nrich problem solving questions

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

Rich Problems – Part 1

Rich problems – part 1, by marvin cohen and karen rothschild.

One of the underlying beliefs that guides Math for All is that in order to learn mathematics well, students must engage with rich problems. Rich problems allow ALL students, with a variety of neurodevelopmental strengths and challenges, to engage in mathematical reasoning and become flexible and creative thinkers about mathematical ideas. In this Math for All Updates, we review what rich problems are, why they are important, and where to find some ready to use. In a later Math for All Updates we will discuss how to create your own rich problems customized for your curriculum.

What are Rich Problems?

At Math for All, we believe that all rich problems provide:

• opportunities to engage the problem solver in thinking about mathematical ideas in a variety of non-routine ways.
• an appropriate level of productive struggle.
• an opportunity for students to communicate their thinking about mathematical ideas.

Rich problems increase both the problem solver’s reasoning skills and the depth of their mathematical understanding. Rich problems are rich because they are not amenable to the application of a known algorithm, but require non-routine use of the student’s knowledge, skills, and ingenuity. They usually offer multiple entry pathways and methods of representation. This provides students with diverse abilities and challenges the opportunity to create solution strategies that leverage their particular strengths.

Rich problems usually have one or more of the following characteristics:

• Several correct answers. For example, “Find four numbers whose sum is 20.”
• A single answer but with many pathways to a solution. For example, “There are 10 animals in the barnyard, some chickens, some pigs. Altogether there are 24 legs. How many of the animals are chickens and how many are pigs?”
• A level of complexity that may require an entire class period or more to solve.
• An opportunity to look for patterns and make connections to previous problems, other students’ strategies, and other areas of mathematics. For example, see the staircase problem below.
• A “low floor and high ceiling,” meaning both that all your students will be able to engage with the mathematics of the problem in some way, and that the problem has sufficient complexity to challenge all your students. NRICH summarizes this approach as “everyone can get started, and everyone can get stuck” (2013). For example, a problem could have a variety of questions related to the following sequence, such as: How many squares are in the next staircase? How many in the 20th staircase? What is the rule for finding the number of squares in any staircase?

• An expectation that the student be able to communicate their ideas and defend their approach.
• An opportunity for students to choose from a range of tools and strategies to solve the problem based on their own neurodevelopmental strengths.
• An opportunity to learn some new mathematics (a mathematical residue) through working on the problem.
• An opportunity to practice routine skills in the service of engaging with a complex problem.
• An opportunity for a teacher to deepen their understanding of their students as learners and to build new lessons based on what students know, their developmental level, and their neurodevelopmental strengths and challenges.

Why Rich Problems?

All adults need mathematical understanding to solve problems in their daily lives. Most adults use calculators and computers to perform routine computation beyond what they can do mentally. They must, however, understand enough mathematics to know what to enter into the machines and how to evaluate what comes out. Our personal financial situations are deeply affected by our understanding of pricing schemes for the things we buy, the mortgages we hold, and fees we pay. As citizens, understanding mathematics can help us evaluate government policies, understand political polls, and make decisions. Building and designing our homes, and scaling up recipes for crowds also require math. Now especially, mathematical understanding is crucial for making sense of policies related to the pandemic. Decisions about shutdowns, medical treatments, and vaccines are all grounded in mathematics. For all these reasons, it is important students develop their capacities to reason about mathematics. Research has demonstrated that experience with rich problems improves children’s mathematical reasoning (Hattie, Fisher, & Frey, 2017).

Where to Find Rich Problems

Several types of rich problems are available online, ready to use or adapt. The sites below are some of many places where rich problems can be found:

• Which One Doesn’t Belong – These problems consist of squares divided into 4 quadrants with numbers, shapes, or graphs. In every problem there is at least one way that each of the quadrants “doesn’t belong.” Thus, any quadrant can be argued to be different from the others.
• “Open Middle” Problems – These are problems with a single answer but with many ways to reach the answer. They are organized by both topic and grade level.
• NRICH Maths – This is a multifaceted site from the University of Cambridge in Great Britain. It has both articles and ready-made problems. The site includes  problems for grades 1–5 (scroll down to the “Collections” section) and problems for younger children . We encourage you to explore NRICH more fully as well. There are many informative articles and discussions on the site.
• Rich tasks from Virginia – These are tasks published by the Virginia Department of education. They come with complete lesson plans as well as example anticipated student responses.
• Rich tasks from Georgia – This site contains a complete framework of tasks designed to address all standards at all grades. They include 3-Act Tasks , YouCubed Tasks , and many other tasks that are open ended or feature an open middle approach.

The problems can be used “as is” or adapted to the specific neurodevelopmental strengths and challenges of your students. Carefully adapted, they can engage ALL your students in thinking about mathematical ideas in a variety of ways, thereby not only increasing their skills but also their abilities to think flexibly and deeply.

Hattie, J., Fisher, D., & Frey, N. (2017). Visible learning for mathematics, grades K-12: What works best to optimize student learning. Thousand Oaks, CA: Corwin Mathematics.

NRICH Team. (2013). Low Threshold High Ceiling – an Introduction . Cambridge University, United Kingdom: NRICH Maths.

The contents of this blog post were developed under a grant from the Department of Education. However, those contents do not necessarily represent the policy of the Department of Education, and you should not assume endorsement by the Federal Government.

Math for All is a professional development program that brings general and special education teachers together to enhance their skills in planning and adapting mathematics lessons to ensure that all students achieve high-quality learning outcomes in mathematics.

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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 .

Mr Barton Maths Podcast

Long-form conversations about teaching and learning with craig barton, tes top 10 resources: problem solving and rich tasks.

The following collection of resources have been assembled by the TES Maths Panel . They can be downloaded for free by registering on the TES website.

Here, I have compiled my top 10 problem solving and rich tasks. Enjoy there are some crackers in here.

Carpet your bedroom – functional maths

Age Range:  11-16  Format:  PPT

An activity for students to design a tessellation of carpet tiles for their bedroom. From the design, they need to work out costs from screenshots of carpet tile prices, calculate the cost of the glue. There are three levels to this task, from a basic bronze version to the gold version which includes an algebra extension related to cost of getting a carpet fitted. Very real world focus and some good discussion if you let the students use their own bedroom.

Rich maths tasks to engage KS3

Age Range:  11-16  Format :  Webpage

This website includes many helpful resources relating to the pedagogy associated with Rich Tasks as well as many sources for further tasks. Each Rich Task has been trialled in the classroom and most are accompanied by a detailed task plan which contains an overview and lesson plan.

Maths investigations

Age Range:  11-16  Format:  PDF

Based on science tasks produced in East Sussex, these investigation starter sheets give an introduction to a topic for investigation and progression guide to allow self- and peer-assessment. Targeted at year 7, these may be suitable for several year groups in KS2/3. Please note that the levelling guide is a first draft and may change.

Maths mystery activity: Thinking skills

This is an ideal thinking skills activity for higher KS3 and KS4 students. This is a mystery activity, where learners are presented with a set of cards containing clues to answer a central focus question, in this case: ‘Does Amelie make it to the catwalk?’. Learners will need to work together (in pairs or threes) and use various strands of mathematics to be able to make sense of the clues and come to a conclusion.

My bedroom: Lesson plans and activities 

This is a complete pack of activities, interactive whiteboards and lesson plans for KS3. This activity can be done in pairs or individually. It has lots of differentiation opportunities from the various materials I have created to provide help in groups, individuals or to the whole class, depending on ability. In addition to this, there are also two good extension activities.

Functional maths skills and enterprise task

Age Range:  11-16  Format:  Excel

This is a business game where players compete to make as much money as possible from an initial investment of £200. All pupils need to do is decide how much to spend on making their product, what the sale price should be and the amount to spend on advertising. When every group has entered their decisions into the spreadsheet the figures are compared and sales are decided. Pupils can then predict their available capital for the next round to earn a cash bonus. It was used across the department with all ages and abilities.

Mathematical Rich Tasks

I created this website with a BECTA grant. There are resources (written, interactive, audio and video) to support teachers and students accessing six rich mathematical tasks, that are ripe for creative exploration!

Holiday planning maths investigation

Age Range:  11-16  Format:  Doc & PDF

A structured problem-solving activity based on booking a holiday for a family of four. Includes a resource booklet containing flight, accommodation, car hire and passport prices, and a task sheet setting students task. Suitable for able KS2 pupils. Also includes differentiation by increasing level of difficulty in the tasks set.

Nrich  differentiated mathematical problem solving

Age Range:  11-18  Format:  WEB

Focussed mathematical problem solving on all the required topics of any mathematical syllabus. A good resource to include in the gifted and talented policy and something that allows the children to access mathematics with real-life problem solving, taking them away from the more difficult abstract questions.

Functional mathematics lessons – projects

Age Range:  11-16  Format:  WEB

Maths functional skills classroom projects to use in lessons. Projects designed for one at the end of each half term for years 7, 8 and 9.

Damian Watson,  Maths Secondary Panel

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Become a Problem-Solving School

Problem-solving Schools is an exciting new initiative to help schools raise the profile of mathematical problem-solving and nurture better problem-solvers.

At the heart of our initiative is our Charter , which offers a framework to help you reflect on your current practice and to agree on areas for development. The Charter highlights five key areas:

• Vision and ethos
• Leadership and PD
• Curriculum, pedagogy and assessment
• Classroom culture
• Problem-solving beyond the classroom.

We invite all schools who are committed to nurturing confident mathematical problem-solvers to apply to become Problem-solving Schools.

Registering your school

During the registration process, you’ll be invited to provide:

• key details about your school
• the results of an initial review of your provision, using the Charter as your guide
• details of three agreed areas for development.

Start registration

What happens next?

After your registration is complete, your school will receive its Welcome Pack and a copy of your initial reflections and action plan.

Your Welcome Pack will include information about the support material and webinars available to you, to help your school move forward on its problem-solving journey. You will also receive a Problem-solving Schools badge to display on letterheads and webpages, highlighting your school’s commitment to nurturing problem-solvers.

Problem-solving Schools status is renewable annually, so we’ll get back in touch during the summer term to support schools to reflect on their progress as Problem-solving Schools.

If you have any questions or comments about the Problem-solving Schools initiative, please email [email protected]

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Learning Path

Maths week pdst problem solving - junior and senior infants learning path.

Click on the Title to open the problem...Upload pictures or answers to the Padlet to enter the Maths Week competition for PDST Numeracy Manuals and a class set of Active Learning Boards. 5 resources in this Learning Path

Incey Wincey Spider

An interactive game encouraging students to move along tracks - forward & back. Nice introduction to negative numbers. Resource can be used on the interactive whiteboard (Flash required) or worksheet downloaded. Teacher guide provided within the link. Dice needed.

How it maps to the curriculum

Strand: Number

Strand unit: Comparing & Ordering

Content objective: This resource should enable a child to:

• use the language of ordinal number: first, last
• use the language of ordinal number: first, second, third, last

Suggestions for use: Use as part of Maths Week Problem Solving activity. Submit problem solving pictures to the Scoilnet-PDST Padlet

Strand unit: Analysis of Number: Numeration

Three Block Towers - Teacher Guide to Activity

Nrich problem solving activity. Link explains the activity, provides worksheets for 3 blocks and 4 blocks challenge and a poster for the problem solving challenge.

Strand: Early Maths Activities

Strand unit: Classifying

Suggestions for use: Problem for Maths Week. Take pictures of the problem solving process and submit the PDST - Scoilnet Padlet.

Strand: Algebra

Lollipop Stick Shapes - PDST

A great problem solving challenge from the Shape and Space Manual, page 72. Pupils are given lollipop sticks to create as many 2D shapes as they can.

Strand: Shape & Space

Strand unit: 2-D Shapes

• combine and divide 2-D shapes to make larger or smaller shapes

Worms (Length) - PDST

A great problem solving challenge from the Measures Manual, page 37. Pupils are given play dough to make a worm. Pupils describe their worm with keywords, long, short, longer than, shorter than..etc

Strand: Measures

Strand unit: Length

Suggestions for use: Maths week problem solving challenge. Submit pictures of the problem solving challenge to the PDST-Scoilnet Padlet.

Noah - Nrich

A problem solving question from Nrich website for the PDST Numeracy - Maths Week Challenge. A very good problem to use when practicing counting.

Suggestions for use: Maths week problem solving challenge. Submit pictures of problem solving techniques to the PDST-Scoilnet Padlet.

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