Reflecting the Emotional Experience of Mathematics, Redefining Success

Mathematics can be tremendously rewarding. The experience of making a new connection, understanding an idea or overcoming a seemingly impossible barrier is inherently joy-filled. I love seeing children battling with the various emotions that arise when doing mathematics – including the excitement of a new discovery. However, learning mathematics can also be very uncomfortable. Therefore, I believe that it’s crucial to help children become literate in the emotional experience of mathematics. And I never tell children that ‘maths will be fun today.’ Here’s why…

When we engage in learning, our brain releases a cocktail of stress chemicals to heighten our level of alertness. This is great for learning but it means that, in terms of our brain chemistry, at the start of a learning experience we have to walk through a temporary ‘door of discomfort’. At this point, I don’t want children to think that they should be enjoying themselves. The joy of a new discovery usually comes on the far side of effort, repetitive practice and maybe after some confusion. It involves discomfort. I want children to recognise and value that discomfort. Notice what it feels like and reframe these moments as part of the learning process – a gateway to success. And, of course, enjoy the moments of breakthrough!

For children to enjoy maths, they certainly need to experience success – to know that effort leads to success. For example, my training explores how children can have high initial success in problem-solving. Where children don’t experience success, I try to explain their difficulties, for example ‘When you can easily recall your times tables, you will have more brain space free to think about the question.’ I always try to address ‘mathematical status differences’ in the class, for example recognising a moment when someone perseveres through a moment of challenge. And I try to be careful in what I celebrate, downplaying the importance of an answer and placing the emphasis on the process of the thinking.

Then, I love to give children the chance to express their understanding in ways that are unique to them. In my training, I like to explore how children can extend sequences of Small Difference Questions, create their own Rank by Difficulty examples or tweak How Many Ways? tasks to increase or decrease the number of possible answers. I think we all have a deep-held desire to be unique. This is a wonderful way to express that in mathematics. It’s a skill that takes training. It’s a hugely worthwhile investment to make.

Mathematics brings up a wide range of emotions. In helping children to recognise and understand the full spectrum of these feelings (and to persevere through them) I believe children will have more mathematical success. I also believe it will also help children to grow as emotionally literate people. I’d love to hear your thoughts. Which ideas resonate with you? What do you disagree with? And how do you help children to navigate the emotions of learning mathematics? All views are welcome!

Shape Puzzles in Y2: small numbers, deep challenge

I’m busy writing I See Problem-Solving – Y2, a resource that I’m super-excited about. It will provide sequences of related questions, tasks and open-ended challenges so children can understand and then explore different problem-solving tasks. I will explain the philosophy behind the resource in a series of future blog posts.

For now, have a look at this sequence of tasks, how it builds children’s understanding of additive reasoning and lays the foundation for algebraic thinking.

Part A: Children are introduced to the idea that a shape represents a number.

Part B and C: Children find the value of each shape. They look for lines made using the same shape. Otherwise, they workout how the sum of a line increases when one more shape is added. Notice the top right example: an extra star is added but the sum for the row does not change. This shows that the star is worth zero!

Part D and E: They apply these principles to find the value of the shapes in these grids, where the sum of each column and row is given.

Part F: Then children can make their own examples.

This blog explains how these ideas can be extended using the I See Reasoning resources in KS2. If you want to trial I See Problem-Solving – Y2 as it is being written, click here to join the I See Maths mailing list

For more information about Gareth Metcalfe’s INSET and twilight maths training click here or for CPD sessions about using the I See Reasoning eBooks. My passion and expertise is in developing children’s ability to reason mathematically and building children as mathematical problem-solvers.

Shape Puzzles in KS2: exploring additive reasoning, laying foundations for algebra

I love using shape puzzles to explore some of the principles of algebraic thinking. The examples in this blog post are from I See Reasoning – Y4 (there are shape puzzles in all the KS2 I See Reasoning eBooks) and I often use these questions with older children too. I’ve found children love completing these questions and love creating their own puzzles!

Step 1: These questions help to uncover the key strategies for working out the value of the shapes.

Left example: the second line has one more circle than the first line and its total is 5 more. Therefore one circle = 5.
Right example: a rectangle is 2 more than a diamond. The child answering this question extended the pattern to show that three diamonds have a sum of 12 and therefore one diamond = 4.

Step 2: We complete shape puzzles using the thought processes from step 1. There are prompts (which can be used or can be hidden) to suggest possible starting points.

Step 3: Children complete different puzzles, explaining their starting points.

Step 4: Time for children to design their own puzzles! I specify two things: there can’t be any rows/columns that are made using only one shape; and the designer of the puzzle must be able to explain a possible starting point.

This webpage, designed by the brilliant Jonathan Hall, enables you to automatically generate these puzzles. And this blog explains how I’ve introduced shape puzzles to children in Y2. A fantastic way to explore some of the big ideas of algebra!

For more information about Gareth Metcalfe’s INSET and twilight maths training click here or for CPD sessions about using the I See Reasoning eBooks. My passion and expertise is in developing children’s ability to reason mathematically and building children as mathematical problem-solvers.

Click here to join the I See Maths mailing list and receive the latest new resources to trial.

Join the Discussion: How Expert Teachers will Rebuild Mathematical Understanding

It’s session 2 of the free Heartbeat of Education series this Thursday (11th March, 6pm-7pm) and it’s going to be a really significant one: how, as Primary teachers, can we ensure that children continue to thrive as mathematicians? And how should our maths lessons be different in this new season?

I believe that this is a time of great opportunity. It gives us the chance to reflect on children’s experience of mathematics and think about the skills and attributes that we truly value and want to build within our mathematicians. What can we do, as teachers, to lay the groundwork for children to have long-term success in mathematics? And how is this more than just helping children to ‘catch up’ on end-of-year targets? We will discuss what should be prioritised and how our teaching might be different in the upcoming weeks and months.

Register here to join the discussion live and to receive the recording of the session. I will be joined by award-winning Infant teacher Toby Tyler, leading teacher and teacher trainer Alison Hogben and the outstanding maths specialist Vicki Giffard. I want our discussion to explore YOUR questions. Here are some of the things that people have asked so far:
How do schools go about getting the balance right between focusing on the ‘Ready to Progress’ criteria as well as fully covering the National Curriculum?
How much weight should be given for retrieval practice if there are clear gaps in learning?
How should I differentiate now there are such gaps between children’s knowledge/experience in maths?

I’d love you to join in and please spread the word. Also, add your questions to the debate. Either post them on social media or email me at I’m looking forward to a lively, thought-provoking and important debate!

Online Training: the present and the future

From next week, I will run my first online training sessions for teachers  – it will be great connecting with educators again! Initially there are four different training events, each with 10am and 7:30pm sessions, so everyone can join in. I’m really excited to explore the opportunities that online CPD can provide.

Sessions will be 90 minutes long, with perhaps 60-70 minutes of content and 20-30 minutes of Q&A and discussion to unpick the themes of the training. This will give us time to explore key ideas in depth whilst leaving participants with a manageable number of take-aways. Future sessions will then develop these themes further. I hope that teams of people join in so they can work alongside their colleagues to implement the ideas.

I’ve already run five parent sessions on Zoom: I’m slowly learning to navigate the technology! So far, everything that can be shown at a training event can also be shown online. And with the ongoing chat and Q&A features, people have been able to interact well and ask questions as we go.

I’m particularly looking forward to the ongoing dialogue that will be created with myself and between participants both during and after. With people joining in from diverse settings, including teachers from overseas, we will learn a lot from each other! As the online training develops, I intend to run a wider variety of sessions and to build future training around the areas that people want to explore further. Sessions can be targeted to specific year groups, topics or aspects of teaching. Online training brings cost and time efficiencies. A recording can be viewed by participants afterwards too.

This form of training is new to me. I’d love to get your ideas on how I can expand or improve my online CPD offering (email This could be about the logistics of accessing sessions, thoughts on the content of training or anything else. At what time would you like sessions to run? What would your dream course title be? And how can we ensure that the impact continues long after the sessions? I would absolutely welcome your feedback. I will plan my training sessions for May soon.

I also regularly send out resources for teachers to trial to people on my mailing list so that I can get teachers’ feedback on my products as they are being written. At the moment I’m writing three new resources. I’m planning to run some free sessions in May where I’ll show some of these resources and ask for people to say what they like and what they would add/change. I’d love to get as many people joining in with these sessions as possible. The more viewpoints I can get the better!

I can’t wait to get started. Hopefully you’ll join me!

Click here to book and for full details about April online training.

Home Learning Lessons: plans for the summer

It’s been a wonderful start to the home learning project, the vision for which is described in this short video. During the school closures, I want children to experience rich, emotionally engaging maths learning. I also wanted them to feel as if they are part of a vibrant, real community of children.

Each day, I post a video lesson for Y3/4 children (all Y3/4 lessons here), and a separate video lesson for Y5/6 children (all Y5/6 lessons here). Each video explores a big mathematical idea in small steps and a range of independent tasks are set for children to complete. Everything is free. In the first six weeks, the videos have had 335k views!

Here’s the outline plan for the rest of the school year. I will continue making the videos for the duration of the UK school closures (however long that lasts):
18th – 22nd May: Measures: Money and Time
1st June – 5th June: Data Handing
8th June – 19th June: Fractions
22nd June – 3rd July: Flexible Calculation
6th July – 17th July: Mathematical Puzzles

It’s been thrilling to hear how much the children have loved the lessons and how it’s helped time-pressured parents too:

I want as many teachers and parents to find out about this project as possible. Tell friends, share with colleagues! Lessons can be uploaded onto school websites; many schools retweet my evening tweets each day from their Twitter feeds. Also, if you like the videos it really helps if you give them a thumbs up on YouTube! Here’s the first Y3/4 lesson and here’s the first Y5/6 lesson – they will give a taste of what the lessons are like!

I have also compiled maths games to play with KS1 and KS2 children and I have been running a series of online training sessions for teachers.

I hope you are all keeping well,

The Plan: Primary Maths Lessons During School Closures

Here’s the plan for the online maths lessons that I will run over the period of the Covid-19 school closures. I hope it will help many children to engage in rich, thought-provoking maths learning during this time. I hope they will be uplifting too!

Every weekday at 9am, two new lesson will be posted on this page via YouTube: one aimed at children in Y3&4 and another for Y5&6. Each video will help children to build the skills needed for the main task. Then children will complete the main task – a challenge or short series of questions – working individually or with adult support. These tasks can also be downloaded from this page by clicking the relevant links. Answers will be provided! The first videos/tasks will be uploaded on Monday 23rd March.

Underneath each video is a short description of the key calculation skills for each lesson. Children may benefit from practising these skills before watching!

A series of games that can be played with children aged 5-8 will also be shared. The first of these videos will be published here on Tuesday 24th March and I plan to release two of these videos per week. I’ve got some great, easy-to-use games to show.

An introductory video has been put together to explain the project – please share this with children and parents. The videos will run until 3rd April (my health permitting), then there will be a two-week break over Easter. I will share a range of games that can be played during that period. Then, all being well, the daily videos will recommence for the duration of the school closures.

I am hoping to be able to run these lessons as ‘live lessons’ soon. I’ve not quite got the technology or expertise for that yet! But watch this space, I’m working on it – I’d love to introduce this and make the lessons more interactive and real.

How you can help
I also want to make the lessons engaging and personal. I’d love to end each video with a message, a joke or a thought from a range of teachers, parents or people from the community. Let’s make the videos feel like ours – help the children feel loved by the wonderful teaching community. Please email, tweet or FB message me with your personal message! And keep sending ideas for maths challenges to include. Spread the word far and wide…

Best wishes

I’m looking into ways that I could deliver free, 45-minute tutorials for parents to show how they could use some of my maths resources at home. If you might be interested, email

I am also exploring possible ways to deliver maths training sessions online to Primary teachers during school closures. Email for more information. 

A favourite multiplication investigation

Here’s a multiplication investigation that I used with a mixed-age Y5/6 class. Before the task we recapped on the grid method (it will become apparent why the grid method was used). Then, the main task was introduced:

Immediately, the children positioned 8 in one of the tens columns and 0 in one of the ones columns. Some children tried 85 x 40, but then we established that the larger digits (8 and 5) need to be placed in the tens columns and the smaller digits (4 and 0) in the ones columns. But where, then, should the digits 4 and 0 be positioned? Which gives the larger product: 84 x 50 or 80 x 54? Will these two calculations give the same answer? This was explored:

We compared these two calculations and asked ‘What’s the same? What’s different?’ The grid method helps to show that multiplying the tens values gives 4000 in both calculations. However, the TO x O parts of the calculation are different. We saw that 80 x 54 gives the largest product. One child even said ‘The short method for multiplying is quicker, but the grid method is better for showing what’s the same and what’s different.’ I show the children an area model to help them to understand the difference between 84 x 50 and 80 x 54:

In the image above, we see the TO x TO part of the calculation. We ask the children to visualise how each image will be changed when the TO x O part of the calculation is also shown (see below):

Of the two calculations, 80 x 54 gives the larger product because we are multiplying a larger tens value by the 4. I also commented that the pair of numbers that are relatively closer together gave the larger product. Then we look at the examples below, comparing pairs of numbers with the same sum that are multiplied:

We see that the pairs of numbers which are closer together have the greater product.

I have run a very similar investigation using a TOxO calculation. Here’s an opening prompt:

And it led into the I See Problem-Solving investigation below. It’s one of the free sample tasks that can be found here:

If you have a go at these investigations or something similar, I’d love to hear about how you get on!

I See Problem-Solving – UKS2 and I See Problem-Solving – LKS2 give a huge bank of rich tasks. There are Worked Examples and Support, Explain and Extend features that help to deepen children’s understanding.

Information can be found here about INSET/Twilight training on developing reasoning and problem-solving. Bookings are currently being taken from late February onwards. For very cost-effective training, you may consider hosting a training event, or running example lessons in your school. 

I See Problem-Solving – LKS2: Support and Challenge

I’m delighted to have released I See Problem-Solving – LKS2. It’s a resource that I’ve lived and breathed in the classroom over the last 10 months! The aim is to help all children to access rich problem-solving tasks, whilst ensuring that all children are challenged and engaged. It is the practical outworking of the research on solving problems from this EEF report (see point 3). 

I See Problem-Solving – LKS2 is comprised of 54 tasks. Each task gives various challenges: the Build prompt introduces the key themes and concepts, before the main Task is presented. Then, there is a Support prompt to help children access the task. The Explain and Extend challenges give rich opportunities for extended exploration.  Here’s an example task, starting from the Build prompt:

This introduces some of the key language that the children will need to understand before they access the main task. Here is the Task:

Children might choose 1, 2 and 3; they could work with 21, 22 and 23. Either way, we can all explore this idea and visual representations can be used to help. If children are need help, they can look at the Support prompt:

The Worked Example shows the solution to the main task step-by-step. The Worked Example can be viewed as a PowerPoint or as a PDF:

To deepen the challenge, the Explain and Extend prompts provide related challenges:

Not all the tasks follow this exact format. Sometimes there are Practise questions:

And there are always questions that extend the challenge:

Information about the resource, plus a link to the Etsy page to buy the resource, is found on the I See Problem-Solving – LKS2 page. There are 5 free example tasks to use too. It costs £24.98 and is available as a digital download of the PDF file. I hope you find this resource super-helpful for engaging children in meaningful problem-solving. It’s certainly given me many great classroom moments already!


Place Value: Seeing the Relative Size of Numbers

In place value, children learn about the value of each digit in a number (e.g. that the 5 in 153 represents 5 tens) – the Deepening Understanding in Column Value blog gives some ideas for extending thinking in these lessons. However, to give children a more complete understanding it’s important that they can also reason about the relative size of numbers. In this blog I will explore how I’ve used a blank number line to develop this form of understanding and look at the wealth of opportunities for reasoning that it can provide.

Consider this task. Children are given a long number line with 0 and 100 at either end and are asked to position 31, 39 and 84 accurately on the number line. Children are challenged to think about whether the lengths between the numbers are appropriately sized.

I have found that children are generally able to order numbers, but that the common mistake is to make the spaces between numbers too similar. In this example, I may ask children to compare the distance between 31 & 39 with the distance between 0 & 31 (which is almost four times longer) and the distance between 84 & 100 (which is exactly twice as long).

In a similar task, children have positioned 4, 7 and 9 on a 0-10 number line. It’s common for children to position 4 by counting four small ‘steps’ on from zero (placing 4 far too close to zero) rather than thinking about the position of 4 relative to the half-way point of the line. Similarly, 7 and 9 are often positioned by counting back from 10, leaving an overly large gap between 4 and 7. With careful modelling, and by looking at the number lines in the classroom, children learn to reason spatially with greater precision.

I’ve included two such tasks in I See Problem-Solving – LKS2 (click on the link for sample tasks), which is due to be released on 29th September. Here’s one of the pages from the Worked Example:

And here is the extension prompt for the task. There’s so much additive and multiplicative reasoning that go into estimating the value of the missing numbers:

I would love to hear about any practical examples of how you are outworked these ideas in the classroom. The blog Deepening Understanding of Column Value gives some more ideas for how to deepen the challenge in place value. Have a great term!

For information about NCETM-accredited training by Gareth Metcalfe, please visit – bookings are being taken from Spring term 2020 onwards.