Promoting Reasoning Part 4: Depth

In this final blog in the ’embedding reasoning’ series, I am sharing some of my favourite strategies for deepening learning. I love the back end of a sequence of lessons, where you can build on children’s growing understanding. Rich conversations emerge and children can apply their skills with increasing flexibility.

Nowadays, there is an increased emphasis on looking for the different ways children might find an answer. I think this is great, assuming we have given children enough tools to find different strategies (and discern the most efficient). Initially I often provide some scaffolds to point children towards different methods (see below-left). I also love ‘rank by difficulty’ as a tool for generating discussion. It helps to draw out the different ways children approach questions and focuses their conversations on key learning points.

To deepen learning, I’m always thinking about ways of stripping back the information that is given within a question. I can always put extra information back in where needed (or specifically requested), but by starting with less I can often have a more open dialogue. Consider the below-right example: I can always add in the squares to the 100-square, or other numbers. But by starting with less information I have a more open discussion about the possible values in the red boxes.

Finally, I love using ‘how many ways?’ as my final question type. Children can access a how many ways task at level 1 (I can find a way) or level 2 (I can find different ways), but the step to be working at level 3 (I know how many ways there are) creates a different kind of challenge. We may have to model how to order thinking systematically as children strive to find all possible answers. Previously taught calculation skills are becoming automated and rich opportunities for reasoning emerge.

All the examples in this blog are from the ‘I See Reasoning’ eBook range:
I See Reasoning – KS1
I See Reasoning – LKS2
I See Reasoning – UKS2

Also check out the other blogs from the series. 

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Promoting Reasoning Part 3: Variation

This blog post, the next in the ‘promoting reasoning’ series, features question types that help children to build on their current knowledge and notice important similarities and differences between questions.

I’m always looking for ways to promote non-counting calculation (there are overlaps here with my ‘visuals’ blog). Prompts like the below-left example helps children to make connections between doubles and near-doubles facts. Children can edit the image to help them see those relationships. Similarly, I love using ‘I know… so…’ question strings. In the below-right example, I hope children will either perform the calculation using the related fact, or they will see the relationship between the three questions.

The examples below probably fit the criteria of ‘variation’ more tightly. Specifically, by keeping all but one aspect of a question/image the same, children’s attention is drawn to the key difference. Consider the below-right example: the left hand image the dominant idea is likely to be ‘one circle’; the right hand image emphasises ‘four quarters’ more. Used together, children’s attention is drawn to four quarters being the same as one whole.

When using the ‘I know… so…’ prompt, I might adjust the amount of variation between the examples depending on where children are up to in their learning. That’s all about knowing your children, the magic ingredient that every great teacher has up their sleeve!

All the examples in this blog are from the ‘I See Reasoning’ eBook range:
I See Reasoning – KS1
I See Reasoning – LKS2
I See Reasoning – UKS2

Also check out the other blogs from the series. 

Promoting Reasoning Part 2: Visual

This is the second blog post in a series looking at building reasoning within the maths curriculum. Here I’m looking at the using visuals in maths questions.

I’m always thinking about the best way to represent maths concepts practically to build understanding. The visual questions I use, therefore, are designed to correspond with the practical/visual models I have already used. The below-left example is one way that I represent the concept of ‘= means the same as’. The below-right example is a prompt that can be used to launch a practical investigation using matchsticks, developing the thought process of ‘how many [divisor] in [dividend]’ as children learn to do division by grouping.

By using questions that correspond directly with children’s practical experiences, the transition between concrete/pictorial to abstract is smoother. I might also use the visual to address common misconceptions (below-right example).

Below are two of my favourite types of visual reps questions to generate discussion. I like using ‘which picture’, where children have to consider which bar model represents a question correctly, and which bar model is showing a common error. Can children explain the mistake? I also love open-ended visuals: the bottom-right example is a particular favourite. More similar examples on this theme on the next blog!

All the examples in this blog are from the ‘I See Reasoning’ eBook range:
I See Reasoning – KS1
I See Reasoning – LKS2
I See Reasoning – UKS2

Also check out the other blogs from the series. 

Promoting Reasoning Part 1: Misconceptions

In this series of blog posts I wanted to share some of my ‘go to’ strategies for interweaving reasoning throughout the maths curriculum. This post is all about specifically addressing common misconceptions.

When planning a sequence of lessons, I tend to spend an eternity considering two things: how to represent each mathematical idea to build children’s conceptual understanding, and the possible misconceptions children may have. Initially I try to break learning down into small, easy-to-digest steps (more about this phase in future blogs).

Then, when I think the children are ready, I try to address those misconceptions directly. I want to focus thinking on the key points that discern right from ‘likely wrong’, deliberately highlighting common errors. Sometimes, like the example below, I might show three possibilities and ask ‘which answer?’. Alternatively, I might ask children to explain given mistakes.

I find that these questions generate great discussion and explanation. I use these examples at different points within a lesson: sometimes as a way of addressing errors from yesterday; often as a final task before children work independently; occasionally as a plenary (although only if very confident that the children will take the correct conclusion away). Having read ‘How I Wish I’d Taught Maths’ by Craig Barton I will start using a couple of them mid-lesson to assess children’s understanding and signpost pupils to appropriate follow-up activities.

Predicting those errors is very much a skill in itself, developed over years of experience. And the process of coming to understand children’s incorrect responses, I find fascinating. Hopefully this technique will help your children to focus on those key learning points, and solidify their conceptual understanding in a range of areas of mathematics.

All the examples in this blog are from the ‘I See Reasoning’ eBook range:
I See Reasoning – KS1
I See Reasoning – LKS2
I See Reasoning – UKS2

Also check out the other blogs from the series. 

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.

Chance favours the connected mind…

My vision is to help children experience maths – visual, deep maths – in all its richness. To this end, I’m passionate about creating a range of user-friendly resources that will help time-pressured teachers to deliver great maths lessons on a day-to-day basis. And to make these resources truly outstanding I need your help!

Future plans, your help
I’ve recently finished writing the range of I See Reasoning eBooks. Much to my amazement, over 800 people signed up to trial I See Reasoning – KS1. The feedback that people gave was extremely helpful. Over the next few years I will write many more resources, and I want to involve as many teachers as possible in the creation of these products (this time with trial materials being sent from the earliest conception of a product). You tell me which tasks really work, and how each idea could be better. I’m working on these ideas next:

  • A resource which represents maths concepts visually using a sequence of clearly constructed, step-by-step images. The first versions of this will be aimed at KS2.
  • A resource helping children to become effective problem-solvers. It will be made up of my favourite problem-solving tasks, broken down with the necessary scaffolds and supports that help children to see the underlying mathematical structure of each problem.
  • A resource for EYFS that is a follow-up to the I See Reasoning range, but delivered in a format more suited to an Early Years provision.

I’m after a team of people who will trial these ideas in their classrooms, let me know what they think of them and tell me how they can be improved. No strings attached, it’s all free, there’s no obligation to reply. The ideas will be plentiful and thought-provoking. You may well get lots of free, useful tasks that never end up being published!

If you are up for it sign up here. Please share this blog too, it would be great to get as many educators on board as possible. Thanks!

I See Reasoning – LKS2

I am delighted to announce that I See Reasoning – LKS2 is now on sale! It will arm teachers with a range of visual and thought-provoking tasks for interweaving reasoning within day-to-day maths lessons.

I See Reasoning – LKS2, the little sister of the hit resource I See Reasoning – UKS2, is a PDF file received as a digital download. It is comprised of 240 questions for deepening mathematical thinking and encouraging purposeful peer discussions. It is a go-to resource for prompts that build understanding and tasks that allow for extended investigations. This blog showcases what to expect from the resource by looking at three typical example questions.

Prompts that show concepts visually

There are a wide range of questions that use visual representations to help children to make connections and develop a conceptual understanding of core concepts. There are lots of ‘read the picture’ examples like the one above, ideas are often represented with bar models and many other images are used.

Prompts that generate discussion around key ideas

There are lots of examples that get children talking about key concepts and identify likely misconceptions. In this example, will children recognise that the size of the angle is represented by the amount of turn rather than the length of the lines? There are thought-provoking images from right across the LKS2 curriculum.

Opportunities for extended exploration

Many of the questions will ask children to calculate in different ways or find multiple solutions. This means challenge is added by getting children to explore the same type of question in more depth, working systematically and flexibly to find all possible answers.

It’s my belief that teachers generally agree on the principles of great maths teaching. However, time-pressured teachers need great resources at their fingertips. That’s what I attribute the unbelievable success of I See Reasoning – UKS2 to. It is ever-increasingly popular and has sold in 12 countries!

I have been so touched by the very many positive messages I have received in recent months. It’s my great pleasure to help teachers create meaningful and engaging maths lessons. I hope I See Reasoning – LKS2 is another piece of this jigsaw.

I See Reasoning – LKS2 is on sale via Etsy here. Information about the resource, plus the free multiplication section, can be accessed on this page.

For further information about training and resources, visit www.iseemaths.com