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Developing Problem-solving Skills

Trial and Improvement at KS1

These lower primary tasks could all be tackled using a trial and improvement approach.

Trial and Improvement at KS2

These upper primary tasks could all be tackled using a trial and improvement approach.

Working Systematically at KS1

Tasks for KS1 children which focus on working systematically.

Working Systematically at KS2

Tasks for KS2 children which focus on working systematically.

Working Backwards at KS1

The lower primary tasks in this collection could each be solved by working backwards.

Working Backwards at KS2

The upper primary tasks in this collection could each be solved by working backwards.

Visualising at KS1

These lower primary tasks all specifically draw on the use of visualising.

Visualising at KS2

These upper primary tasks all specifically draw on the use of visualising.

Reasoning and Convincing at KS1

The tasks in this collection can be used to encourage children to convince others of their reasoning, using 'because' statements.

Reasoning and Convincing at KS2

The tasks in this collection can be used to encourage children to convince others of their reasoning, by first convincing themselves, then a friend, then a 'sceptic'.

Conjecturing and Generalising at KS1

The tasks in this collection encourage lower primary children to conjecture and generalise.

Conjecturing and Generalising at KS2

The tasks in this collection encourage upper primary children to conjecture and generalise.

South Australia

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Nrich - a website with maths activities and games

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Resources to develop mathematical reasoning and problem solving. Nrich aims to enrich the mathematical experiences of all learners and embeds rich mathematical tasks into everyday classroom practice.

Structure and features

Activities and games can be explored independently or together with families.

The website covers topic such as:

• measurement
• problem solving

The 'Thinking Mathematically' sections provide additional educational activities.

Links and files

Teacher notes.

The resources are grouped into levels. Level 1 aligns with early years, while level 4 aligns with lower secondary. Teachers can use the search functionality to find tasks that match the topic that they are teaching.

The tasks are not mapped to the Australian Curriculum but align well with the mathematical concepts.

Page last updated: 12 Oct 2022

learningathome [at] sa.gov.au

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• NRich Maths Problem Solving Day
• News & Events
• Latest News

We were very lucky to have a visit from the NRich Problem Solving Roadshow on Thursday 2nd March.  NRich, which promotes problems solving, reasoning and deeper thinking in mathematics, is based at Cambridge University.

All of the classes got a chance to work on some fiendishly difficult mathematical problems.  Have a look at the pictures section to see more from the day.

Unfortunately not the ones with chocolate chips.

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Problem Solving and Reasoning

• Key Information

Within the new maths curriculum, there is now a strong emphasis on providing children with opportunities to use and apply their skills in a range of problems, investigations and puzzles. Maths is not just about knowing how to add, subtract, multiply and divide, it is more about using these skills out of context to solve a range of different problems. Through our maths lessons, we are providing children with the opportunity not just to talk about maths but to talk maths (Mike Askew, Transforming Primary Maths). These are two quite different skills; children can talk about what they are doing in maths but talking about, 'why, why not and what if' is something else.  At St. Christopher's, we use a range of resources to support the children in developing their use of maths talk:

• White Rose Hub Materials (new),
• NCETM Mastery Materials (new),
• Rising Stars: Problem Solving and Reasoning,
• Brain Buster,
• Brain Snack,
• Resources from NCETM,
• Resources from NRICH,
• Learning Wales and
• Resources from guidedreasoning.co.uk.

If you would like to have a go at applying your skills, why not visit the NRICH website and have a go at some of the challenges. WARNING: Some of them are tough so persevere!

There are may different ways to reason about something in maths but below is an example of what reasoning looks like in St Christopher's. These strategies are used to promote deeper understanding and application of a skill in all domains of the maths curriculum. Below, you will also find an example of the strategies in each year group. This is just a sample of what is available at our school. These challenges can be found within the daily maths lesson but also on our working walls. 

Reasoning Strategies Explained...

• reasoning strategies.docx

Year Group Reasoning Strategies

• reasoning strategies Y2.docx
• reasoning strategies Y3.docx
• reasoning strategies Y4.docx
• reasoning strategies Y5.docx
• reasoning strategies Y6.docx

Unfortunately not the ones with chocolate chips.

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Website CMS

A 'sessionid' token is required for logging in to the website and a 'crfstoken' token is used to prevent cross site request forgery. An 'alertDismissed' token is used to prevent certain alerts from re-appearing if they have been dismissed. An 'awsUploads' object is used to facilitate file uploads.

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The Joint Mathematical Council of the United Kingdom

Addressing the five ‘big questions’ in problem-solving with NRICH

The importance of ensuring learners acquire the problem-solving skills which will enable them to thrive both socially and economically in their increasingly automated world is widely recognised (Luckin et al., 2017). Nevertheless, government inspectors have reported serious concerns about the quality and quantity of problem-solving in our schools (Ofsted, 2015). This summer schools were challenged to reflect on ‘Five big questions for problem-solving’ (EEF, 2021). In this blog, we will consider each of those five questions and explore the ways that the NRICH team is supporting schools to address them.

Question one: Do teachers in your school select genuine problem-solving tasks for which pupils do not already have a ready-made method available?

Too often, learners are presented with routine word problems which merely require the application of a known algorithm. ‘Genuine’ problems enable them to make their own problem-solving decisions by choosing their own strategies and enabling them to compare their approach with those of other learners, thus developing their problem-solving efficiency and flexibility. At NRICH , our award-winning activities allow learners to develop these key skills alongside the confidence to tackle genuine problems. Moreover, our ‘ low threshold, high ceiling ‘ approach enables everyone to get started on the problem while ensuring a suitable level of challenge too, making them ideal for whole-class teaching.

Question two: Are pupils given the opportunity to see – through multiple worked examples – to use, and to compare different approaches to solving a problem?

Many problems can be explored in more than one way. Working flexibly, making connections between different areas of the curriculum and reflecting on various problem-solving approaches are key steps towards becoming a more fluent mathematician. NRICH encourages learners to develop these skills in these two ways:

Our primary , secondary and post-16 Live Problems invite learners to explore and submit their ideas to the team. We review each submission that we receive and publish a selection on our website showcasing different approaches and the reasoning behind them.

Our NRIC H online activities sometimes feature ‘hide and reveal’ buttons showcasing different starting points towards a solution for learners to explore further for themselves. This approach enables learners to widen their range of strategies for solving unfamiliar problems and develop alternative approaches to explore when they get stuck using their first-choice strategy.

Question three: Are pupils encouraged to use visual representations to support them to solve a problem?

One of the most important approaches towards solving an unfamiliar problem is drawing a good diagram. Learning to draw diagrams is a skill which we encourage learners of all ages to develop alongside their other mathematical skills and knowledge. From sketching graphs to drawing a bar model, good diagrams can help learners clarify their understanding and identify possible ways forward.

Our four steps towards problem-solving feature highlights the importance of drawing a diagram to enable young learners to get started on a problem. We often highlight a useful diagram, table or sketch graph in the solutions chosen for publication. As learners progress through their learning, the team model more specific drawing skills, such as sketching a graph to help solve a STEP problem.

Question four: Are pupils supported to monitor, reflect on, and communicate their reasoning and choice of strategies, possibly through the use of prompt questions?

NRICH  encourages learners to reflect on their learning using this approach inspired by the Strands of Mathematical Proficiency model introduced by Kilpatrick et al. (2001).

Our approach uses child-friendly language that teachers and parents can share with students five key ingredients that characterise successful mathematicians. At NRICH , we believe that learning mathematics is about much more than just learning topics and routines. Successful mathematicians understand the curriculum content and are fluent in mathematical skills and procedures, but they can also solve problems, explain their thinking and have a positive attitude about themselves as learners of mathematics.

With this in mind, we have created  this self assessment tool  to help learners recognise where their mathematical strengths and weaknesses lie. We hope learners will explore NRICH activities and then take time to reflect on their own mathematical capabilities using our model.

Question 5: Is professional development time allocated to develop teachers’ pedagogical understanding of problem-solving, with particular support for early career teachers?

NRICH supports teachers to maximise the potential of our activities by offering free, regular professional development for teachers .  Each session is delivered online, enabling teachers to access the support wherever they are based, reducing teacher travel and cover costs for schools. We also record the sessions and upload them to our website so that schools can access them for future professional development days or staff/department meetings in their settings.

The live sessions are led by NRICH team members and they link directly to our latest primary , secondary and post-16 Live Problems. This approach enables teachers to consider the possibilities of the activities with the NRICH team before exploring them the next day with their own classes. Later, they are invited to share their classwork with our team for possible publication on the NRICH website.

The five ‘big questions’ provide excellent starting points for evaluating the teaching and learning of problem-solving in different settings. I hope that this blog shares an insight into the different ways that NRICH can support schools to address the five questions for themselves by engaging with our activities, Live Problems and teacher webinars.

Dr Ems Lord FCCT

Director of NRICH

Centre for Mathematical Sciences

University of Cambridge

Selected references

EEF. (2021). EEF Blog: Integrating evidence into maths teaching – guiding problem-solving. Accessed from https://educationendowmentfoundation.org.uk/news/eef-blog-integrating-evidence-into-mathematics-guiding-problem-solving /

Kilpatrick, J. Swafford, J., & Findell, B. (2001). Adding it up: Helping children learn mathematics (Vol. 2101). J. Kilpatrick, & National research council (Eds.). Washington, DC: National Academy Press.

Luckin, R., Baines, E., Cukurova, M., Holmes, W., & Mann, M. (2017). Solved! Making the case for collaborative problem-solving. Accessed from http://oro.open.ac.uk/50105/1/solved-making-case-collaborative-problem-solving.pdf

Ofsted. (2015). Better Maths Conference Spring Keynote 2015. Accessed here https://www.slideshare.net/Ofstednews/better-mathematics-keynote-spring-2015

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Learning Path

Exploring problem solving with 5th and 6th class learning path.

This learning path gives a brief overview of how to develop a classroom culture of sharing knowledge, valuing mistakes and providing cognitively challenging tasks. Problem solving strategies can be explicitly taught using this learning path with an introduction to the concept of Low Threshold High Ceiling Tasks also. 11 resources in this Learning Path

Developing Maths problem solving

a pre assessment to see how children are able to respond and complete problem

How it maps to the curriculum

Strand: Number

Strand unit: Operation: Addition

Suggestions for use: Assessment of student learning

Same But Different

The images on this website can be used for developing classroom discussion, critical thinking and collaboration. Key components for problem solving.

Strand: Useful Websites

Suggestions for use: This can website can be used by teachers as a mini problem-solving lessons (10 mins approx.) to develop a classroom climate conducive to problem-solving. Teachers should encourage all pupils to ‘have a go’ and ‘value all contributions’. Promoting higher order skills of reasoning and discussion. Success is based on effort and skills rather than answers.

Low Threshold High Ceiling

A low threshold high ceiling task is one which is designed to be mathematically accessible, and to have built-in extension opportunities. In other words, everyone can get started and everyone can get stuck. In this updated feature, NRich brings together their favourite low threshold high ceiling tasks, as well as two articles which will support you in creating a low threshold high ceiling classroom.

Suggestions for use: Developing a positive classroom climate in maths through mathematical discussion and collaboration

Problem Solving with 5th and 6th Class - Sandwiches (Trial and Improvement)

The problem is particularly valuable as it gives students an opportunity to work on a proof to explain why something is impossible.

Strand: Algebra

Suggestions for use: Using number to predict, generalise and verify

Problem Solving with 5th and 6th Class - Tea Cups (Working Systematically)

The problem lends itself for small group work, so that the learners have an opportunity to decide on approaches. Learners can collaborate, share and discuss the different solutions, and each method's strengths and weaknesses.

Suggestions for use: Allowing learners to read the accompanying story with the problem can support student’s comprehension and decision making about which pieces of information are relevant. Development of higher order maths skills: Applying and problem-solving, Communicating and expressing, Integrating and connecting, Reasoning

Problem Solving with 5th and 6th Class - Tables without tens (Pattern spotting)

This problem provides an interesting way of revising multiplication tables. It is also very useful for getting learners to predict what they think they will find out and spot pattern between times tables.

Strand unit: Number Theory

Content objective: This resource should enable a child to:

• identify common factors and multiples
• identify factors and multiples
• identify simple prime and composite numbers

Suggestions for use: times tables

Problem Solving with 5th and 6th Class - Counting Cards (Working Backwards)

While this problem provides the “trick” element to it, it is firmly rooted in mathematical concepts and problem solving strategies.

Problem Solving with 5th and 6th Class - Baravelle (Visualising)

The aim of this problem is to encourage discussion about the different ways of seeing, and to pose questions that can form the focus of further investigation.

Strand: Shape & Space

Strand unit: 2-D Shapes

• make informal deductions about 2-D shapes and their properties
• plot simple co-ordinates and apply where appropriate
• tessellate combinations of 2-D shapes
• use 2-D shapes and properties to solve problems

Suggestions for use: Allows for creative pattern design. A good follow up problem can be found here https://nrich.maths.org/2132/index (Inside 7 squares)

Problem Solving with 5th and 6th Class - Planning a school trip (Reasoning)

This problem will encourage learners to organise information, identify redundant information and to check their work.

Suggestions for use: The activity lends itself to collaborative working, both for children who are inexperienced at working in a group and children who are used to working in this way. By working together on this problem, the task is shared and therefore becomes more manageable than if working alone. A number of cross curricular links can be used to extend this lesson.

Problem Solving with 5th and 6th Class - Magic V's (Conjecture)

Task card involving placing numbers 1 to 5 in the V shape so that the two arms of the V have the same total. Short task.

Strand unit: Operations: Addition & Subtraction

Suggestions for use: Print and display on maths station or use separately in a teacher led problem solving activity. Magic V gives opportunities for children to make conjectures, prove these conjectures and make generalisations. They will be practising addition and subtraction, and applying their knowledge of odd/even numbers. Supporting video clip on the problem can be found at this link https://www.youtube.com/watch?v=-JrZcMbsNdA

Strand unit: Operations

• add and subtract whole numbers and decimals (to three decimal places) without and with a calculator

Problem Solving with 5th and 6th Class - Money Bags (Conjecture)

This problem is a good example of a challenge which does not require high-level mathematics, but does need a systematic approach. It also lends itself to a focus on different ways of recording and learners can discuss on the merits of the different ways of recording findings.

Strand: Measures

Strand unit: Money - Euro

• compare 'value for money' using unitary method
• explore value for money

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‘AI means maths problem-solving skills are more important than ever’

Cambridge bolsters classroom learning with new 'Problem-Solving Schools' initiative

By Stephen Bevan Published: 16th November 2023

Credit: Phil Boorman

Mathematicians at the University of Cambridge are supporting UK schools to help prioritise problem solving in maths – a key skill that is likely to become ever more critical with the rise of automation and artificial intelligence.

The new Problem-Solving Schools initiative, developed by the University’s Faculty of Mathematics, aims to create ‘a movement of problem-solving schools’ by providing free learning resources and teacher training to refocus attention on the skill.  Along with fluency and reasoning, problem solving has been central to the National Curriculum for maths since it was introduced in 2014, but often does not receive the same amount of attention in the classroom.

In the summer, Ofsted published new guidance encouraging schools to focus more consistently on teaching problem solving, and emphasised the importance of teaching skills that “equip [pupils] for the next stage of education, work and life”.

Dr Ems Lord, Director of NRICH , which provides thousands of free online mathematics resources for ages three to 18, and is launching Problem-Solving Schools, said: “It's fair to say that many schools feel increasingly confident supporting fluency and reasoning skills, and there’s a lot of support out there. What’s been missing is the problem-solving aspect, and that’s been repeatedly picked up by Ofsted. It’s not being prioritised, often because of a lack of training for teachers and a lack of access to sufficient, high-quality resources to support it.

Dr Ems Lord at the University's Maths Faculty. Credit: Nathan Pitt

“Some schools are not covering it as well as others, so it means we’re in this very patchy landscape and at the same time we have AI coming in, with everyone thinking about how that will impact future roles and careers. And it’s looking increasingly likely that students who are good problem solvers, and have good teamwork skills, are the ones who are going to thrive.”

Although AI is developing rapidly, Dr Lord says at present problem solving isn’t one of its strong points. And business analysts believe that in the future jobs which computers cannot perform ­– that require uniquely human skills such as critical thinking ­– will become more significant and those with these skills will be in even more demand.

“I can put our problems into an AI system, some it can solve, some it gives ridiculous answers to. But how would someone know which is which unless they know how to solve the problem themselves – or even know what question to ask to get the answer they’re after?

“Problem-solving is not about memorising facts, it’s about being confronted with something for the first time and thinking, ‘Right, how do I use my skills to approach this?’ And these are transferrable skills, for all aspects of life, which will help children in the future, not just at work but also socially. We want our young people to have the curiosity and confidence to question things, so if they come across some data or a graph in the media, or wherever, they have the experience and skills to know what a good graph looks like, and they can analyse it for themselves.

“It’s such an important area that we have to get right, and at the moment we’re not doing it. The whole point of learning maths is to be able to solve problems.”

Dr Lord says the Problem-Solving Schools initiative aims to help embed the skill in classrooms by providing themed resources and webinar training on how to best use them – to support teachers who might be lacking in confidence themselves, or are unsure how to refocus how they teach the Curriculum.

The webinar series will also include tips on engaging parents with maths so they can help support their children in the subject. In a recent study , NRICH’s Solving Together project, which offers family-friendly homework activities, was found to significantly increase parental involvement in the subject.

'Problem-solving is not about memorising facts, it’s about being confronted with something for the first time and thinking, ‘Right, how do I use my skills to approach this?'

- Dr Ems Lord, Director of NRICH

Pupils using NRICH maths resources. Credit: University of Cambridge

In addition, a Charter for schools to sign up to is also being introduced. It puts problem solving at the heart of maths learning, from the commitment of the school’s leadership team, to values in the classroom – where good problem-solving behaviour is encouraged, and where it’s ok to make mistakes – to how activities can be widened out to the local community.

The NRICH team has developed the programme in consultation with schools, and has actively sought the views of colleagues in the Department for Education, and the National Centre for Excellence in the Teaching of Mathematics – the Government’s maths body set up to improve mathematics teaching in England.

“Many of the resources given to teachers up to this point have focused on fluency, and if a teacher isn’t mathematically trained they tend to revert to where they feel safe, how they were taught,” says Dr Lord. “We need to break the mould on that, we need to make sure there are good resources available for problem-solving learning, and free training, so it isn’t a case of ‘we should be doing this’, but, ‘why wouldn’t we be doing this?’

“We’ve created a complete, wraparound package. We’re looking for schools across the country to sign up to the Charter, create a movement of problem-solving schools and change the agenda.”

Professor Bhaskar Vira, Pro-Vice-Chancellor for Education at the University of Cambridge, said: “Problem-Solving Schools is an exciting initiative that builds on the University’s work to support schools around the country through outreach and learning. NRICH’s high quality resources will help maths teachers embed problem solving in the classroom, as part of Cambridge’s mission to contribute to society through education, learning and research, and equip pupils with this key skill for the future.”

As part of the Problem-Solving Schools launch, NRICH is developing its resources, which have been supporting learners since the outreach programme’s launch 25 years ago , and recently made a huge contribution to the national effort during the COVID-19 lockdowns. Between March and September 2020, nrich.maths.org registered a 95% increase in UK visits compared to the previous year. In the 2020–21 school year alone, the site attracted just under 33 million page views. In spring 2020, the UK Government highlighted NRICH resources to schools and the team contributed to the BBC’s heavily used Bitesize maths resources.

And as the team launches its newest initiative, it continues to support post-pandemic catch-up work, by helping fill gaps in knowledge and focusing on students’ attitude to maths.

“It’s not just about doing the maths, it’s about enjoying it and finding it worthwhile – understanding the applications,” says Dr Lord. “If our materials are just about covering subject knowledge it’s really hard for student to enjoy what they’re doing.

“It’s a bit like having never seen Messi score a goal. If all you’ve done is go to football practice, where the coach puts down markers and tells you to dribble through them for an hour, and you come back the next week and do exactly the same thing, you kind of wonder why you’re doing it.

“But if you go to football practice and then switch on the TV and see a Messi wonder goal – it’s like ‘Aah – that’s what it’s all about!’ And I sometimes think that’s what’s missing when we talk about maths – the sheer moments of awe and wonder that you can have, and that feeling when you solve a problem which is absolutely fantastic!”

Credit: University of Cambridge

The text in this work is licensed under a Creative Commons Attribution 4.0 International License .