Research School Network: Solving Problematic Problem Solving How to help children solve problems with mathematical knowledge

Teaching problem solving has long been one of the more problematic
elements in the Maths Curriculum. Just how do we get children to apply their mathematical knowledge to solve problems in different contexts and with different structures?

Take this problem, for example, and imagine the typical range of responses from a class of Year 3 children.

You might see:

The Panicker: the child who stares blankly both the problem on the board and their empty whiteboard. They then turn to hopefully look at their partner, searching their eyes for a suggestion of what to do. Their partner stares back at them.

The Copier: the child who wants to please, they quickly copy the question down in lovely big writing to fill their whiteboard and in doing so, looks very busy indeed. Their partner takes the pen and waits for the teacher to say what to do next.

The Underliner: the child who copies the question down and quickly follows this by underlining the following important, key information– ​‘older’, ​‘total’. They then look at their partner expecting them to know what to do next. Their partner suggests that ​‘total’ means they need to add.

The Adder: this child has done problems like this before, they write ​‘30’ on a whiteboard, because they are usually required to add the numbers. They hold the whiteboard aloft, like a trophy, expecting a nod of approval from the adult in the room. Their partner can now breathe; they have an answer!

The Reader: They take their time, they read the problem over and over, they’re in Year 3 – surely, they need to do something other than add. Then they spot it, they whisper excitedly to their partner, who nods, relieved that they are sat on this table. ​‘We need two answers.’ Their partner nods again and they agree what the answers must be. They write ​‘26’ and ​‘30’ on their whiteboard, and whisper to the Adder that they need to say how old the brother is too.

Recognise anyone?

Why do children find this so difficult when they are often able to do the maths involved with ease?

Recommendation 3 in the EEF Guidance Report for Improving Mathematics in Key Stages Two and Three defines a problem-solving situation as one in which pupils do not have a readily available method to find a solution. Sometimes problem-solving can mean offering children routine word-problems, where children are required only to carry out a given procedure to arrive at a solution. The guidance report suggests that for children to succeed as problem solvers they need to tackle genuine problem-solving tasks where they do not have well-rehearsed, ready-made methods to find solutions. Rather, they need to be able to flexibly draw on a variety of problem-solving strategies.

The guidance report lists several factors which teachers should consider when developing problem solving skills. These include using visual representations to gain an insight into the structure of a problem, using worked examples to analyse different strategies and organising teaching so problems with similar structures are taught together. The focus, in effect, moves away from simply finding answers to word problems, to teaching children strategies to make sense of unfamiliar situations and tackle them intelligently.

As a school, we wanted to see if we could improve on what we already do. Initially, we used the guidance report to identify the strategies on which we might focus. We then looked to the resources produced by Gareth Metcalfe to give teachers practical ways to develop these strategies in the classroom. We used ​‘I See Problem Solving’ for Year 2, Lower Key Stage 2 and Upper Key Stage 2.

In the guidance report, they use an example of simplifying the problem as a strategy. Identifying a simpler, but related problem and discussing the solution to expose the underlying mathematical structure in the harder problem. This was something which we felt we could develop through both key stages, ensuring teachers make links between the similar problems explicit.

Often, we present children with a variety of different problems, set in different contexts with each one being very different to the one which came before. With each new problem, children struggle to find a starting point, and don’t know how to proceed other than to underline key information, and this can sometimes be misleading. If they do not achieve success early in the process, they lack the confidence to move on and all too often, their working memory is taken up with large numbers or a great number of tasks to do. Gareth Metcalfe suggests giving children an experience of success early in the problem-solving cycle, building their confidence.

Gareth Metcalfe also advises using a ​‘slow reveal’ strategy. Initially, rather than looking at the problem as a whole and trying to understand both the context and the structure, slow reveal allows children to build their understanding. Many children want to find the answer quickly. Covering information and numbers ensures children attend to the mathematical structure underlying the problem. Discussion is generated when children only have a small amount of information to work with. What do we know? What don’t we know? What are we trying to work out? As more information is revealed, how does this change what we know?

In a staff meeting, we looked at a problem and explored how we might use both strategies: simplify and slow reveal. We also considered which of the recommendations we intended to focus on and decided to explore teaching problems with similar structures together and using visual representations to provide an insight into the structure of a problem. We then thought about how this might work in practice; how might we break a problem down so that all children are successful?

Take the problem from recommendation 3, for example.

What the children might find difficult? What might they find confusing and what mistakes they might make?

A simpler problem might be:

This uses smaller numbers, removes one of the parameters and opens the possibility of more than one answer. The underlying structure remains the same. In this simpler problem, there remains a great deal to think about so at first, the children might only see this: ​‘A sister is older than her brother’, with the rest of the problem covered. Discussion might include how we would represent this. What would a bar model look like? What do we know and what is important?

Revealing the question next, ​‘What could their ages be?’ gives all children a chance to be successful early in the problem-solving process. Look at the bar model, what numbers could we use? Now some numbers are included in the bar model, reveal that they are added, what totals can we make? Finally, ​‘The total is 10’ can then be revealed with children understanding that the bar model will not change but the numbers will. Children will have a much clearer understanding of what it is they need to do.

Having worked through the simpler problem, using visual representations, and slowly revealing information, children will hopefully approach the harder problem with more confidence. Discussion will then focus on the similarities and differences in the two problems. Rather than being met with panic, children will approach with understanding.

The guidance report also suggests that teachers should consider organising teaching so that problems with similar structures and different contexts are presented together, and, likewise, that problems with the same context but different structures are presented together. The next step might be to change the context but keep the structure the same, to explore differences and similarities and to use representations to identify connections. Follow up problems might be:

The response from teaching staff has been very positive. Taking whole lessons or a series of lessons to explore problem solving allows time for quality discussion of approaches and strategies. Children’s confidence improves as does their resilience. Hopefully gone are the days when a problem written on the board results in a class of Panickers, Copiers, Underliners, Adders and Readers: essentially these children are approaching the problem without a strategy. We might soon be able to introduce a class of Understanders!


Mathematical Problem Solving – Gareth Metcalfe.
The Dynamic Deputies Podcast.
https://podcasts.apple.com/gb/podcast/mathematical-problem-solving-gareth-metcalfe/id1449384975?i=1000552270060
(Accessed 10 January 2023).

EEF. (2017) Improving Mathematics in Ket Stages Two and Three. https://educationendowmentfoundation.org.uk/education-evidence/guidance-reports/maths-ks‑2 – 3
(Accessed 10 January 2023).

Metcalfe, G. I See Problem Solving – Year 2: Maths Tasks for Building Problem Solvers. Available at: https://www.iseemaths.com/problem-solving-y2/

Metcalfe, G. I See Problem Solving – LKS2: Maths Tasks for Teaching Problem Solving. Available at: https://www.iseemaths.com/problem-solving-lks2/

Metcalfe, G. I See Problem Solving – UKS2: Maths Tasks for Teaching Problem Solving. Available at: https://www.iseemaths.com/problem-solving-uks2/