Learning to Problem-Solve (and the role of I See Problem-Solving)

It’s easy to give children maths problems to solve; it’s much harder (and very time consuming) to systematically teach problem-solving so children become more competent problem-solvers. Here’s some thoughts on how my teaching of problem-solving has evolved and how I See Problem-Solving fits within this vision.

I have always loved engaging children in rich maths problems. The EEF research (recommendation 3) highlights the importance of engaging children in non-routine maths tasks. However, it also points out that such tasks can create a heavy a cognitive load for many children. Now I try I think more carefully about the sub-skills involved in solving a problem, and build up to the introduction of a problem-solving task using more carefully chosen examples. Consider this question:

Here’s how I built up to using this task. First of all, we clarify how to calculate the sum of four numbers and the difference between the smallest/largest numbers using this question:

Then we model the process of finding all possible answers with this prompt (answers: 5&5, 4&6, 7&3). We also highlight a likely mistake: 8&2 (the difference between the smallest/largest number no longer 4).

Then the final ‘pre-task’ question is introduced, focusing on the most difficult part of the calculation:

Now we get into the main task (sum of 4 numbers is 23, difference between smallest/largest = 4, all numbers different, how many ways?). Where appropriate, children use whiteboards and counters to access the task. When we find the answer 3, 4, 7, 9 we consider whether there is another solution that can be found keeping the 3&9 as the smallest/largest numbers.

As the lesson progresses, we explore systems for finding all possible answers and some children move on to the explain/extend tasks (which are variations on the main task). If you want to use Task 13, check out the free sample resources on this page. The solution to the problem is shown step-by-step by the pre-made worked example.

One little caveat: some children may benefit more from going straight into the question; others may need this extra scaffolding before getting into the main task. As ever, it’s about knowing your children.

There have been lots of great maths problems written. I hope that I See Problem-Solving takes things to the next level by presenting related problems and reasoning tasks in a coherent order, and by clearly representing the mathematics within the tasks.

I also hope that this blog gives food for thought about how to introduce problem-solving tasks so more children experience success. Enjoy using the resources!


I See Problem Solving – UKS2: providing challenge

I See Problem Solving – UKS2 is designed to transform the teaching of problem-solving in mathematics. Its design addresses Recommendation 3: teach strategies for solving problems from the recent EEF report. It will give all children the opportunity to understand and answer non-standard questions, whilst also providing appropriate challenge. This blog focuses on how extra challenge is provided to deepen and extend children’s learning in each task.

Each task begins with the main prompt question. For some children, answering this question may be their ‘Everest’; others will need more challenge.

First of all, a number of the tasks include a ‘how many ways’ prompt, with the challenge coming from finding all possible answers. Here’s a typical example of a ‘how many ways’ task:

And here are two other initial prompts:

Each task comes with an Explain prompt, where children have to unpick likely misconceptions, solve related problems, rank questions by difficulty or agree/disagree with different opinions. The prompts here are similar to those from I See Reasoning – UKS2 and will extend the thinking from the initial prompt. Here are the Explain prompts for the above examples:

For many children, the real challenge will come from the Extend prompt for each task. These tasks are related to the initial prompt, but the challenge is taken to the next level. These are the corresponding Extend examples (and two of the friendlier ones!):

This blog explains how extra support is also provided in each task.

I See Problem Solving – UKS2 includes a huge range non-standard problem-solving tasks spanning right across the curriculum. It costs £24.98.

I See Problem Solving – UKS2: providing support

I See Problem Solving – UKS2 is designed to transform the teaching of problem-solving in mathematics. Its design addresses Recommendation 3: teach strategies for solving problems from the recent EEF report. It will give all children the opportunity to understand and answer non-standard questions, whilst also providing appropriate challenge. This blog focuses on how extra support is provided to help children to ‘see’ the structure of the problems and to experience success.

The tasks are designed to be used at the end of a sequence of lessons, so children have developed their basic skills in that curriculum area. To start with, children are given the initial prompt – a question where there is not an obvious ‘standard’ approach to work out the answer. Here are two examples of the initial prompts:

There is then a ‘support’ prompt for each task which the children may choose to use. This will help children to understand the mathematical structure of the task. It may show a part-completed bar model, give some suggestions or offer a ‘way into’ the task. This will help all children to access the task and be more likely to taste success.

Then there is a ‘worked example’ to accompany each task. This is a series of images that shows the solutions modelled step-by-step, helping children to see the ‘deep structure’ of each problem. You will be able to download this for free as a PDF and/or as a PowerPoint file (available at iseemaths.com when the product is released). Here is a page from each of the worked examples from our two example tasks:

This blog explains how deeper levels of reasoning and extra challenge are then built into each task.

I See Problem Solving – UKS2 includes a huge range non-standard problem-solving tasks spanning right across the curriculum. It costs £24.98.

One of my favourite investigations

This is one of my very favourite mathematical investigations from I See Reasoning – UKS2: there’s a great pattern to explore. When I was in Y6 it was one of my ‘go to’ tasks for this time of year. Here’s our first discovery:

Despite having the same sum, the numbers give different products. And the further apart the numbers get, the smaller the product. But look at this:

There’s a pattern to how the products decrease: 1 less, 4 less, 9 less, 16 less. This is a pattern of square numbers. How odd! I wonder… is this the case for this example only? Or would it work for any example where the sequence starts from a square number? So the exploration continues, and we see that the pattern is repeated (e.g. 10×10=100, 11×9=99, 12×8=96, 13×7=91).

Eventually, I would challenge the children to use this knowledge to perform calculations. For example, consider 23×17. We know 20×20=400, so it follows that 23×17=391 (9 less than 400).

A beautiful pattern to explore!


The Vision: I See Problem-Solving

My philosophy has always been simple: be firmly rooted in educational research; find ways to apply evidence-based principles in the classroom; share the ideas that work best with other teachers. This approach led to me write the I See Reasoning eBooks, and it has driven the creation of my next resource, I See Problem-Solving UKS2.

In reference to problem-solving in maths, the latest EEF research states:
‘Explicit training appears essential; these sub-skills do not appear to derive from practice without direction and oversight.’

It also says: ‘Teachers should deliberately select visual representations that are appropriate to the problem’ and continues ‘provide prompts to encourage learners to monitor and reflect during problem solving.’

My belief is that I See Problem-Solving UKS2 will help us to explicitly teach problem-solving skills, helping all children understand the mathematical structure of each question. The resource will unpick a wide range of multi-step problems from right across the UKS2 maths curriculum.

Each task centres on a main question, like the example below:

Tasks are made more accessible by the ‘scaffold’ prompt which children have the option of accessing. This might be a part-completed worked example, a related example or some other prompt to break down the question:

The ‘explain’ prompt will provide a context for deeper thought or discussion, for example using ‘explain the mistake’ examples:

There is also an ‘extension’ prompt to provide further challenge based on the same task:

The mathematical structure of the problem is shown step-by-step and very visually by the ‘worked example’. This will be made available as a PDF or as a PowerPoint file. The worked examples can be displayed like ‘flip books’, showing each stage of the problem. Click here to see the worked example for this task – click through the pages rather than scrolling up and down for maximum effect!

I See Problem-Solving UKS2 is currently in production. It is being trialled by a large team of teachers who are sending me feedback on how the tasks can be improved. To join the team, register here.

I’m writing the resource over the 2018 summer holidays, will trial a few tasks in early September then I will aim for a September release – watch this space!

Logic Squares: next move

Logic Squares is my favourite maths app: it helps children to learn that the = sign means ‘same as’ rather than ‘makes’ and it gets children using number facts with flexibility. Click here for more details and examples.

Here’s a free resource, helping to break down the thinking behind completing a level. I hope you’ll find it a thought-provoking activity whether or not you use the app.

You will find 10 part-finished example Logic Squares grids. On each grid one square is highlighted: children will then discuss and explain which number they would put in the highlighted square.

The selected squares are the key ones to completing each level – which of the remaining numbers can be used?
Can it be done in different ways? Children could then complete the level using the available numbers at the bottom.

Logic Squares, made by Alan Peat ltd, is available for iPad on the App Store.

I See Reasoning – KS1

I’m delighted to announce that I See Reasoning – KS1 has now been released! It arms teachers with 281 thought-provoking prompts for embedding reasoning in every area of the KS1 maths curriculum.

The eBook’s creation has been a personal odyssey, beginning 18 months ago teaching a quite wonderful year 1 class. I’ve always been passionate about the power of visual, rich tasks to engage children in deep mathematical thought; designing tasks that achieve this for children in KS1 has been an awesome challenge. It’s led me to work side-by-side with some great teachers and share many memorable classroom moments.

So here’s what to expect. First of all, there are loads of prompts that help children to distinguish between right and ‘likely wrong’, helping to generate meaningful talk. Words are kept to a minimum – simplicity is king!

There’s a raft of questions presented with visual supports that encourage non-counting calculation strategies. Scaffolds are provided to focus children’s thinking onto key ideas or strategies.

There are a wide range of challenges, giving children the opportunity to build conceptual understanding and apply their skills in various ways.

And questions are diverse, covering all areas of the curriculum.

I believe I See Reasoning – KS1 will help to enrich maths lessons on a day-to-day basis. It’s been a joy to write and it’s my great privilege to share. I hope it gives you many great classroom moments!

I See Reasoning – KS1 is a digital download (£14.99). Buyers receive their eBook (as a PDF file) attached to an email from Etsy. There is no need to have an Etsy profile.

To see more information about the resource and to get access to the free addition sample section click here.

Also check out I See Reasoning – LKS2 and I See Reasoning – UKS2.

Full details about my maths training can be found at www.iseemaths.com and you can follow me on social media at Gareth Metcalfe Primary Maths (Facebook) , @gareth_metcalfe (Twitter) and I See Maths on Pinterest.