garethmetcalfe Maths discussion

Adapting ‘I-We-You’ to deepen mathematical thinking

The principles of ‘I-We-You’ are central to my maths teaching: give children the tools to think with clear models and examples; build understanding and address misconceptions in the guided practice; then children work with high success in their independent practice.

Issues can arise, though, if an ‘I-We-You’ approach leads to a more procedural approach to teaching. Mathematics is not about following a set of pre-determined instructions. I might need a prescriptive approach if you are teaching me to wire a plug. Step-by-step instruction can help me to learn how to do a column subtraction. But it isn’t how I can become a mathematician. And it won’t lead to me becoming emotionally invested in my mathematics. Maths lessons should allow children to play around with key ideas, become curious and make connections.

There is another big challenge in the ‘We’ phase of the lesson. How can we provide questions/tasks that build understanding and extend thinking for all children? If children are given a question to answer in the ‘We’, some children may finish almost immediately. Other children may need more time and more scaffolding.

How can we hold all these ideas together? Is it possible to provide high-quality modelling and engage children in effective guided practice, whilst allowing space for children to be curious and make their own discoveries? I believe we can! In the ‘I’, think about exploring concepts rather than modelling steps. For questions in the ‘We’, think about using the slow release of information (the subject of this blog) or using pairs of questions (the subject of a future blog) to engage all children in discussion. Consider the question below:

Following an ‘I-We-You’ approach could mean modelling two questions that are near-replicas of this one. Then, I could give this question to the children to answer on their whiteboards. What problems do I see? Firstly, some children will be able to answer this question within seconds. The rest of the class, then, feel a pressure to catch up. Speed is emphasised, the answer becomes the focus. Secondly, will the children be able to answer a different coordinates question when the ‘steps to success’ no longer apply?

Here is a subtly different approach. In the ‘I’ phase, start by giving example coordinates that are and are not on the green and orange lines, noting the x/y coordinate that stays the same:

Then, looking at page 1 of the example below, I ask children to predict what the hidden coordinate could be, before revealing (9,7) and noting that the y coordinate is still 7. Next, introduce the red line (page 3). Ask children to predict which other coordinates could be on the red line. I could compare the positions of (6,4) and (4,6). Are they inside or outside of the bottom-left rectangle?

My main focus for this blog is to look at the slow release of the information for the question below, in the ‘We’ phase. Rather than presenting the whole question to the children, start at page 1. Ask the children what the red coordinate could/could not be. Note that x > y and spatially reason about the different possible coordinates. This acts as our estimate. Then, introduce the rectangle from the question (page 2) and ask the children ‘what information will we be given that will mean we can answer the question?’ Explore all possibilities. In doing so, children have the time and space to think about the structure of the task before the question itself is introduced. The necessary information is slowly revealed (pages 3 and 4) and children can now give the answer having had time to think about the structure of the question. To extend, coordinates on the vertices, edges and inside/outside the rectangle can be given.

The power of the slow reveal is that children get to ‘play around’ with the big ideas before they have to give an answer. Children have more time to process the information, which is revealed in stages. Challenge exists as we consider ‘is that definitely what the missing information will be?’ Reasoning is emphasised and children use their imaginations!

Please share your thoughts, objections or related examples! Is there another maths curriculum area that you would like me to consider on a future blog? Or other similar example tasks that you could share? I hope that this blog can spark some interesting conversations and collaborations!

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 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 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 discussion

Mathematical Reasoning Routines

We all have a very limited attention: as you might be aware, children can’t think about many different things at once! So establishing routines that promote mathematical reasoning – routines that children become familiar with – will allow children’s attention to be focused on the key learning in the lesson. Thinking about these routines in advance can therefore be very important.

And so much better if these routines are consistent throughout the school. In Thinking Deeply About Primary Mathematics by Kieran Mackle, I loved Matt Swain’s routine for how children hold up their whiteboards. The children always hold their whiteboards to their chests; the teacher tells the children to put their boards down one table at a time. When children are familiar with routines like this, their attention isn’t wandering to ‘will Mr Swain see my answer?’ but is held on the content of the lesson.

Here are four routines that I think support learning in a primary maths classroom:

Pair work: short independent thinking slots
In pair work, I often ask children to start by working on a task individually before discussing with their partner. This promotes different methods/thought processes and lessens the risk of one partner becoming too dominant in a conversation. The length of time that I would expect children to work independently will increase as they get older, but it’s something I try to establish with all children. In most contexts, I’d have periods of silence when working independently – children find it more difficult to block out background noise than adults. I have found that these short periods of individual thinking make children value their collaboration time more.

Re-state the views of others
In group or whole class discussions, I generally try to spend longer drawing out the detailed thinking of a child or a small number of children. It’s important, though, that all children are actively thinking about what is being discussed. As a result, I routinely ask children to re-state the opinion of the person that has been speaking. This helps children to follow a conversation rather than just thinking about what they would like to say or to give their opinion. It also opens children up to different ways of thinking or different methods.

Doubt at the point of answer
I want children to focus on the process of their thinking and encourage them to reason. I don’t want children overly focused on whether answers are right or wrong. As a result, I tend to react with indifference when children give an answer. This gives children a reason to explain their thinking and it shows them that the thing I value is their thought process. Also, where a child has answered some questions and has made a few mistakes (but doesn’t hold a clear misconception) I often tell them how many questions they have got correctly/incorrect and ask them to find their mistakes. This gives the child more thinking to do than when the questions are marked and they simply correct their mistakes.

Consistency in question types
I like to have a consistent bank of question types, using common headings, throughout the maths curriculum. These common question types are woven throughout my I See Reasoning eBooks (this blog explains some of the Y3 & Y4 techniques and this blog explains about some of the Y5 & Y6 techniques). So when building understanding, children are used to being given an Explain the Mistakes task; they know that they will be asked to explain links between questions when answering Small Difference Questions and they have become used to working systematically when given a How Many Ways? challenge. By establishing these norms, we can focus more of the children’s attention to the maths content of the task, rather than having to explain how to approach each new technique. I hope the eBooks are super-useful for this!

I’m happy to host training events on Building Reasoning Routines and Building Problem-Solvers for the 2021-2022 school year and I’m working alongside teachers to implement these ideas in the classroom. Please get in touch by emailing iseemaths@hotmail.com if you are interested in receiving support in these areas. I will also keep sharing new resources for people to trial for those people signed up to my mailing list.

Also, please share your favourite school or classroom routines, however big or small. How do they create a positive learning culture? How do they help to direct children’s limited attention in a productive way? I’d love to pick up and share new ideas!

garethmetcalfe Maths discussion

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 iseemaths@hotmail.com. I’m looking forward to a lively, thought-provoking and important debate!

garethmetcalfe Maths discussion

I See Reasoning for Y3 and Y4: the big vision for deepening mathematical thinking!

I’m delighted to have released the eBooks I See Reasoning Y3 and I See Reasoning Y4. They are breakthrough resources for building conceptual understanding; for helping children to notice patterns and relationships; and for deepening challenges. They are comprehensive and user-friendly.

Free Sample: I See Reasoning Y3 Division and Multiplication and Division

Free Sample: I See Reasoning Y4 Division and Multiplication and Division

These eBooks are a big upgrade on I See Reasoning – LKS2. First of all, between them there are 872 questions in the two eBooks, compared to the 240 tasks in the original eBook, I See Reasoning – LKS2. In each section of the new eBooks, mathematical concepts are shown using different images and representations:

Common misconceptions are highlighted and addressed:

Then there are a range of questions for highlighting patterns, generating discussion and digging deeper. Can children see the relationships between the Small Difference Questions? And find all answers to How Many Ways tasks?

Each eBook costs £24.98 and only one copy is needed per school. I believe that this represents amazing value – hopefully it means that my resources can impact many children. In-depth online or in-person CPD on embedding reasoning within sequences of lessons can also be arranged. To receive updates on all future events and to receive free resources, join the I See Maths mailing list community. Also, here are the links for I See Reasoning Y5 and I See Reasoning Y6.

I hope the eBooks will inspire many children to enjoy deep, rich mathematical experiences and that they will give you many great classroom moments!

garethmetcalfe Maths discussion

Why I See Reasoning – Y5 and Y6 is new and unique!

I’m delighted to have  released the eBooks I See Reasoning – Y5 and I See Reasoning – Y6. They are an exciting development from anything I’ve done before and will enrich all children’s mathematical thinking. Here’s what makes them unique:

Detailed breakdown of small steps
For children to understand the individual parts of mathematical processes, I’ve introduced lots of new questions for breaking down learning into small pieces, focusing children’s thinking on specific points. For example, Next Step questions get children to analyse specific parts within calculations and Part-Complete Examples support children as they first learn to use methods. As ever, a range of misconceptions are addressed with Explain the Mistakes examples.

Opening up patterns and developing flexible thinking
There are lots of sequences of Small Difference Questions which highlight key mathematical relationships and give children surprises. For example, when children realise that different questions give the same answer, we can explore why. There are so many other patterns to uncover! There’s also a massive range of tasks that promote flexible thinking and using different strategies:

Explores big mathematical ideas (including word questions!) and allows children to create
Each topic is explored from a wide range of different angles. We look at different contexts for rounding; algebraic ideas are explored through shape puzzles; concepts are interleaved as children calculate angles between the hands of a clock at different times. There are ‘numberless’ word questions, where children explore different question structures without numbers, tasks where children are invited to create their own questions or extend sequences and How Many Ways? tasks to open up investigations!

Comprehensive
I See Reasoning – Y5 has 362 tasks and I See Reasoning – Y6 has 396 tasks, compared to the 176 tasks of the predecessor, I See Reasoning – UKS2. The tasks cover every area of the curriculum and they incorporate the ideas from the latest DfE Mathematical Guidance. And answers are given for every question!

The eBooks cost £24.98 each and only one copy of each eBook is needed per school. I believe this represents amazing value!

Click here to order I See Reasoning – Y5 and click here to order I See Reasoning – Y6.

I hope I See Reasoning makes a huge impact on your teaching and helps all children to think mathematically. Please spread the word!

My very best wishes to everyone for the new term,
Gareth

garethmetcalfe Maths discussion

Counters & bar models used to unpick a classic PS question

This question is taken from the Y3 Autumn term White Rose Progress Check assessment:

I’ve really enjoyed exploring this question type (although, I have to admit, never yet with children as young as Y3). I want children to see and feel the structure of this type of problem, building up to being able to answer a question like the example above in small steps. Then, by working through a series of related questions, children will learn how to use efficient problem-solving strategies. They will also come to see that questions with different ‘surface features’ can have a very similar mathematical ‘deep structure’.

To start with, using double-sided red/blue counters, children attempt the question below:

Often, children start with 8 counters – 4 red, 4 blue. Then, they turn over two blue counters. They realise (with a nudge) that the difference between the number of red/blue counters is incorrect. With a bit more cajoling, we see only one counter needed turning over. At this point I line the counters up above/below each other. I suggest, rather than starting with the correct number of counters, we could start with the correct difference. Have 2 more red counters than blue counters; keep adding a red & blue until you have 8 counters.

That technique, or other methods, are then be practised using the question below. We note that this question is worded slightly differently, but see that the red/blue counters can still be useful:
This time, many children start by laying out four blue counters. We note that ten more counters are needed (5 blue, 5 red). Other children get 14 counters and experiment with how many to turn over. We look at these different approaches. Then, I draw a bar model around the counters (like for the original example), drawing a dotted line to highlight the difference of 4 counters.

Now it’s time for a worked example and another ‘different surface, same deep structure’ question. In this case, I model how to answer the question using the ‘start from the difference’ technique:
Having shown that the difference between the prices is 10p, the cost of the rubber can be calculated by halving 30p (a common incorrect answer to this question is pencil=30p, rubber=10p).

Children then attempt questions that have a very similar structure, still regularly using the counters. Some children are given slightly extended challenges:
Here’s another lesson example of how to break down the problem-solving process. I See Problem-Solving – UKS2 is designed to give teachers the tools to teach problem-solving systematically too. Work will start on the LKS2 version in January 2019. I can’t wait!