nrich mathematics problem solving

‘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 .

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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Solving Problems with the Unknown in any position

Beginning as early as 1st grade, students should have experience solving addition and subtraction problems with the unknown in any position . Traditionally, we have tended to focus on result unknown problems, such as 3 + 2 = ⬜. But students also need to be able to solve problems such as 3 + ⬜ = 5. So exactly how do we go about building that understanding?

This post contains affiliate links, which simply means that when you use my link and purchase a product, I receive a small commission. There is no additional cost to you, and I only link to books and products that I personally use and recommend.

number bonds

Flexibility with numbers begins in Kindergarten when students learn all the combinations for the numbers through 10. These combinations are often referred to as number bonds . Here you see a common way to show number bond relationships. In this case, you see all the ways to make 5.

You may notice that number bonds look a lot like fact families, and they are similar. The difference is how we approach building an understanding of the relationship between the numbers. When we taught fact families, it was typically done as a rote skill. Students knew that they needed to have two addition equations and two subtraction equations using the same numbers, yet they didn’t have an understanding of what the equations represented. We knew that because we would often see unreasonable results, like 2 – 5 = 3.

Students in Kindergarten need to first work with number bonds in a totally concrete way. You can find lots of different games, many with free downloads, in this post .

We can use number bond cards to help students understand the part/whole relationship of the numbers that make up a number bond. We want to start out with result unknown cards and provide students the concrete support of counters.

Here we see that they start out by putting teddy bear counters on the two parts that are known—3 bears on one part and 2 on the other. Next, students move the bears representing the two parts to the unknown whole section, finding that the whole is 5.

After lots of practice with result unknown , students can move to working with the cards with the unknown as one of the parts. Keep in mind that this won’t happen at the same time for all children. Differentiation is critical. We begin by placing the 5 bears on the whole. Next, we move 2 of the bears to the part we know. Finally the remaining 3 bears are moved to the unknown part. Be sure to provide plenty of guided practice during small group instruction before asking students to work with part unknown cards independently.

3 reads protocol

Using word problems makes abstract concepts more concrete because they put the numbers in a familiar context. However, we need to make sure that we help students develop reading comprehension skills to allow them to understand what the numbers represent in the context of the story. Enter a powerful strategy called the 3 Read Protocol.

The Three Reads Protocol, not surprisingly, involves reading a word problem three times, with each read having a different purpose.

But here are some things that might surprise you.

The problem in Read 1 has NO numbers and NO question. When you take out the numbers, students have to focus on the words. This helps them learn to make mental pictures of what’s taking place, which helps them understand what math to do.

Read 2 provides the numbers, but still does not have a question. Now the focus shifts to the numbers and what they represent in the story.

Finally, in Read 3, students come up with questions that could complete the word problem.

While this sample problem is a result unknown problem (but could also be a comparison problem, right?), you would gradually introduce stories that have the unknown in other positions.

For more information on 3 Reads and how to incorporate it into your instructional routine, check out this post .

Part/whole models

Another tool that can be used to help students understand that the unknown can be in any position is a part/whole diagram. To illustrate, let’s revisit the word problem from the last section, but now let’s make at a part unknown problem.

Now let’s listen in on what it would sound like to incorporate part/whole thinking.

TEACHER: [displays the word problem and a blank part/whole diagram] Let’s read this problem together and decide how each number fits into our part/whole diagram. First we’ll read the whole problem and talk about what’s happening in this story. Then we’ll read each sentence and add the numbers to our diagram. [teacher and students read the story]

TEACHER: Who is this story about? [Juliet and her grandmother] What’s happening in the story? [Juliet is saving money for a video game. She already has some money. Her grandmother gives her money for her birthday.]

TEACHER: Okay, let’s go back to the first sentence:  Juliet has saved $15 for a video game.  Is the $15 she had already saved the whole, her total money, or is it part of her money. [part of her money] What part is it? [the part she had already saved] Great! Let’s add that to our part/whole diagram and label it money she had . Sound good?

TEACHER: Next sentence:  Her grandmother gave her some  [teacher shrugs her shoulders when she says some ] money for her birthday. Huh, do we know how much her grandmother gave her? [no] So that is our unknown in this problem! Is the money her grandmother gave her part of her money or all of the money, the total? [part] Can we label that part  money her grandmother gave her?  If we don’t have a number for the money her grandmother gave her, what should we put in that part of our diagram? [a question mark]

TEACHER: Next sentence:  Now she has $33. Is that all of her money, the whole, or one of the parts? [all of her money] Why don’t we label the whole All of her money  and add the $33 to our diagram.

TEACHER: It seem like we’re getting really close to solving this problem! Let’s read the question and make sure it matches where we placed the unknown in our diagram:  How much money did her grandmother give her? Does that match our diagram? [yes] Yes, because we have our question mark in the part labeled money her grandmother gave her .

TEACHER: Great work! Now work with your partner to solve the problem.

Let me add an important note—I used a missing part problem for this example. Keep in mind that your students would have been using the part/whole diagram and this process for talking through the problem on less complicated, result unknown problems extensively before moving on to unknown parts.

Another note—be careful of calling this a subtraction problem. Yes, most students will use subtraction to solve the problem, but they could also use a counting up strategy ( 18 and 2 more is 20. It’s 10 more to 30, and another 3 to 33. So her grandmother gave her $15 ). Allow for flexible strategies in solving ALL problems!

I hope that gives you some fresh ideas for tackling this tricky concept!

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