesson objectives as questions to elementary school children..

A  ‘light bulb’ idea from my own teaching and learning was to redesign learning objectives, fashioning them into a question for learning. For example, instead of ‘to identify multiples of a number’, I’ll use ‘why is a square number a square number?’. Another example is: instead of ‘to use ratio to describe the relationship between two quantities,’ ask students ‘in a recipe, if the ratio of sugar to flour is 3:5, what does that mean?’

Phrasing LOs as a question instantly engages and enthuses children, they wonder what the answer is. It also ensures they show their reasoning in a model or image when they answer.

In this instance, children knew the process of calculating square numbers but could not articulate or mathematically reason why it worked until after the session.

It seems denying children answers allows them time to use their thinking skills, struggle and learn.

Ban the word ‘yes’ in math lessons

One of the simplest strategies I have found to make reasoning inseparable from mathematical learning is to ban the word ‘yes’ from the classroom.

Instead, asking children to reason their thoughts and explain why they think they are right can allow for greater learning gains and depth of understanding. Admittedly, this is still a work in progress and easier said than done.

To facilitate this, I always tell my children that I am not the answer key.

Using my example of square numbers, I allowed children time to struggle and wrestle with my question without providing an answer or giving hints. Instead, I challenged the students to unpack their understanding at the beginning of the lesson and then brought together mathematical ideas during a whole class discussion.

After a short discussion on how children might show or visualize a square number we began to show a model using arrays:

The children working at greater depth were encouraged to consider cubed numbers and show how they might be represented using multi-link cubes without input from me. This ensured the students made links between math concepts, mathematical vocabulary, and learning.

Use ‘sometimes, always, never’ classroom activities

‘Sometimes, always, never’ activities are another great way to foster reasoning and problem-solving skills. Take the image below:

Here, the children must sort the fraction statements into always, sometimes or never true. The next day, they move on to the lesson with the title phrased as a question. Not ‘to identify patterns’, but ‘how does this pattern work?’ with a pattern already presented on the board.

The children, instantly engaged, began conjecturing, made predictions and thought about the next patterns in the sequence. This lesson was inspired by an Nrich activity- a math education project run by the University of Cambridge) .

5 tips for developing mathematical reasoning in the elementary school

Small changes will not provide a framework to properly embed reasoning in the classroom. But, when implemented alongside ideas such as those mentioned above, these tips can help instil greater depth in math in your class for all ability levels.

1. Start lessons with a question.

2. start lessons with a provocative mathematical statement or mind bender.

Challenge your class to provide the mathematical proof. Examples include: 

• “N will always = N” 
•  “Multiples of 9 always have the digital sum of 9”.
• “When multiplying decimals the number of decimals places in the answer will be the total number of decimal places in the two numbers being multiplied. (For example, the answer to 2.5 x 3.21, will have 3 decimal places.)”
• “A square is always a rectangle, but a rectangle is not always a square.”

3. Present answers to exam questions as a puzzle

Use puzzles to generate discussions and make connections. Students can use their repertoire of math skills to explore the relationships between the numbers: Does the line signify addition, subtraction, multiplication, etc?

Puzzles can also be presented on a simple number line. When framed like this, children like to ‘come up’ with what the question could be:

4. Grouping children in threes

Three is the magic number when working through problems;

• Child one talks through the problem.
• Child two writes down everybody’s reasoning.
• Child three actively listens and watches.

5. Include reasoning prompting posters

Display visual mathematical reasoning prompts around the classroom. For example, the image below can help children in the first stages of formulating thoughts, predictions and assertions.

Your students will need an in-depth understanding of mathematical facts and concepts to succeed. They will also need to use and apply that knowledge to various contexts, classroom discussions, workbooks and in their homework.

As such, it’s clear that we must provide students with a strong foundation of reasoning skills to give them the very best shot at the assessments they must face.

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Ultimate Guide to Metacognition [FREE]

Looking for a summary on metacognition in relation to math teaching and learning?

Check out this guide featuring practical examples, tips and strategies to successfully embed metacognition across your school to accelerate math growth.

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9 times tables fluency, reasoning and problem solving

Subject: Mathematics

Age range: 7-11

Resource type: Worksheet/Activity

Last updated

29 November 2018

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A worksheet for the nine times table focusing on fluency, reasoning and problem solving.

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