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https://www.tes.com/teaching-resources/blog/fluency-reasoning-and-problem-solving-primary-maths

This blog discusses many aspects of reasoning but has a focus on verbal reasoning. It also discusses the need to understand mathematical language and to be able to verbally communicate this understanding to demonstrate mastery.

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https://www.ncetm.org.uk/resources/45776

The NCETM identifies that to be able to efficiently use reasoning to develop a mastery approach to mathematics, there must be differentiation for children to access their mathematics. This does not mean separating out the mathematically weak from the able, but more the ability to reason at different levels.

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Practice is a vital part of conceptual understanding and encourage reasoning and mathematical thinking
https://www.ncetm.org.uk/files/69314786/Secondary+Teaching+for+Mastery+December+2017.pdf

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The aims of the mathematics national curriculum, fluency (that blend of conceptual understanding and
procedural flexibility), mathematical reasoning and problem solving, capture the best in mathematics
education that we would surely want for any pupil. The word mastery is not used in the programmes of
study, but the principles cited by Charlie are at one with them. Likewise, the messages from Ofsted’s
report, Mathematics: made to measure, emphasise the importance of developing conceptual
understanding (not just procedural proficiency on its own) and giving all pupils the chance to solve
problems and reason about their work. So, there’s synergy in what we are all aiming for.

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"Mathematical reasoning 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, reasoning is the bridge between fluency and problem solving. It allows pupils to use the former to accurately carry out the latter."

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Fluency, Reasoning and Problem Solving: What They REALLY Look Like

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How to teach and support pupils in fluency reasoning and problem solving in every KS2 maths lesson and how practice can be developed across each term.

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

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This focuses on problem solving and reasoning in the early years. From reading this article it links to many of the COEL in which is used in the EYFS. Active learning and creating and thinking critically, links alot to Mastery Mathematics

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Critical thinking doesn't have to be a daunting prospect, says Judith Dancer; there are simple, effective and exciting ways to encourage children's mathematical investigation and exploration...
Maths is a subject many adults lack confidence in. Having struggled with it at school they often avoid it, wherever possible, when grown up. But if maths seems scary for some people, then problem solving in mathematics can cause even more anxiety. There is no 'safety net' of knowing the 'correct answer' beforehand as problem solving lends itself to investigation and exploration with lots of possible tangents. Understandably this is often the area of maths where many practitioners feel least confident, and where young children, who are not restrained by right answers, feel the most enthused and animated.

The non-statutory Development Matters Guidance, as part of 'creating and thinking critically' in the Characteristics of Effective Learning, identifies that practitioners need to observe how a child is learning, noting how a child is:

● thinking of ideas;

● finding ways to solve problems;

● finding new ways to do things;

● making links and noticing patterns in their experience;

● making predictions;

● testing their ideas;

● developing ideas of grouping, sequences, cause and effect;

● planning, making decisions about how to approach a task, solve a problem and reach a goal;

● checking how well their activities are going;

● changing strategy as needed;

● reviewing how well the approach worked.

All of these elements are, at one time or another, part of the problem identifying and solving process - although not at the same time and in the same problem.

Role of the adult

Problem solving in mathematics for young children involves them understanding and using two kinds of maths:

● Maths knowledge - learning and applying an aspect of maths such as counting, calculating or measuring.

● Maths thinking skills - reasoning, predicting, talking the problem through, making connections, generalising, identifying patterns and finding solutions.

The best maths problems for children are the ones that they identify themselves - they will be enthused, fascinated and more engaged in these 'real', meaningful problems. Children need opportunities to problem solve together; too, as they play, they will often find their own mathematical problems. One of the key roles of practitioners is to provide time, space and support for children - to develop situations and provide opportunities in which children can refine their problem-solving skills and apply their mathematical knowledge.

Effective practitioners can support children's developing problem-solving strategies through:

● Modelling maths talk and discussion - language is part of maths learning because talking problems through is vital. Children need to hear specific mathematical vocabulary in context. Practitioners can promote discussion through the use of comments, enabling statements and open- ended questions.

● Providing hands-on problem solving activities across all areas of the setting - children learn maths through all their experiences and need frequent opportunities to take part in creative and engaging experiences. Maths doesn't just happen in the maths learning zone!

● Identifying potential maths learning indoors and outdoors - providing rich and diverse open-ended resources that children can use in a number of different ways to support their own learning. It is important to include natural and everyday objects and items that have captured children's imaginations, including popular culture.

Problem solving possibilities

Spell it out

This experience gives children lots of opportunities to explore calculating, mark making, categorising and decisions about how to approach a task.
What you need to provide:
● Assorted containers filled with natural materials such as leaves, pebbles, gravel, conkers, twigs, shells, fir cones, mud, sand and some 'treasure' - sequins, gold nuggets, jewels and glitter.

● Bottles and jugs of water, large mixing bowls, cups, a 'cauldron', small bottles, spoons and ladles.

● Cloaks and wizard hats.

● Laminated 'spells' - e.g. "To make a disappearing spell, mix 2 smooth pebbles, 2 gold nuggets, 4 fir cones, a pinch of sparkle dust, 3 cups of water".

● Writing frameworks for children's own spell recipes, with sparkly marker pens and a shiny 'Spell Book' to stick these in and temporary mark-making opportunities such as chalk on slate.

The important thing with open-ended problem-solving experiences like this is to observe, wait and listen and then, if appropriate, join in as a co-player with children, following their play themes. So if children are mixing potions, note how children sort or categorise the objects, and the strategies they use to solve problems - what happens if they want eight pebbles and they run out? What do they do next?

When supporting children's problem solving, practitioners need to develop a wide range of strategies and 'dip into' these appropriately. Rather than asking questions, it is often more effective to make comments about what you can see - e.g. "Wow, it looks as though there is too much potion for that bottle". Acting as a co-player offers lots of opportunities to model mathematical behaviours - e.g. reading recipes for potions and spells out loud, focusing on the numbers - one feather, three shells...

Going, going, gone

We all know that children will engage more fully when involved in experiences that fascinate them - so if a particular group has a real passion for cars and trucks, consider introducing problem-solving opportunities that extend this interest.

This activity offers opportunities for classifying, sorting, counting, adding, subtracting, amongst many other things.
What you need to provide:
● Some unfamiliar trucks and cars and some old favourites - ensure these include metal, plastic and wooden vehicles that can be sorted in different ways.

● Masking tape and scissors.

● Sticky labels and markers.

Mark out some parking lots on a smooth floor, or huge piece of paper (lining paper is great for this), using masking tape. Line the vehicles up around the edge of the floor area. Encourage one child to select two vehicles that have something the same about them. Ask the child, "What is the same about them?" When the children have agreed what is the same - e.g. size, materials, colour, lorries or racing cars - the child selects a 'parking lot' to put the vehicles in. So this first parking lot could be for 'red vehicles'.

Another child chooses two more vehicles that have something the same - do they belong in the same 'parking lot', or a different parking lot? E.g. these vehicles could both be racing cars. What happens when a specific vehicle could belong in both lots? E.g. it could belong in the set of red vehicles and also belongs in the set of racing cars. Support the children as they discuss the vehicles, make new 'parking lots' with masking tape, and create labels for the groups, if they choose. It's really important to observe the strategies the children use - where appropriate, ask the children to explain what they are doing and why.

If necessary, introduce and model the use of the vocabulary 'the same as' and 'different from'. Follow children's discussions and interests - if they start talking about registration plates, consider making car number plates for all the wheeled toys outdoors, with the children. Do the children know the format of registration plates? Can you take photos of cars you can see in the local environment?

Camping out

Constructing camps and dens outdoors is a good way to give children the opportunity to be involved in lots of problem-solving experiences and construction skills learning. This experience offers opportunities for using the language of position, shape and space, and finding solutions to practical problems.
What you need to provide:
● Materials to construct a tent or den such as sheets, curtains, poles, clips, string.

● Rucksacks, water bottles, compass and maps.

● Oven shelf and bricks to build a campfire or barbecue.

● Buckets and bowls and water for washing up.

Encourage the children to explore the resources and decide which materials they need to build the camp, and suggest they source extra resources as they are needed. Talk with the children about the best place to make a den or erect a tent and barbecue. During the discussion, model the use of positional words and phrases. Follow children's play themes - this could include going on a scavenger hunt collecting stones, twigs and leaves and going back to the campsite to sort them out. Encourage children to try different solutions to the practical problems they identify, and use a running commentary on what is happening without providing the solution to the problem. Look for opportunities to develop children's mathematical reasoning skills by making comments such as, "I wonder why Rafit chose that box to go on the top of his den."

If the children are familiar with traditional tales, you could extend this activity by laying a crumb trail round the outdoor area for children to follow. Make sure that there is something exciting at the end of the trail - it could be a large dinosaur sitting in a puddle, or a bear in a 'cave'.

Pick it up

Children rarely have opportunities to investigate objects that are really heavy. Sometimes they have two objects and are asked the question, "Which one is heavy?" when both objects are actually light. This experience gives children the chance to explore really heavy things and explore measures (weight) as well as cooperating and finding new ways to do things.
What you need to provide:
● A 'building site' in the outdoor area - include hard hats, builders' buckets, small buckets, shovels, spades, water, sand, pebbles, gravel, guttering, building blocks, huge cardboard boxes and fabric (this could be on a tarpaulin).

● Some distance away, builders' buckets filled with damp sand and large gravel.

● Bucket balances and bathroom scales.

With an open-ended activity such as this, it is even more important to observe, wait and listen as the children explore the building site and the buckets full of sand and gravel. Listen to the discussions the children have about moving the sand and the gravel to the building site. What language do they use? Note the strategies they use when they can't lift the large buckets - who empties some of the sand into smaller buckets? Who works together collaboratively to move the full bucket? Does anyone introduce another strategy, for example, finding a wheelbarrow or pull-along truck?

Where and when appropriate, join in the children's play as a co-player. You could act in role as a customer or new builder: "How can I get all this sand into my car?"; "How much sand and gravel do we need to make the cement for the foundations?" Extend children's learning by modelling the language of weight: heavy, heavier than, heaviest, light, lighter than, lightest; about the same weight as; as heavy as; balance; weigh.
Judith Dancer is an author, consultant and trainer specialising in communication and language and mathematics. She is co-author, with Carole Skinner, of Foundations of Mathematics - An active approach to number, shape and measures in the Early Years.

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Problem-solving Activities for Early Years Settings | Learning and Development | Teach Early Years

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Critical thinking doesn't have to be a daunting prospect, says Judith Dancer; there are simple, effective and exciting ways to encourage children's mathematical investigation and exploration... Maths is a subject many adults lack confidence in. Having struggled with it at school they often avoid it, wherever possible, when grown up.

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This document covers all three aspects of mastery and how there role comes into play in the areas of the early years curriculum.  It also talks about the CPA approach too.

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Fluency in maths - means that pupils can memorise more than one procedure to work out something. For example, 2+1=3 0+3=3 etc.

http://gargrave.n-yorks.sch.uk/data/documents/EYFS-fluency_1.pdf

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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

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Mathematical fluency is a fairly broad concept. The basics – 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

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Fluency Definition: What is Fluency in Maths?

One of the biggest changes in practice was a new-found focus on fluency. So what does this really mean?

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

Fluency gives pupils the ability to delve deeper into Maths; to develop number sense and choose the most appropriate method for the task at hand; to be able to apply a skill to multiple contexts.

https://thirdspacelearning.com/blog/what-is-fluency-in-maths-definition-ks2/

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How Does Mathematical Fluency Fit with Reasoning and Problem Solving?

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

To enable them to achieve this, pupils must develop their conceptual understanding, mathematical thinking and use of mathematical language. This is where fluency and reasoning come in. To find out more, read our blog on the relationship between fluency, reasoning and problem solving.

https://thirdspacelearning.com/blog/what-is-fluency-in-maths-definition-ks2/

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The children need to be able to explain how they have reached their answer. E.g. through them working out (by drawring dots).
https://www.barrscourtprimaryschool.co.uk/wp-content/uploads/2015/12/EYFS-reasoning.pdf

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So, personally, I disagree, as I don't think this is taking into account children with ASD, SLCN, EAL, shyness or low confidence... but what do I know?

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Mastery Moments - video shorts: 1. Answers in complete sentences

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It's important for pupils in maths lessons to give answers to questions using full sentences--not just one-word answers. Watch this 2 minute video to find out why.

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Reasoning being the bridge between fluency and problem solving...

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Reasoning being the bridge between fluency and problem solving...

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So, reasoning is said to be the bridge between fluency and problem solving, but isn't fluency learning by rote and then the reasoning is applying the rote learning to solve the problems? Feel free to disagree!

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Reasoning

The way pupils speak and write about mathematics transforms their learning. Mastery approaches use a carefully sequenced, structured approach to introduce and reinforce mathematical vocabulary. Pupils explain the mathematics in full sentences. They should be able to say not just what the answer is, but how they know it’s right. This is key to building mathematical language and reasoning skills.

https://www.tes.com/teaching-resources/teaching-for-mastery-in-primary-maths