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Fluency, Reasoning and Problem Solving: What This Looks Like In Every Maths Lesson

Neil Almond

Fluency reasoning and problem solving have been central to the new maths national curriculum for primary schools introduced in 2014. Here we look at how these three approaches or elements of maths can be interwoven in a child’s maths education through KS1 and KS2. We look at what fluency, reasoning and problem solving are, how to teach them, and how to know how a child is progressing in each – as well as what to do when they’re not, and what to avoid.

The hope is that this blog will help primary school teachers think carefully about their practice and the pedagogical choices they make around the teaching of reasoning and problem solving in particular.

Before we can think about what this would look like in practice however, we need to understand the background tothese terms.

What is fluency in maths?

Fluency in maths is a fairly broad concept. The basics of mathematical fluency – as defined by the KS1 / KS2 National Curriculum for maths – involve knowing key mathematical facts and being able to recall them quickly and accurately.

But true fluency in maths (at least up to Key Stage 2) means being able to apply the same skill to multiple contexts, and being able to choose the most appropriate method for a particular task.

Fluency in maths lessons means we teach the content using a range of representations, to ensure that all pupils understand and have sufficient time to practise what is taught.

Read more: How the best schools develop maths fluency at KS2 .

What is reasoning in maths?

Reasoning in maths is the process of applying logical thinking to a situation to derive the correct problem solving strategy for a given question, and using this method to develop and describe a solution.

Put more simply, mathematical reasoning is the bridge between fluency and problem solving. It allows pupils to use the former to accurately carry out the latter.

Read more: Developing maths reasoning at KS2: the mathematical skills required and how to teach them .

What is problem solving in maths?

It’s sometimes easier to start off with what problem solving is not. Problem solving is not necessarily just about answering word problems in maths. If a child already has a readily available method to solve this sort of problem, problem solving has not occurred. Problem solving in maths is finding a way to apply knowledge and skills you have to answer unfamiliar types of problems.

Read more: Maths problem solving: strategies and resources for primary school teachers .

We are all problem solvers

First off, problem solving should not be seen as something that some pupils can do and some cannot. Every single person is born with an innate level of problem-solving ability.

Early on as a species on this planet, we solved problems like recognising faces we know, protecting ourselves against other species, and as babies the problem of getting food (by crying relentlessly until we were fed).

All these scenarios are a form of what the evolutionary psychologist David Geary (1995) calls biologically primary knowledge. We have been solving these problems for millennia and they are so ingrained in our DNA that we learn them without any specific instruction.

Why then, if we have this innate ability, does actually teaching problem solving seem so hard?

Mathematical problem solving is a  learned skill

As you might have guessed, the domain of mathematics is far from innate. Maths doesn’t just happen to us; we need to learn it. It needs to be passed down from experts that have the knowledge to novices who do not.

This is what Geary calls biologically secondary knowledge. Solving problems (within the domain of maths) is a mixture of both primary and secondary knowledge.

The issue is that problem solving in domains that are classified as biologically secondary knowledge (like maths) can only be improved by practising elements of that domain.

So there is no generic problem-solving skill that can be taught in isolation and transferred to other areas.

This will have important ramifications for pedagogical choices, which I will go into more detail about later on in this blog.

The educationalist Dylan Wiliam had this to say on the matter: ‘for…problem solving, the idea that pupils can learn these skills in one context and apply them in another is essentially wrong.’ (Wiliam, 2018)So what is the best method of teaching problem solving to primary maths pupils?

The answer is that we teach them plenty of domain specific biological secondary knowledge – in this case maths. Our ability to successfully problem solve requires us to have a deep understanding of content and fluency of facts and mathematical procedures.

Here is what cognitive psychologist Daniel Willingham (2010) has to say:

‘Data from the last thirty years lead to a conclusion that is not scientifically challengeable: thinking well requires knowing facts, and that’s true not simply because you need something to think about.

The very processes that teachers care about most—critical thinking processes such as reasoning and problem solving—are intimately intertwined with factual knowledge that is stored in long-term memory (not just found in the environment).’

Colin Foster (2019), a reader in Mathematics Education in the Mathematics Education Centre at Loughborough University, says, ‘I think of fluency and mathematical reasoning, not as ends in themselves, but as means to support pupils in the most important goal of all: solving problems.’

In that paper he produces this pyramid:

This is important for two reasons:

1)    It splits up reasoning skills and problem solving into two different entities

2)    It demonstrates that fluency is not something to be rushed through to get to the ‘problem solving’ stage but is rather the foundation of problem solving.

In my own work I adapt this model and turn it into a cone shape, as education seems to have a problem with pyramids and gross misinterpretation of them (think Bloom’s taxonomy).

Notice how we need plenty of fluency of facts, concepts, procedures and mathematical language.

Having this fluency will help with improving logical reasoning skills, which will then lend themselves to solving mathematical problems – but only if it is truly learnt and there is systematic retrieval of this information carefully planned across the curriculum.

Performance vs learning: what to avoid when teaching fluency, reasoning, and problem solving

I mean to make no sweeping generalisation here; this was my experience both at university when training and from working in schools.

At some point schools become obsessed with the ridiculous notion of ‘accelerated progress’. I have heard it used in all manner of educational contexts while training and being a teacher. ‘You will need to show ‘ accelerated progress in maths ’ in this lesson,’ ‘Ofsted will be looking for ‘accelerated progress’ etc.

I have no doubt that all of this came from a good place and from those wanting the best possible outcomes – but it is misguided.

I remember being told that we needed to get pupils onto the problem solving questions as soon as possible to demonstrate this mystical ‘accelerated progress’.

This makes sense; you have a group of pupils and you have taken them from not knowing something to working out pretty sophisticated 2-step or multi-step word problems within an hour. How is that not ‘accelerated progress?’

This was a frequent feature of my lessons up until last academic year: teach a mathematical procedure; get the pupils to do about 10 of them in their books; mark these and if the majority were correct, model some reasoning/problem solving questions from the same content as the fluency content; set the pupils some reasoning and word problem questions and that was it.

I wondered if I was the only one who had been taught this while at university so I did a quick poll on Twitter and found that was not the case.

I know these numbers won’t be big enough for a representative sample but it still shows that others are familiar with this approach.

The issue with the lesson framework I mentioned above is that it does not take into account ‘performance vs learning.’

What IS performance vs learning’?

The premise is that performance in a lesson is not a good proxy for learning.

Yes, those pupils were performing well after I had modeled a mathematical procedure for them, and managed to get questions correct.

But if problem solving depends on a deep knowledge of mathematics, this approach to lesson structure is going to be very ineffective.

As mentioned earlier, the reasoning and problem solving questions were based on the same maths content as the fluency exercises, making it more likely that pupils would solve problems correctly whether they fully understood them or not.

Chances are that all they’d need to do is find the numbers in the questions and use the same method they used in the fluency section to get their answers – not exactly high level problem solving skills.

Teaching to “cover the curriculum” hinders development of strong problem solving skills.

This is one of my worries with ‘maths mastery schemes’ that block content so that, in some circumstances, it is not looked at again until the following year (and with new objectives).

The pressure for teachers to ‘get through the curriculum’ results in many opportunities to revisit content just not happening in the classroom.

Pupils are unintentionally forced to skip ahead in the fluency, reasoning, problem solving chain without proper consolidation of the earlier processes.

As David Didau (2019) puts it, ‘When novices face a problem for which they do not have a conveniently stored solution, they have to rely on the costlier means-end analysis.

This is likely to lead to cognitive overload because it involves trying to work through and hold in mind multiple possible solutions.

It’s a bit like trying to juggle five objects at once without previous practice. Solving problems is an inefficient way to get better at problem solving.’

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Fluency and reasoning – Best practice in a lesson, a unit, and a term

By now I hope you have realised that when it comes to problem solving, fluency is king. As such we should look to mastery maths based teaching to ensure that the fluency that pupils need is there.

The answer to what fluency looks like will obviously depend on many factors, including the content being taught and the year group you find yourself teaching.

But we should not consider rushing them on to problem solving or logical reasoning in the early stages of this new content as it has not been learnt, only performed.

I would say that in the early stages of learning, content that requires the end goal of being fluent should take up the majority of lesson time – approximately 60%. The rest of the time should be spent rehearsing and retrieving other knowledge that is at risk of being forgotten about.

This blog on mental maths strategies pupils should learn in each year group is a good place to start when thinking about the core aspects of fluency that pupils should achieve.

Little and often is a good mantra when we think about fluency, particularly when revisiting the key mathematical skills of number bond fluency or multiplication fluency. So when it comes to what fluency could look like throughout the day, consider all the opportunities to get pupils practicing.

They could chant multiplications when transitioning. If a lesson in another subject has finished earlier than expected, use that time to quiz pupils on number bonds. Have fluency exercises as part of the morning work.

Read more: How to teach times tables KS1 and KS2 for total recall .

What about best practice over a longer period?

Thinking about what fluency could look like across a unit of work would again depend on the unit itself.

Look at this unit below from a popular scheme of work.

They recommend 20 days to cover 9 objectives. One of these specifically mentions problem solving so I will forget about that one at the moment – so that gives 8 objectives.

I would recommend that the fluency of this unit look something like this:

LY = Last Year

This type of structure is heavily borrowed from Mark McCourt’s phased learning idea from his book ‘Teaching for Mastery.’

This should not be seen as something set in stone; it would greatly depend on the needs of the class in front of you. But it gives an idea of what fluency could look like across a unit of lessons – though not necessarily all maths lessons.

When we think about a term, we can draw on similar ideas to the one above except that your lessons could also pull on content from previous units from that term.

So lesson one may focus 60% on the new unit and 40% on what was learnt in the previous unit.

The structure could then follow a similar pattern to the one above.

Best practice for problem solving in a lesson, a unit, and a term 

When an adult first learns something new, we cannot solve a problem with it straight away. We need to become familiar with the idea and practise before we can make connections, reason and problem solve with it.

The same is true for pupils. Indeed, it could take up to two years ‘between the mathematics a student can use in imitative exercises and that they have sufficiently absorbed and connected to use autonomously in non-routine problem solving.’ (Burkhardt, 2017).

Practise with facts that are secure

So when we plan for reasoning and problem solving, we need to be looking at content from 2 years ago to base these questions on.

Now given that much of the content of the KS2 SATs will come from years 5 and 6 it can be hard to stick to this two-year idea as pupils will need to solve problems with content that can be only weeks old to them.

But certainly in other year groups, the argument could be made that content should come from previous years.

You could get pupils in Year 4 to solve complicated place value problems with the numbers they should know from Year 2 or 3. This would lessen the cognitive load, freeing up valuable working memory so they can actually focus on solving the problems using content they are familiar with.

Read more: Cognitive load theory in the classroom

Increase complexity gradually.

Once they practise solving these types of problems, they can draw on this knowledge later when solving problems with more difficult numbers.

This is what Mark McCourt calls the ‘Behave’ phase. In his book he writes:

‘Many teachers find it an uncomfortable – perhaps even illogical – process to plan the ‘Behave’ phase as one that relates to much earlier learning rather than the new idea, but it is crucial to do so if we want to bring about optimal gains in learning, understanding and long term recall.’  (Mark McCourt, 2019)

This just shows the fallacy of ‘accelerated progress’; in the space of 20 minutes some teachers are taught to move pupils from fluency through to non-routine problem solving, or we are somehow not catering to the needs of the child.

When considering what problem solving lessons could look like, here’s an example structure based on the objectives above.

Fluency, Reasoning and Problem Solving should NOT be taught by rote 

It is important to reiterate that this is not something that should be set in stone. Key to getting the most out of this teaching for mastery approach is ensuring your pupils (across abilities) are interested and engaged in their work.

Depending on the previous attainment and abilities of the children in your class, you may find that a few have come across some of the mathematical ideas you have been teaching, and so they are able to problem solve effectively with these ideas.

Equally likely is encountering pupils on the opposite side of the spectrum, who may not have fully grasped the concept of place value and will need to go further back than 2 years and solve even simpler problems.

In order to have the greatest impact on class performance, you will have to account for these varying experiences in your lessons.

Read more: 

• Maths Mastery Toolkit : A Practical Guide To Mastery Teaching And Learning
• Year 6 Maths Reasoning Questions and Answers
• Get to Grips with Maths Problem Solving KS2
• Mixed Ability Teaching for Mastery: Classroom How To
• 21 Maths Challenges To Really Stretch Your More Able Pupils
• Maths Reasoning and Problem Solving CPD Powerpoint
• Why You Should Be Incorporating Stem Sentences Into Your Primary Maths Teaching

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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.

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

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

Using reasoning to support fluency and problem solving 

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

What does progress in mathematical reasoning look like?

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

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

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

The role of relationships in thinking mathematically

Quantitative relationships.

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

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

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

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

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

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

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

Structural relationships : 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Solution to  Changes from Saturday to Sunday and the result

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

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

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

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

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

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

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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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• Progression Maps

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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• 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

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.

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Fluency, Problem Solving and Reasoning | Maths Focus at Britannica

• 25th November 2020

At Britannica, our vision is to deepen the children's mathematical understanding through a mastery approach. To achieve this, we have embedded within our Mathematics Scheme of work a focus on Fluency, Problem Solving and Reasoning skills.

• Problem-solving is at the heart of mastering Mathematics. While there is nothing new about using problem-solving questions to consolidate understanding, mastery gets teachers to rethink the traditional lengthy word-problem format. Instead, problem-solving questions are often open-ended, with more than one correct answer.
• Verbal reasoning demonstrates that pupils understand Mathematics. Talk is an integral part of mastery as it encourages students to reason, justify and explain their thinking.
• Fluency, reasoning, and problem solving underpin the deepening of understanding. Fluency alone doesn't give students the chance to delve deeper into Mathematics. They may well be able to answer the questions, but can they also justify their answer or explore other possibilities?

Mathematics is all around us, and it is essential to everyday life, from the spiral in a sea shell, the symmetry in a butterfly to the shapes in the world around us. A high-quality Mathematics education, therefore, provides a foundation for understanding the world, the ability to reason Mathematically, an appreciation of the beauty and power of Mathematics, and a sense of enjoyment and curiosity about the subject.

At Britannica, we aim to make children's learning in Mathematics relevant, practical, creative, exciting and engaging. Mathematics teaches children to calculate, communicate, reason and solve problems. It enables children to understand and appreciate relationships and pattern both number and space in their everyday lives. We aim for children to achieve their fullest potential in this subject.

Students are encouraged to delve deeply into their understanding of Mathematics and how it relates to the world around them. Our Mathematics teaching actively encourages risk-taking, which enables pupils to explore and try new ideas without the fear of failure. This is fundamental to building pupils' self-esteem within Mathematics. Throughout history, the study of Mathematics stems from intrigue and curiosity, with people's desire to pose and solve problems relating to the real world or purely within mathematics itself. We aim for our students to appreciate this and use their Mathematics to explore and question the way the world works and also to apply their reasoning skills.

Finally, underpinning all of this is making Mathematics fun and providing engaging lessons that the children will cherish and build their love for learning from!

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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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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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Intervention based on science of reading and math boosts comprehension and word problem-solving skills

by University of Kansas

New research from the University of Kansas has found that an intervention based on the science of reading and math effectively helped English learners boost their comprehension, visualize and synthesize information, and make connections that significantly improved their math performance.

The intervention , performed for 30 minutes twice a week for 10 weeks with 66 third-grade English language learners who displayed math learning difficulties, improved students' performance when compared to students who received general instruction. This indicates that emphasizing cognitive concepts involved in the science of reading and math are key to helping students improve, according to researchers.

"Word problem-solving is influenced by both the science of reading and the science of math. Key components include number sense, decoding, language comprehension and working memory. Utilizing direct and explicit teaching methods enhances understanding and enables students to effectively connect these skills to solve math problems . This integrated approach ensures that students are equipped with necessary tools to navigate both the linguistic and numerical demands of word problems," said Michael Orosco, professor of educational psychology at KU and lead author of the study.

The intervention incorporates comprehension strategy instruction in both reading and math, focusing and decoding, phonological awareness, vocabulary development, inferential thinking, contextualized learning and numeracy.

"It is proving to be one of the most effective evidence-based practices available for this growing population," Orosco said.

The study, co-written with Deborah Reed of the University of Tennessee, was published in the journal Learning Disabilities Research and Practice .

For the research, trained tutors implemented the intervention, developed by Orosco and colleagues based on cognitive and culturally responsive research conducted over a span of 20 years. One example of an intervention session tested in the study included a script in which a tutor examined a word problem explaining that a person made a quesadilla for his friend Mario and gave him one-fourth of it, then asked students to determine how much remained.

The tutor first asked students if they remembered a class session in which they made quesadillas and what shape they were, and demonstrated concepts by drawing a circle on the board, dividing it into four equal pieces, having students repeat terms like numerator and denominator. The tutor explains that when a question asks how much is left, subtraction is required. The students also collaborated with peers to practice using important vocabulary in sentences. The approach both helps students learn and understand mathematical concepts while being culturally responsive.

"Word problems are complex because they require translating words into mathematical equations, and this involves integrating the science of reading and math through language concepts and differentiated instruction," Orosco said. "We have not extensively tested these approaches with this group of children. However, we are establishing an evidence-based framework that aids them in developing background knowledge and connecting it to their cultural contexts."

Orosco, director of KU's Center for Culturally Responsive Educational Neuroscience, emphasized the critical role of language in word problems, highlighting the importance of using culturally familiar terms. For instance, substituting "pastry" for "quesadilla" could significantly affect comprehension for students from diverse backgrounds. Failure to grasp the initial scenario could impede subsequent problem-solving efforts.

The study proved effective in improving students' problem-solving abilities, despite covariates including an individual's basic calculation skills, fluid intelligence and reading comprehension scores. That finding is key, as while ideally all students would begin on equal footing and there would be few variations in a classroom, in reality, covariates exist and are commonplace.

The study had trained tutors deliver the intervention, and its effectiveness should be further tested with working teachers, the authors wrote. Orosco said professional development to help teachers gain the skills is necessary, and it is vital for teacher preparation programs to train future teachers with such skills as well. And helping students at the elementary level is necessary to help ensure success in future higher-level math classes such as algebra.

The research builds on Orosco and colleagues' work in understanding and improving math instruction for English learners. Future work will continue to examine the role of cognitive functions such as working memory and brain science, as well as potential integration of artificial intelligence in teaching math.

"Comprehension strategy instruction helps students make connections, ask questions, visualize, synthesize and monitor their thinking about word problems," Orosco and Reed wrote. "Finally, applying comprehension strategy instruction supports ELs in integrating their reading, language and math cognition…. Focusing on relevant language in word problems and providing collaborative support significantly improved students' solution accuracy."

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Geometry in the Primary Curriculum

The Mathematics National Curriculum for primary children in England aims to ensure that all pupils:

• become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately.
• reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language
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In this article, part of our Jaunts into Geometry Feature , we will look at how these aims of problem solving, reasoning and fluency can be embedded in geometry, using NRICH tasks. In the classroom, this might be achieved in two ways, depending on the focus of a particular lesson or series of lessons, by exploring:

• Problem solving, reasoning and fluency in a geometrical context
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In conclusion You may wish to consider the activities you use currently to explore geometrical ideas.  How are they already promoting problem solving, reasoning and fluency?  Might they benefit from being adapted?  What is the 'coverage' like across the geometrical strands? We hope you will be inspired to embed problem solving, reasoning and fluency into geometry in your classroom. Further reading Take a look at our Problem Solving Feature , which includes the article Using NRICH Tasks to Develop Key Problem-solving Skills . For further support on developing learners' mathematical reasoning, see our Reasoning Feature . You may also be interested in our Number Fluency Feature . References DfE (2013) Mathematics programmes of study: Key stages 1 and 2 National Curriculum in England https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/335158/PRIMARY_national_curriculum_-_Mathematics_220714.pdf Here is a PDF version of this article.

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Multiplying numbers by 10- fluency, reasoning and problem solving

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Title: pecc: problem extraction and coding challenges.

Abstract: Recent advancements in large language models (LLMs) have showcased their exceptional abilities across various tasks, such as code generation, problem-solving and reasoning. Existing benchmarks evaluate tasks in isolation, yet the extent to which LLMs can understand prose-style tasks, identify the underlying problems, and then generate appropriate code solutions is still unexplored. Addressing this gap, we introduce PECC, a novel benchmark derived from Advent Of Code (AoC) challenges and Project Euler, including 2396 problems. Unlike conventional benchmarks, PECC requires LLMs to interpret narrative-embedded problems, extract requirements, and generate executable code. A key feature of our dataset is the complexity added by natural language prompting in chat-based evaluations, mirroring real-world instruction ambiguities. Results show varying model performance between narrative and neutral problems, with specific challenges in the Euler math-based subset with GPT-3.5-Turbo passing 50% of the AoC challenges and only 8% on the Euler problems. By probing the limits of LLMs' capabilities, our benchmark provides a framework to monitor and assess the subsequent progress of LLMs as a universal problem solver.

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