garethmetcalfe Maths discussion

I See Reasoning – Y1 and Y2: Exploring Concepts, Creating Reasoning Habits

We want KS1 children to develop a deep understanding of Y1 & Y2 maths content. We also want young children to be able to explain their thinking, identify common errors, estimate, explore ideas and think creatively. The tasks in the I See Reasoning – Y1 and I See Reasoning – Y2 eBooks help to give children these experiences, inspiring a wide range of mathematical conversations and explorations.

These new eBooks have replaced I See Reasoning – KS1. They follow the same format as the original eBook, but include many, many more examples (365 tasks in the Y1 eBook and 392 tasks in the Y2 eBook) and they have a range of new types of reasoning questions. Here are some of the key ideas:

Non-counting strategies, estimation, reasoning

A HUGE focus is placed on children explaining answers using non-counting strategies. This includes ‘how many dots’ questions, where children describe their non-counting strategies. It involves calculations that border 10 or subtractions with small differences. The emphasis is not ‘what’s the answer?’ but instead ‘how did you know?’ or ‘what do you visualise?’ The questions are highly visual and don’t require too much reading.

Misconceptions, visuals, patterns

The questions introduce the key I See Reasoning question structures. Children will learn to spot mistakes, explain mistakes, compare questions and spot patterns. They will be challenged to explain what they noticed and find all of the answers. In doing so, children will be trained in the routines of thinking mathematically, routines that can be extended in KS2. This will help to build reasoning tasks into every maths lesson, giving schools a progressive approach to how reasoning is taught.

Exploration

There are lots of questions, of many different forms, for exploring mathematics. This includes estimation tasks, open challenges or questions with different possible answers. There are also a range of spatial reasoning tasks, for children being able to visualise items from different perspectives.

Depth

There are also a wide range of tasks to add challenge! These are very diverse and sometimes require children to find multiple answers or explain their thinking. These tasks are highly varied depending on the area of the maths curriculum that they cover.

The introductory price of the eBooks is £30 each (including VAT). From 1st January 2026, they will cost £35 each (including VAT).

I See Reasoning – Y1 and I See Reasoning – Y2 lay the foundations for children to experience maths as a thinking, exploring, explaining subject. If you click on the links, you can view a sample section of each resource. I hope that they inspire the children in your class and give you many fantastic classroom moments!

garethmetcalfe Maths ideas

The New Deconstructing Word Questions – Y6

The updated version of Deconstructing Word Questions – Y6 is now completed! Full information about the resource, including a free sample task, can be found here. It provides a coherent, detailed approach to teaching children to answer word questions and gives a wide range of thought-provoking challenges.

This resource has been re-released in April 2025. If you purchased the original resource, you can have this new version for free! Just email iseemaths@hotmail.com and attach the original version of the resource (as proof of purchase) or give the order number for your original order. Then, we will reply by sending you the new resource.

Why has the resource been updated and re-released?
Since writing the original version, I have written Deconstructing Word Questions resources for Year 2, Year 3, Year 4 and Year 5. All of these resources followed a specific lesson structure:
Build 1 – teaching prompts
Task A – pair discussion task
Build 2 – teaching prompts
Task B (Version 1 and 2) – questions
Extend – deeper challenge
I have re-written the year 6 resource so that it also follows this lesson structure. This means that the resources give a totally consistent whole-school approach to teaching children to answer word questions. This video shows how the resources can be used to teach a lesson:

I want schools to have a whole-school vision for teaching children to answer multi-step word questions. Therefore, I am running 90-minute online INSET sessions on 1st and 2nd September to communicate this vision. It would be great to have you involved!

The ability to answer word questions is one part of how we can build children as mathematical problem-solvers. The full vision, including detailed guidance and video exemplification, can be found on this page. I believe it gives a practical, exciting vision for how we can build all children as mathematical problem-solvers!

garethmetcalfe Maths discussion

Deconstructing Word Questions: the vision

Imagine this: you are asked to describe the strengths that the children in your school have as mathematicians. You say ‘they are brilliant at understanding and answering word questions!’ When asked to elaborate, you say ‘the children read questions carefully and pick out the important information.’ Or perhaps ‘the children are great at spotting multi-step questions.’ Maybe even ‘they show their understanding in different ways.’

In reality, so many children struggle to answer multi-step or non-standard word questions. So how do we go from giving children word questions to teaching all children to answer word questions? What does a consistent approach look like?

This has become my mission. For the last 3 years, I have been writing Deconstructing Word Questions for Y2 – Y6. Each task has been trialled in a number of different schools, being honed with the help of some amazing teachers. The eBooks are on sale here.

The golden thread that runs through every technique, every activity, is focusing children’s thinking on the deep structure of each question. It is about taking the attention away from calculating answers to understanding the steps involved. Here are four ways that this is achieved.

1. Slowly revealing information in questions
Children predict what the hidden words/information could be, as in the examples below (Y2 and Y5). Then, the information is revealed. This means children have thought about the structure of the question before they answer the question.

2. Using equipment or bar models
Sometimes, children are asked to represent questions with counters. Sometimes, children are asked ‘which bar model represents the question?’ (left-side example, Y3). For some questions, children are given part-complete bar models to fill which act as a scaffold (right-side example, Y4).


3. ‘Minimally different’ questions
Children analyse pairs of questions that are very subtly different. The children look at how the questions are the same/different. This helps children notice the subtle but all-important differences in the wording of questions (left-hand example, Y2). This variation is used in the questions that children answer (right example, Y3).


4. Depth
Lots of techniques are used to extend children’s thinking. This includes explaining which approach is correct (left example, Y2) or in giving the information that is missing in a question (right example, Y4).


There is a trial task for each year group to try out. Click on the links below for the resources and for the short ‘how to’ video:
Deconstructing Word Questions – Y2
Deconstructing Word Questions – Y3
Deconstructing Word Questions – Y4
Deconstructing Word Questions – Y5
Deconstructing Word Questions – Y6

The Vision: Building Problem-Solvers maps out a holistic vision for building children as problem-solvers. There are 10 videos to exemplify the key principles shared.

I hope Deconstructing Word Questions helps many children to grow as mathematical problem-solvers.

garethmetcalfe Uncategorized

UPDATED BLOG: My 2023-2024 Writing Challenge

I have managed to post an example task for a streak of 107 days this academic year. However, at this point I’m going to redirect my efforts as I have decided to focus on writing Deconstructing Word Questions Y2-Y5. I see this as being my greatest possible contribution. It’s hard to properly communicate the vision for these resource using photos – it’s better done using short videos. I will spend the Spring term focusing on writing the resources and trialling them in lots of different classrooms. Then, in the summer term, I will post a video a week explaining how, I believe, we can transform the teaching of word questions. I will also send out lots of free resources to trial to the people on my mailing list. I am so excited about what is to come!

The resources will be released some time between September 2024 and January 2025, depending on the outcomes from the classroom trials.

Below is my original blog post:

Day 1: The Mountain Pass Logic Puzzle and The Mountain Pass Answers

I’m passionate that all children get to experience the true richness of mathematics and for maths to be an intellectually and emotionally rich pursuit. To this end, over the last 10 years I’ve strived to create maths tasks that generate conversations, give space for curiosity and provide opportunities for extended exploration. I hope that my resources are inspirational and simple to integrate with your maths curriculum.

I have so many plans for new resources! I See Reasoning – Y1 and I See Reasoning – Y2 are being written: they will be comprehensive resources for building number sense and embedding reasoning in daily lessons. Deconstructing Word Questions will be written for Y1-Y5 after the successful launch of the Y6 eBook. Then I will go back to my roots: writing a range of logic puzzles and problem-solving ‘task families’, so problem-solving skills can be built coherently.

To get into the routine of writing new resources every day, I have set myself a challenge: to post a new task every school day on social media for the whole of the 2023-2024 academic year. Images or videos will be posted at 7:30pm every day on Twitter, Facebook and on my NEW INSTAGRAM FEED. At the time of writing, I have exactly 0 Instagram followers!

I am actively seeking your feedback on all my posts. Specifically, what would you change about each task? Or if you use any of the tasks in the classroom, what age of children did you use the task with and what happened? I won’t always respond immediately – I can struggle with insomnia so I’m usually off social media after 8pm – but I will read every comment. Feedback can be emailed to iseemaths@hotmail.com

I have taken so much from the feedback given by teachers about my work – it helps me to reject my bad ideas and improve my half-baked ones! I love to receive any suggested improvements or comments.

The first post will be on 4th September. Until then, have a great summer – Gareth

garethmetcalfe Maths discussion

Learning content, developing habits of thinking

I’ve taken a lot from listening to Dylan Wiliam speaking over the years. One of his insights has particularly resonated with me: the idea that the improvement in learning that a child derives from being in the class of a highly effective teacher extends well beyond the time that the child is in that teacher’s class. For example, if a child experiences expert teaching in Y3, they are likely to make more rapid progress in Y4 and beyond too.

It is, therefore, hard to judge the true effectiveness of teaching from end-of-year maths assessments. It might give an indication of progress made in the content goals for that year but it won’t give the full story of the long-term impact of that teaching. The lens is too narrow. If all our efforts are placed on getting children to achieve their ‘content goals’ for the current day/block/year, we may always be limited in our impact. We want out teaching to help children to learn the content and to build children’s capacity to learn other new content.

I have come to think of each maths lesson as an opportunity to develop a child’s content knowledge and their habits of thinking. Of course, surface knowledge is important as reasoning doesn’t happen in a vacuum. However, I generally look at the content of the lesson as the context through which I will build the mathematical habits of mind. This may involve helping children to represent ideas visually, explain misconceptions or spot patterns. Perhaps children will be challenged to create their own examples. Or maybe a task will require an element of perseverance and self-regulation. This approach may have a short-term cost in the speed at which content knowledge is acquired, but this is likely to be a worthwhile investment. It can give the children a richer experience of being a mathematician.

My simple encouragement is to be aware of the need to develop content knowledge and build mathematical habits of mind. To focus on the detail as well as being aware of the bigger picture. Content knowledge is more tangible and it is easier to assess. The habits of thinking that are developed, though, will play a powerful role in children’s long-term mathematical success. We want all children to develop these habits, not just pupils whose attainment is already relatively strong. It will also require us to make thoughtful choices about what we don’t teach. Where time is limited, content is usually prioritised.

A starting point might be to establish What, as a school, do you consider to be your ‘mathematical habits of mind’. What are the characteristics that you want to build within children over time? How are they explicitly taught and made visible? And how do we promote and celebrate children’s progress is developing these habits? My aim, of course, with the I See Reasoning eBooks is to write questions and tasks that help children to develop these habits. Schools generally have a clear plan for how children learn their number bond facts. I want schools to have the same clarity in how children reason mathematically and grow as creative, independent thinkers.

If you have any thoughts on this blog, please share them with me by emailing iseemaths@hotmail.com or commenting below – Gareth.

garethmetcalfe Maths discussion

Emotional Regulation in Learning Mathematics

We all know, from personal experience, that mathematics evokes a broad range of emotions. In maths lessons, we’ve probably all had to navigate through frustration and confusion, doubt and even embarrassment; but then we may have also experienced the joy of a new discovery and the immense satisfaction that maths can bring.

For so many people, though, the uncomfortable emotions involved in learning maths are so dominant. We know how destructive this can be. And we know how common it is for adults to admit to children that ‘I was never any good at maths.’

Much has been done. Mistakes are embraced as opportunities to grow our brain; learning is broken into small steps so working memory isn’t overwhelmed; extra support helps children to keep pace. I wonder, though, if there’s still more ground to take in terms of helping children to self-regulate their emotional response in maths lessons.

Acknowledge the uncomfortable emotions that come with learning maths.
Firstly, I think it’s important to recognise that learning maths is, biologically, uncomfortable. Our bodies are always looking to maintain homeostasis. However, to prime us for learning, our brains release a chemical called epinephrine (adrenaline released in the brain). This heightens our focus and therefore supports learning, but it means that learning can be uncomfortable. Also, at a very primitive level, we are a social species. The desire to maintain our status ‘in the group’ is wired deep within us. Whilst I can encourage children to not compare themselves to their friends, at a very basic level it’s natural that doing maths could make us feel somewhat threatened. I think we should be open about this. Normalise it. Help children to see that they are not ‘doing it wrong’ if sometimes they feel this way! It’s just a product of our human wiring.

Recognise the physical manifestations of different emotions
Also, I think I could have been much more granular about the different emotions that can be experienced when learning maths. Consider the emotions frustration, embarrassment and apathy. All three are very different, with very different physical manifestations. For example, frustration could be thought of as being uncomfortable but helpful: it might drive us towards action and to a deeper level of focus. Embarrassment (which is experienced by children of across the attainment spectrum) has a much stronger social component. Processing embarrassment may require children to step away momentarily and regain their normal perspective. In contrast, apathy may be more associated with a reduction in psychological arousal: stepping out of apathy may require more action.

How can we help children to transition through some of these emotional states? By noting the physical sensations. By explaining that these responses are natural. And by emphasising that these emotions are temporary.

Helping children to transition through emotions
I love to celebrate the breadth of emotions that doing maths can generate. Before a lesson starts, I like to give an example emotion that children might experience (either an uncomfortable or a pleasant emotion). We can note how that emotion feels, why we have that response and what we can do if we feel that way. This dialogue helps children to see that these emotions are normal and that they are passing. Children shouldn’t feel ‘like they are doing it wrong’ if they experience a certain emotion in a maths lesson. It’s NORMAL to feel anxiety when learning maths! But, of course, we want children to transition through that state of feeling anxious, not to remain stuck in that emotional experience.

And, of course, there’s so much to enjoy on the other side! Relief, satisfaction, creativity, pride, motivation, surprise, joy, discovery…

The next steps
I am always looking for ways to outwork these ideas into something tangible that teachers can use in the classroom. Of course, the way in which we have these conversations will depend on the age, maturity and nature of the children. But maybe there are some prompts that I could create that would open up some great discussions in your classroom around emotional regulation in maths lessons? Or maybe you have a different perspective on this topic, or different insights to offer? I would love to know! Highlight my ignorance, help me to understand your context, raise your questions, give me your best ideas.

I want all children to experience the true emotional richness of learning mathematics.

garethmetcalfe Maths ideas

Deconstructing Word Questions in UKS2: see the vision, share your views!

I’m passionate about helping all children to become effective problem-solvers. Over the last 3 years, I’ve done a huge amount of work on helping children to deconstruct multi-step word questions using techniques similar to those described here by Brian Bushart or here by The Erikson Institute for Early Math.

The focus is all about getting children to focus on the deep structure of a question by removing the need for children to find the single definitive answer. It’s far too big a topic for me to write about in a blog, but this video gives an insight by showing how a SATS question can be deconstructed:

I’m going to embark on a big project to give teachers the resources to outwork these ideas in the classroom, initially in UKS2, and I’d like to get input/inspiration from as many people as possible before I start work on the resource.

As such, I’m holding two free 30-minute Zoom sessions where I will spend 15 minutes explaining my ideas, then I will provide an open forum for people to give their thoughts and suggestions. And no pressure to contribute! You are very welcome to join in without having to interact. I will also share lots of free resources in the upcoming months to everyone on my mailing list as I create and trial the resource:
Register for the session on Tuesday 25th January, 12:30pm-1:00pm
Register for the session on Wednesday 26th January, 6:30pm-7:00pm

I hope this project will help children to flourish as mathematical problem-solvers. I’d love to see you there!

I See Problem-Solving – Y2 has now been released!

garethmetcalfe Maths ideas

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!

garethmetcalfe Education General

Using lessons from neurobiology to enhance learning

I’ve always been fascinated by cognitive psychology. I remember, perhaps 6 years into my teaching career, thinking that I should really know more about the science of our brains and how we learn. Since then, so much more is known about optimising learning, like the relationship between working memory and long-term memory or the importance of spaced practice.

Over the last few months, I have been listening to a fascinating podcast by Professor Andrew Huberman who works at the Stanford University School of Medicine called The Huberman Lab. A number of things from the podcast have really resonated with me, and I wondered if they could have application in our schools to further enhance learning. Here are just a few of them.

Among the many science-based tools and protocols shared, Professor Huberman describes how important it is after a bout of learning – and ideally immediately after – for deep rest to take place to improve the transfer into long-term memory. After a 90-minute bout of piano practice, for example, learning can be enhanced by having just a few minutes of resting with your eyes closed. I know that this exact protocol won’t apply directly in primary schools. But might another protocol take its place?

He also describes the importance of utilising the visual system, including having at least some periods of time each day for looking into the far distance. As well as being important for eyesight, your brain has significant rest when you are not focusing on things that are close up. Looking into the distance after a lesson could potentially enhance the transfer of learning into long-term memory. This is also why looking at an electronic device is unlikely to give your brain a good break after a bout of learning. A large proportion of our brain activity is generated by the visual system, so knowing how to optimise it to enhance learning is perhaps a less explored aspect of learning.

I was also fascinated by Professor Huberman talking about adult learning. He describes how, before the age of about 25, we are wired for brain plasticity. However, after that age we need to work harder for our brain to adapt. At the start of a bout of learning, our brain releases Epinephhrine – the name given to adrenalin in the brain. This increases alertness but can initially feel uncomfortable. Learning new things is hugely rewarding and joyful. However, in terms of our brain chemistry, it also comes with some discomfort. My current belief is that we need to recognise that learning is both enriching and uncomfortable. This is the biological experience of learning.

One final example. I listened to Professor Tim Spector, on the Feel Better, Live More podcast, questioning how frequently children should be eating snacks. He talked about the benefits that can be gained from stabilising children’s insulin response from them not eating too regularly and how that can improve concentration. He talked about how Italian and French children have different patterns for eating within a day. Do we know the optimal approach in this regard for maximising learning and improving health?

I don’t claim to be an expert on any of these subjects. I don’t suggest that you implement any changes based on this blog post either: I’m not knowledgeable enough on any of these subjects to be a prominent voice. However, I think there is scope for improving learning by listening to the real experts in the respective fields. I’m sure that there will be so many other lessons that we can take away too.

I’d love to hear your opinions on any of these subjects. All comments are welcome: what you agree with, disagree with, other possible areas of interest and other sources of information that I can learn from. I pick up my messages on social media and my email address is iseemaths@hotmail.com. I can be slow to respond, but I read every message! I would love to hear your thoughts.

garethmetcalfe Maths ideas

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.