problem solving strategies nrich

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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6 Strategies for Increasing Critical Thinking with Problem Solving

By Mary Montero

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For many teachers, problem-solving feels synonymous with word problems, but it is so much more. That’s why I’m sharing my absolute favorite lessons and strategies for increasing critical thinking through problem solving below. You’ll learn six strategies for increasing critical thinking through mathematical word problems, the importance of incorporating error analysis into your weekly routines,  and several resources I use for improving critical thinking – almost all of which are free! I’ll also briefly touch on teaching students to dissect word problems in a way that enables them to truly understand what steps to take to solve the problem.

This post is based on my short and sweet (and FREE!) Increasing Critical Thinking with problem Solving math mini-course . When you enroll in the free course you’ll get access to everything you need to get started:

• Problem Solving Essentials
• Six lessons to implement into your classroom
• How to Implement Error Analysis
• FREE Error Analysis Starter Kit
• FREE Mathematician Posters
• FREE Multi-Step Problem Solving Starter Kit
• FREE Task Card Starter Kit

Introduction to Critical Thinking and Problem Solving

According to the National Council of Teachers of Mathematics, “The term “problem solving” refers to mathematical tasks that have the potential to provide intellectual challenges for enhancing students’ mathematical understanding and development .)”

That’s a lot of words, but I’d like to focus in on the word POTENTIAL. I’m going to share with you strategies that move these tasks from having the potential to provide a challenge to actually providing that challenge that will enrich their mathematical understanding and development. 

If you’re looking for an introduction to multi-step problem solving, I have a free multi-step problem solving starter kit for that! 

I also highly encourage you to download and use my free Mathematician posters that help students see what their “jobs” are as mathematicians. Giving students this title of mathematician not only holds them accountable, but it gives them greater confidence and gives me very specific verbiage to use when discussing math with my students. 

The impacts of Incorporating Problem Solving

When I made the shift to incorporate problem solving into my everyday instruction intentionally, I saw a distinct increase in student understanding and application of mathematical concepts, more authentic connections to real-world mathematics scenarios, greater student achievement, and notably increased engagement. There are also ripple effects observed in other areas, as students learn grit and a growth mindset after tackling some more challenging problem-solving situations. I hope that by implementing some of these ideas, you see the very same shift.

Here’s an overview of some problem solving essentials I use to teach students to solve problems.

Routine vs. Non-Routine Problem Solving

Routine problems comprise the vast majority of the word problems we pose to students. They require using an algorithm through one or more of the four major operations, have relevance to real-world situations, and often have a distinct answer. They are solvable, and students can use several concrete strategies for solving, like “make a table” or “draw a picture” to solve.

Conversely, non-routine problem-solving focuses on mathematical reasoning. These are often more open-ended and allow students to make generalizations about math and numbers. There isn’t usually a straight path leading to the answer, there isn’t an algorithm readily available for finding the solution (or students are going to have to come up with the algorithm), and it IS going to require some level of experimentation and manipulation of numbers in order to solve it. In non-routine problems, students learn to look for patterns, work backwards, build models, etc. 

Incorporating both routine and non-routine problems into your instruction for EVERY student is critical. When solving non-routine problems, students can use some of the strategies they’ve learned for solving routine problems, and when solving routine problems, students benefit from a deeper understanding of the complexity of numbers that they gained from non-routine problems. For this training, we will focus heavily on routine problems, though the impacts of these practices will transition into non-routine problem solving.

Increasing Critical Thinking in Problem Solving

When tackling a problem, students need to be able to determine WHAT to do and HOW to do it.  Knowing the HOW is what you likely teach every day – your students know how to add, subtract, multiply, and divide. But knowing WHAT to do is arguably the most essential part of solving problems – once students know what needs to be done, then they can apply the conceptual skills – the algorithms and strategies – they’ve learned and will know how to solve. While dissecting word problems is an excellent starting point, exposing students to various ways to examine problems can help them figure out the WHAT. 

Being faced with a lengthy, complex word problem can be intimidating to even your most adept students. Having a toolbox of strategies to use when you tackle problems and seeing problems in various ways can enable students to get to the point where they feel comfortable knowing where to begin.

Shifting away from keywords

While it isn’t best practice to rely solely on operation “keywords” to determine what operation needs to occur when solving a problem, I’m not ready to fully ditch keyword-based instruction in math. I think there’s a huge difference between teaching students to blindly rely on keywords to determine which operation to use for a solution and using words found in the text to guide students in figuring out what to do. For that reason, I place heavy emphasis on using precise mathematical vocabulary , including specific operation keywords, and when students become accustomed to using that precise mathematical vocabulary every day, it really helps them to identify that language in word problems as well.

I also allow my students to dissect math word problems using strategies like CUBES , but in a way that is more aligned with best practice. 

Six Lessons for Easy Implementation

Here are six super quick “outside the box” word problem, problem solving lessons to begin implementing into your classroom. These lessons shouldn’t replace your everyday problem solving, but are instead extensions that will help students tackle those tricky problems they encounter everyday. As a reminder, we look at all of these lessons in the FREE Increasing Critical Thinking with problem Solving math mini-course .

Lesson #1: What’s the Question?

In this lesson, we’ll encourage students to see. just how many different questions can be asked about the same statements or information. We start with a typical, one-step, one-operation problem. Then we cross out or cover up the answer and ask students to generate possible questions.

After students have come up with a variety of questions, ask them to determine HOW they would solve for each one.

Reveal the question and ask students how they would solve this one and see if any of the questions they came up with match.

This activity is important because it demonstrates to students just how many different questions can be asked about the same statement or information. It’s perfect for your students who automatically pick out numbers and start “operating” on them blindly. I’ve had students come up with 5-8 questions with a single statement!

I like to do this throughout the year using different word problems based on the skill we’re focused on at the time AND skills we’ve previously mastered, but be careful not to only use examples based on the skill you’re teaching right then so their brains don’t automatically go to the same place.

These 32 What’s My Operation? task cards will help your student learn and review which operations to use for different types of word problems! They’re perfect to use as a quick assessment, game of SCOOT, math center activity, or homework.

Lesson 2: Similar Scenarios

In this lesson, students will evaluate similar scenarios to determine the appropriate operations. Start with three similar scenarios requiring different operations and identify what situation is happening in each scenario (finding total, determining an amount, splitting or combining, etc.).

Read all three-word problems on a similar topic. Determine the similarity of all of them and determine which operation would be used to solve them. How does the situation/action of the problem help you determine what step to take?

I also created these differentiated word problem task cards after noticing my students struggling with which operation to choose, especially when given multiple problems from a similar scenario. They encourage students to select the appropriate operation for each word problem.

Lesson 3: Opposing Operations

In this lesson, students will determine relevant information from a set of facts, which requires a great deal of critical thinking to determine which operation to use. Give students a scenario and a variety of facts/information relating to the scenario as well as several questions to answer based on the facts . Students will focus on determining HOW they will solve each question using only the relevant information. 

These Operation Fascination task c ards engage students in critical thinking about operations. Each card has a scenario, multiple clues and facts to support the scenario, and four questions to accompany each scenario. The questions are a variety of operations so that students can see how using the same information can solve multiple problems.

Lesson 4: Next Level Numberless

In this lesson, we’ll take numberless word problems to the next level by developing a strong conceptual understanding of word problems. Give students scenarios without numbers and have them write a question and/or insert numbers using a specific operation and purpose . This requires a great deal of thinking to not only determine the situation, but to also figure out numbers that fit into the situation in a way that makes sense.

By integrating these types of math problems into your daily lessons, you can significantly enhance your students’ comprehension of word problems and problem-solving. These numberless word problem task cards are the ideal to improve your students’ critical thinking and problem-solving skills. They offer a variety of numbered and numberless word problems.

Lesson 5: Story Situations

In this lesson, we’ll discuss the importance of students generating their own word problems with a given set of information. This requires a great deal of quantitative reasoning as students determine how they would use a given set of numbers to create a realistic situation. Present students with two predetermined numbers and a theme. Then have students write a word problem, including a question, using the given information. 

Engage your students in additional practice with these differentiated division task cards that require your students to write their OWN word problems (and create real-world relevance in their learning!). Each task card has numbers and a theme that students use to guide their thinking and creation of a word problem.

Lesson 6: No Scenario Solving

In this lesson, we’ll decontextualize problem solving and require students to create the situation, represent it numerically, and solve. It’s a cognitively demanding task! Give students an operation and a purpose (joining, separating, comparing, etc.) with no other context, numbers, numbers, or theme. Then have students generate a word problem.

For additional practice, have students swap problems to identify the operation, purpose, and solution.

Implementing Error Analysis

Error analysis is an exceptional way to promote thinking and learning, but how do we teach students to figure out which type of math error they’ve made? This error analysis starter kit can help!

First, it is very rare that I will tell my students what error they have made in their work. I want to challenge them to figure it out on their own. So, when I see that they have a wrong answer, I ask them to go back and figure out where something went wrong. Because I resist the urge to tell them right away where their error is, my students tend to get a lot more practice identifying them!

Second, when I introduce a concept, I always, always, always create anchor charts with students and complete interactive notebook activities with them so that they have step-by-step procedures for completing tasks right at their fingertips. I have them go back and reference their notebooks while they are looking at their errors.  Usually, they can follow the anchor chart step-by-step to make sure they haven’t made a conceptual error, and if they have, they can identify it.

Third, I let them use a calculator. When worst comes to worst, and they are fairly certain they haven’t made a conceptual mistake to identify, I let them get out a calculator and start computing, step-by-step to see where they’ve made a mistake.

IF, after taking these steps, a student can’t figure out their mistake (especially if I find that it’s a conceptual mistake), I know I need to go back and do some individual reteaching with them because they don’t have a solid understanding of the concept.

This FREE addition error analysis is a good place to start, no matter the grade level. I show them the process of walking through the problem and how best to complete an error analysis task.

Digging Deeper into Error Analysis

Once students show proficiency in the standard algorithm (or strategies), I take it a step further and have them dive into error analysis where they can show a “reverse” understanding as they evaluate mistakes made and fix them. Being able to identify an error in someone else’s work requires higher order thinking not found in most other projects or activities and certainly not found in basic math fact completion.

First, teach students the difference between a computational error and a conceptual error. 

• Computational is when they make a mistake in basic math facts. This might look as simple as  64/8 does not equal 7. Oops!
• A Conceptual or Procedural Error is when they make a mistake in the procedure or concept. 
• I can’t tell you how many times students show as not proficient on a topic when the mistakes they are making are COMPUTATIONAL and not conceptual or procedural. They don’t need more review in how to use a strategy… they need to slow down and pay closer attention to their math facts!

Once we’ve introduced the types of errors they should be looking out for, we move on to actually analyzing these errors in someone else’s work and fixing the mistake.

I have created error analysis tasks for you to use with you students so they can identify the errors, types of errors, rework the problem, and create their own version of the problem and solve it. I have seen great success with incorporating these tasks into ALL of my math units. I even have kids beg to take their error analysis tasks out to recess to finish! These are great resources to start:

• Error Analysis Bundle
• 3rd Grade Word Problem of the Day
• 4th Grade Word Problem of the Day
• 5th Grade Word Problem of the Day

The final step in using error analysis is actually having students correct their OWN mistakes. Once I have instructed on types of errors, I will start by simply telling them, Oops! You’ve made a computational error here! That way they aren’t furiously looking through the procedure for a mistake, instead they are looking to see where they computed wrong. Conversely, I’ll tell them if they’ve made a procedural mistake, and that can guide them in figuring out what they need to look for.

Looking at the different types of errors students are making is essential to guiding my instruction as well, so even though it takes a bit longer to grade things like this, it is immensely helpful to me as I make adjustments to my instruction.

Resources and Ideas for Critical Thinking

I’ve compiled a collection of websites for complex tasks with multiple, open-ended answers and scenarios. The majority of these tasks are non-routine and so easy to implement. I often post these tasks and allow students short bursts of time to strategize and plan for a solution. Consider using the tasks and problems from these sites as warm-ups, extensions of your morning meeting, during enrichment groups, or on a Problem of the Week board. I also highly encourage you to incorporate these non-routine problems into your core instruction time for all students at least once or twice a month.

• NRICH provides thousands of FREE online mathematics resources for ages 3 to 18. The tasks focus on developing problem-solving skills, perseverance, mathematical reasoning, the ability to apply knowledge creatively in unfamiliar contexts, and confidence in tackling new challenges..
• Open Middle offers challenging math word problems that require a higher depth of knowledge than most problems that assess procedural and conceptual understanding. They support the Common Core State Standards and provide students with opportunities for discussing their thinking. All problems have a “closed beginning,” meaning that they all start with the same initial problem, a “closed-end” meaning that they all end with the same answer, and an “open middle” meaning that there are multiple ways to approach and ultimately solve the problem.
• Mathcurious offers interactive digital puzzles. Each adventure is dedicated to exploring the world of math and sharing experiences, knowledge, and ideas.
• Robert Kaplinsky shares math strategies, lessons, and resources designed to create problem solvers. The lessons are detailed and challenging!
• Mathigon “The mathematical playground” offers free manipulatives, activities, and lessons to make online learning interactive and engaging. The digital manipulates are a must-use!
• Fractal Foundation uses fractals to inspire interest in science, math and art. It has numerous fractal activities, software to help your students create their own fractals, and more.
• Greg Fletcher 3 Act Tasks contain engaging math videos with guiding questions. You can also download recording sheets to go with each video.

Mary Montero

I’m so glad you are here. I’m a current gifted and talented teacher in a small town in Colorado, and I’ve been in education since 2009. My passion (other than my family and cookies) is for making teachers’ lives easier and classrooms more engaging.

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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.

Our Newsletter Provides Ideas for Making High-Quality Mathematics Instruction Accessible to All Students

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• The Problem with Word Problems March 11, 2024
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The Problem-Solving Schools' Charter

The NRICH team has developed this Charter to help you reflect on how you currently promote mathematical problem-solving in your school

Values and ethos

We have a shared belief that:

• Mathematical ability is not fixed: everyone can learn and make progress
• Problem-solving often involves taking wrong turns and making mistakes: every learner has the right to struggle and the right to enjoy success
• Everyone should have the opportunity to develop the skills and attitudes necessary to become confident problem-solvers
• Problem-solving can motivate learners to learn new mathematics, apply previous learning and make mathematical connections

Leadership and professional development

In our setting:

• Our staff promote positive attitudes towards problem-solving
• Time is set aside to discuss problem-solving in our meetings
• Our displays, newsletters, website, and social media content celebrate problem-solving for all
• Our monitoring system ensures that priority is given to problem-solving and mathematical thinking
• We engage with printed, online and face-to-face professional development opportunities offered by subject organisations

Curriculum, pedagogy and assessment

We are committed to:

• Regularly embedding non-standard problem-solving opportunities in our maths curriculum for all
• Ensuring that problems, and classroom support, offer opportunities for all to experience both struggle and success
• Allocating time to developing key problem-solving skills and positive attitudes
• Including non-standard problems in our internal/formative assessments
• Liaising with other subjects so that meaningful cross-curricular links can be made

Classroom culture

• Create a safe environment in which learners explore, take risks, and appreciate the value of learning from their mistakes
• Celebrate multiple approaches to solving problems and discuss the merits of the different strategies offered
• Provide frequent opportunities for individual and collaborative problem-solving, where learners are given both thinking time, and opportunities to share ideas and insights
• Celebrate the mathematical thinking of every learner

Problem-solving beyond the classroom/school

We encourage:

• Learners to engage with school Maths Club(s) and high quality maths books, ideally stocked by the school library
• Learners to take advantage of printed, online and off-site mathematical enrichment opportunities
• Parents and carers to engage with problem-solving through family homeworks and in-school events, while recognising that not every adult has had a positive experience of maths
• Our learners to appreciate, and learn more about, the achievements of a diverse range of mathematicians

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Maths problem solving – Why schools should prioritise this skill

Problem-solving is a crucial, yet somehow underserved component of maths lessons – and that needs to change, says Dr Ems Lord…

Maths teachers will be all too familiar with the three core focuses of the curriculum – fluency, reasoning and problem-solving. They might, however, find themselves struggling over that last one…

At NRICH’s many events and workshops, we’ve repeatedly heard teachers tell us that they’re struggling to consistently deliver high-quality problem-solving opportunities for their classes. Why?

Sometimes it’s because they lack the time to prioritise problem-solving within the timetable. They may worry that there’s a lack of consistency in approach across their school, or sometimes struggle to afford resources within their limited budget. Yet without problem-solving skills, students will be missing out on what makes maths so exciting for so many.

Why is problem-solving so important?

While exploring numbers can be very fulfilling, possessing amazing number skills isn’t always enough on its own. Learners must also be able to consider, choose and draw upon different strategies in order to successfully make decisions and solve problems.

Today’s workplaces demand employees who can solve non-routine problems in collaborative environments. To succeed, students will therefore need to develop resilience, be able to work collaboratively, think creatively and flexibly, and perhaps most importantly, be able to solve problems.

Collaborative problem-solving is becoming increasingly recognised as an important 21st century skill, as highlighted in the OECD’s PISA 2015 Collaborative Problem Solving assessment . That said, problem-solving doesn’t just empower students in the long term, but can also play a crucial role in helping them discover for themselves the joy of mathematics in the present day.

Imagine attending football practice, but only ever learning how to dribble. Or taking piano lessons, but only ever playing scales. Problem-solving can provide those real ‘lightbulb moments’ for students in maths lessons – it’s watching Messi score a goal, or listening to your favourite performer play live.

Problem-solving presents students with a mechanism for connecting the dots, as their maths skills and fluency in the subject become more purposeful and meaningful over time. They get to really see the value in what they’re doing, and once that value is better understood, they’ll become ever more engaged and motivated to learn.

Highlighted by Ofsted

Problem-solving is an area frequently picked up on by Ofsted, and was highlighted in the regulator’s July 2023 maths subject report. Said report called for schools to not just spend more time teaching specific problem-solving strategies, but for students to be given more opportunities to actually put those strategies into practice.

But where can schools turn to for support in implementing these recommendations, given their limited budgets and resources? One avenue is the nationwide Problem-Solving Schools programme recently launched by NRICH – part of the University of Cambridge, and one of the world’s largest maths outreach projects.

Problem-Solving Schools aims to ensure all schools are equipped to nurture students’ problem-solving skills through self-assessments of their existing provision, implementing whole-class problem-solving activities and rolling out dedicated teacher support – all provided entirely free of charge.

Problem-Solving Schools Charter

The programme begins with the Problem-Solving Schools Charter , which hundreds of schools have already signed up to. The Charter presents a framework to help you reflect on your current practice and identify areas for development, while highlighting five key areas:

• Values and ethos
• Leadership and professional development
• Curriculum, pedagogy and assessment
• Classroom culture
• Problem-solving beyond the classroom

NRICH believes that every child can become a better problem-solver, and invites readers to sign up to that ambition as part of Problem-Solving Schools.

Once you’ve reflected on your current provision and practice and identified your action points, NRICH will then provide free classroom resources and support. Our low-threshold, high ceiling activities will allow you to work through maths problems with whole classes, giving students the time and support they need to consider problems in depth, and attempt different strategies for resolving them.

We can also provide wraparound support for teachers, consisting of webinars, videos and teacher notes to help build confidence. Embark on the journey towards improving problem-solving in your setting by registering as a Problem-Solving School .

Dr Ems Lord is the Director of NRICH. current Chair of The Mathematical Association , a Fellow of The Institute of Mathematics and its Applications and a Founding Fellow of the Chartered College of Teaching . Download a maths problem-solving KS2 lesson plan from NRICH.

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More From Forbes

Success in sustainability: two cognitive strategies for effective problem-solving.

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Thomas Lim is the Vice-Dean of Centre for Systems Leadership at SIM Academy. He is an AI+Web3 practitioner & author of Think.Coach.Thrive!

Systems thinking and critical thinking are distinct yet complementary cognitive tools essential for effective problem-solving. Systems thinking allows businesses to understand and address the broad impacts of their actions on an interconnected system, while critical thinking sharpens decision-making, ensuring that outcomes are viable, ethical and based on solid reasoning.

Systems thinking provides a holistic perspective, focusing on how various components of a system interact and affect each other within a broader context. It emphasizes understanding the interconnections, dynamics, long-term impacts and patterns within systems to predict future behaviors and develop sustainable solutions.

This approach is particularly valuable in complex environments like organizational change, environmental management, and technological systems, where understanding the big picture is crucial.

On the other hand, critical thinking adopts a more analytical approach, concentrating on evaluating information and arguments, identifying logical inconsistencies, and making reasoned judgments. It involves dissecting complex problems into manageable parts, emphasizing evidence-based decision-making and rigorous evaluation of ideas and assumptions.

Critical thinking is key in activities that require clear, structured thinking, such as logical reasoning, decision-making, and solution evaluation, often focusing on scrutinizing existing solutions and preventing errors.

Together, these methodologies enhance decision-making and problem-solving by providing both macro and micro analytical perspectives to the challenge at hand.

Integrating both of these ways of thinking into sustainability initiatives offers organizations a robust framework for tackling complex challenges through a structured yet flexible approach. It helps organizations transform their approach to sustainability from fragmented efforts into a coherent, strategic agenda that drives real change.

Here's how organizations can implement these two ways of thinking effectively:

Step 1: Articulating Vision And Current Reality

Begin by defining a clear sustainability vision and objectively assess the current state to identify gaps. What is the desired future and the existing barriers or deficiencies preventing its realization? Engaging in this step ensures that all stakeholders have a unified understanding of the objectives and challenges.

For instance, a government agency might aim for sustainable urban development while recognizing current inefficiencies in urban infrastructure. The systemic structure would take into consideration manpower availability, lifetime cost of building projects and green funding availability.

Step 2: Structuring Decisions Based On Evidence

Detail decisions across all levels from strategic to tactical, ensuring that each decision aligns with the overarching sustainability goals.

This step often involves using decision hierarchies to maintain clarity and relevance at every level, thus preventing duplications and identifying gaps in strategies.

For example, a multinational corporation might structure decisions around reducing its carbon footprint through supplier engagement programs. Using critical thinking methodologies, they could create an analytical and evidence-based workflow and test assumptions on handoffs to ensure compliance.

Step 3: Prioritizing Challenges At Different Levels Of Perspectives

Identify the most impactful sustainability challenge and focus resources and efforts on areas where they can make the most significant difference to help maximize impact.

For example, a healthcare provider may prioritize waste reduction in its facilities by improving waste segregation and processing and develop the necessary systems and processes in keeping with the new disposal methods. They may modify or eliminate altogether outdated policies, leading to new behaviors of pattern over time in personnel involved.

Step 4: Developing Nested Solutions

Use both systems and critical thinking to create comprehensive, innovative and interconnected solutions. This might involve using systems diagrams to visualize problems and how they relate and employing logical reasoning to evaluate potential solutions for effectiveness and feasibility.

Remember the government agency aiming for sustainable urban development? In this scenario, they may create a stakeholder map aligning and enabling various parties to translate purpose into strategy. This would allow them to co-create multifaceted urban plans that integrate green spaces and renewable energy solutions. As a result, corresponding tactics and activities happen in an integrated, not haphazard, way.

Step 5: Crafting A Theory of Success

Develop a clear and actionable theory of success that outlines the key actions and leverage points. This theory should detail how the proposed solutions will address the identified challenges and lead to the desired change, identifying where small interventions could lead to significant systemic improvements.

In the case of the multinational corporation, their leverage was in incentivizing suppliers to adopt low-carbon technologies. Their theory of success was not in "shifting the burden" but in creating a positive reinforcement loop where they focused on the quality of relationships for long-term commitment.

Step 6: Implementing And Adjusting The Strategy

Put the strategies into action while establishing mechanisms for ongoing monitoring and adaptation. This includes setting up feedback loops to continuously gather data on the effectiveness of the interventions and making necessary adjustments based on empirical evidence and changing conditions.

For the healthcare provider addressing waste management challenges, this might involve adjusting waste management procedures based on ongoing feedback and outcomes. They might realize they are oftentimes reactive in their problem solving and therefore intend to conduct an intentional analysis and internalize and operationalize key insights.

As businesses become more complex and interconnected, the ability to think both systemically and critically isn’t just an advantage; it’s essential to survival and success in an interconnected world.

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