How can Cuisenaire Rods help us to develop maths fluency?

They can of course be used to add and subtract numbers with children using counting strategies and number tracks. However, Cuisenaire Rods are a great resource to help expose the structure behind calculation strategies and support children to move on from being reliant on counting.

In this blog, we will explore how Cuisenaire Rods can support:

• understanding of reordering and adding complements to key benchmarks;
• using compensation when adding and subtracting;
• exploring the strategy of equal difference.

One top tip before we begin…

If children are unfamiliar with the value of each rod, lining them up with a ruler can be really useful! The videos embedded in this blog model this.

Reordering and complements to benchmarks

What is the easiest way to find the total of 3 + 5 + 7? 

Choosing an appropriate calculation strategy is all about making the maths easier for ourselves! In this case, using what we know about complements to 10 (a helpful benchmark) may be a useful strategy. The following video shows how we might model and expose this strategy.

Once the calculation strategy has been exposed and understood using 1-digit numbers, this can then be applied to calculations using larger numbers

So, in Year 2 (and subsequently), we could use this strategy to explore complements to 100. The green rod was worth 3 ones but let’s say it is now worth 3 tens or 30. The yellow rod is worth 5 tens and the black rod is worth 7 tens. 

30 + 50 + 70 could be reordered so that the complement to 100 is found (30 and 70) before adding the 50. Children should be encouraged to reason that: ‘I know that 7 tens and 3 tens is 10 tens or 100, and then I will add the other 5 tens’.

Each time we model, it would be worth reminding the children that we are using a strategy to make the maths easier because we are using our known facts.

In Year 4 (and subsequently), exploration can progress to fractions and decimals.

Finding complements and reordering could be a useful strategy for Year 4 pupils learning to add fractions with the same denominators.

In Year 5, we can apply this to adding decimal amounts. In these next two videos, rather than representing a one, the white Cuisenaire Rod is used to represent tenths and then fifths. 

Compensation (9 is near 10)

What is compensation?

When we add 9, we may think to ourselves, ‘9 is close to 10, so I can add 10 and then subtract 1’. We are changing the amount we have added (10 instead of 9) and so we need to compensate after adding 10 by subtracting 1.

‘I have added 10 which is too many. 9 is 1 less than 10 so I need to subtract 1.’

How can we use Cuisenaire Rods to expose this strategy? 

What about if we were to subtract?

Let’s take the calculation 15 – 9. Many children may be able to recall this as a known fact. However, this is a useful calculation to start with when wanting to expose the structure behind the strategy. The following images and language show how this could be modelled using Cuisenaire Rods: 

‘9 is 1 less than 10 so subtracting 9 is very close to subtracting 10. I can subtract 10 from 15’.

When modelling this, we could use the orange rod to represent the 10. Then replace this with the blue and white rods to show that 10 is also represented by 9 and 1. 

‘15 subtract 10 is 5. I can use my place value knowledge for this. 1 ten and 5 ones subtract 1 ten, leaves 5 ones’.

‘I have subtracted 10 which is 1 more than 9 so I need to add 1 back on. 5 + 1 = 6’.

’15 – 9 is equal to 15 – 10 + 1.
15 – 9 equals 6’. 

Once the calculation strategy is exposed and understood, children can explore more complex calculations. For example, using compensation to calculate 3562 – 198 could look and sound like this:

‘Subtracting 198 is like subtracting 200 and then adding 2 back on.

3562 – 200 is 3362. 3362 + 2 is 3364. Therefore, 3562 – 198 = 3364’

Exploring equal difference

If we needed to subtract 9 from 17, perhaps it would be easier to represent the calculation as 18 – 10.

How does this work?

To make the calculation easier, we have added 1 to both numbers and this maintains the same difference. The difference between 17 and 9 is equal to the difference between 18 and 10. This is called ‘equal difference’ and is an alternative method to compensation.

In the compensation strategy, we adjust after we have performed the subtraction. Whereas in equal difference, we adjust first, and then perform the subtraction. 

It is important children understand that if we add or subtract the same amount from both numbers (the minuend and the subtrahend) then the difference remains the same.

In the example here, by adding 2 to both 28 and 13 we get 30 – 15 which the children may be able to use known number facts to solve. Alternatively, we could subtract 3 from both numbers to get 25 – 10.

This is about finding more manageable ways of solving the calculation and that choice may be different for different children.

The following video shows how equal difference could be explored using Cuisenaire Rods before this knowledge is then applied to calculations. 

Now that we have explored these strategies, are there other strategies that you could use Cuisenaire Rods to explore? I wonder what near doubles would look like with Cuisenaire Rods? Or ‘Think 10’?

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As educators, we are very familiar with the term ‘feedback’. Its plays a vital role in our own CPD and providing feedback – in all its forms - takes up a large proportion of our time. Feedback can be transformational. We know that “providing feedback is well-evidenced and has a high impact on learning outcomes,” (Education Endowment Foundation (EEF), T&L Toolkit but we also recognise that when “Done badly, it can even harm progress.” (EEF, Teacher Feedback to Improve Learning)

Before we explore forms of effective feedback in the writing classroom, let’s remind ourselves of the purpose of providing feedback: to move learning forward. This blog by the Education Endowment Foundation (EEF), provides a series of recommendations to support teachers and leaders to ensure that pupils are making effective use of the feedback they receive. As Dylan Wiliam states, “no matter how well the feedback is designed, if students do not use the feedback to move their own learning forward, it’s a waste of time. It is essential that the feedback is productive.” (Helping or Hindering? 2014)

In classrooms where writing is valued, there can be a beautiful synergy between the joy of writing and the art of providing reflective, motivating feedback. When used thoughtfully and constructively, this feedback has the power to inspire, motivate, and guide young writers, contributing to improved self-esteem and the development of writer identities.

The joy of self-expression

“Writing is something that is both personal and intensely social, both analytical and emotive." (Young & Ferguson 2021)

If writing is a form of communication and expression, then it is our job as educators to ensure that our young writers feel safe and motivated to leave a little piece of themselves on their page. The art of providing feedback to writers is being able to mentor and guide them to make improvements, whilst allowing them to maintain agency and autonomy so that they can still feel a sense of ownership and responsibility. Writing is a powerful human act, and it is imperative that our feedback remains human too – whatever form it is delivered in.

Seeking feedback in supportive settings

Feedback is most valuable when it provides actionable suggestions for improvement. Instead of simply pointing out errors, offer guidance on how to address them and provide a purpose for this.

We can begin by shifting the spotlight from just the final product to the entire writing process. Acknowledge the effort put into generating ideas, planning, drafting and revising. Highlight the value of revision as a tool for improvement, reinforcing the idea that writing is an evolving craft that requires time and dedication. Use feedback as an art that shapes the budding authors of tomorrow.

Do children know that real authors add, remove and substitute words and full sentences all of the time, and often as a result of feedback from others? Do they see making improvements as a positive step to improving written outcomes and readability for a real reader? And as a result of all of this, do they seek feedback themselves?

In all its forms, feedback should be delivered by a trusted source, with ample time and opportunity to respond and use the feedback provided.

“I’m ready for my feedback!”

“Feedback is what happens second, after high-quality teaching and careful selection of assessment tasks that reveal how well students have understood the learning.” (Dylan Wiliam, 2021)

High quality instruction and effective formative assessment will reduce the work that feedback needs to do. First, it may be worth evaluating the impact of your writing curriculum. This blog on what makes an effective writing curriculum will help you to make further considerations.

Let’s consider how we can use timely and specific feedback for writing in the classroom:

For you, my reader

When writers have a clear goal in mind, it enables them to reflect on their intent and effect throughout the writing process. Do children know who they are writing for? Do they have their audience in mind throughout?

When the answer is yes, this can enhance engagement and motivation hugely. Whether it’s a friend, teacher, parents, or a broader audience, having a specific audience in mind gives young writers a sense of direction. The feedback we provide should also relate to this notion.

Evidence shows that feedback is less effective when it is about the person. So instead of using generic comments such as, “fantastic writing!” or “you’re a natural!”, let’s relate this feedback to the impact the writing has on their reader. Be specific about what you appreciate in their writing. Highlight specific sentences, phrases, or details that stand out. For instance, you might highlight a specific area and say, “I love this vivid description of the setting in your story—it really transported me there,” and give specific examples.
When improvements or adaptations are required, provide a purpose here, too. For example, if the child wrote:
 

Always start with a specific piece of positive praise – “I love how you have thought about giving your reader information about how the dog ran and where it was running to. This is essential.”

Then, provide one clear suggestion that guides them to improve further, with their reader in mind:

“How could we develop this even further to help our reader understand the reasons for the dog’s urgency and speed?" Or “Which words - verbs/noun choices - could be tweaked here to help the reader understand and empathise more?”
 

“Fantastic! Now your reader will really understand the dog’s reason for sprinting. We understand him more and can relate to his excitement.”

Keep reading for more about this ‘conferencing’ approach and other feedback approaches.

We’re live in 3, 2, 1…

Live marking has recently been deemed by many as the more superior form of ‘marking’. Whether this feedback is delivered as a whole class, with target groups or individuals, in written form or verbal, it can allow us to address misconceptions as they arise and adapt our teaching as necessary.  In addition to reducing teacher workload, live marking or ‘fast feedback’ is considered a useful tool for quick wins. However, the EEF advises that “teachers should judge whether more immediate or delayed feedback is required (EEF Feedback Recommendations, 2021) to ensure that feedback is personalised and impactful”.

Live oral feedback can allow pupils to make impactful changes or reflect deeply as they write in the moment.

Many schools implement pit stops/mini plenaries within whole-class feedback to provide opportunities for whole-class improvement. Here, children can check and ‘live mark’ themselves during the writing process.

On the contrary, delayed feedback might provide pupils with time away from their writing. Returning at a later point may help them to evaluate their initial choices more effectively.

Continue to consider the timing of your feedback and when it is most impactful for individual learners. Ask yourself: Is it moving learning forward? Is it addressing the learning gaps that your pupils have?
Is it serving the needs of struggling writers?

Conferencing 

Conferencing can be used within, or away from, ‘live marking’ in the moment. It provides a short and supportive moment for teacher-pupil dialogue. So, what might a ‘conference’ look like? 

1. First, put the writer at ease. Start with simple, open-ended questions that prompt effective dialogue.
   “How’s it going?”  “Tell me about your writing so far!” “Is there anything you’d like to read aloud?” “Anything I can help with?”
2. Encourage them to read their writing aloud. You can then ask more specific questions to prompt the writer to think about the impact on their reader. “What did you want your reader to imagine/feel here exactly?” “I felt quite fearful reading this bit. Was that your intention? How could we add to this emotion?”
3. Using the knowledge gained from the discussion, provide one piece of really clear feedback - based on the success criteria - for them to implement. 
4. Make them feel proud and successful – be specific and human throughout.

If pupils require further support (perhaps with the transcriptional elements of their writing) this can then inform your assessment and future feedback or support. 

Read more about conferencing over at the Writing for Pleasure Centre.

Written feedback

Whether in marks, scores, comments or through use of tools such as sticky notes and speech bubbles, its success will vary depending on when and how it is delivered and of course, when children have time to act on it. Remember: is it timely? is it specific? Read more about this in the EEF Guidance Report.

Here are some impactful examples of written feedback that you may wish to try with your writers:

“You can’t edit a blank page”

True – but a blank page can certainly be useful for providing feedback during the drafting process. Some teachers find that leaving a blank page next to pupils’ writing is beneficial for recording personal notes and peer/teacher feedback. This is particularly useful if you didn’t have time to feed back verbally during live marking and/or want to provide some focused suggestions in the form of sticky notes or when you are away from the child. Remember to follow this up with dialogue and give the writer time to respond.

Build independence with signpost marking 

A simple arrow, highlighted area or asterisks in the margin can point children in the right direction to areas that may need a further look. Again, time and further dialogue may be needed here too.

Box marking 

A simple highlighted box around a small, focused area within longer pieces of writing can be used to provide specific and focused feedback during live marking moments or even conferencing. Within this box, you may guide writers to make compositional or transcriptional improvements. Using the knowledge gained from this focused feedback will support them when editing or proofreading the rest of their piece independently or with a partner.

Peer perspectives

Foster a collaborative writing community by incorporating peer feedback into the process. Encourage students to share their work with classmates, promoting a culture of constructive critique. This not only provides additional perspectives but also cultivates a supportive atmosphere where students learn from each other's strengths and areas for improvement. However, this needs to be planned carefully within supportive environments, so children feel safe and successful taking part.

Like any form of teaching, children need to be explicitly taught how before having some time to practise. Prepare them by:

• Walking them through it: model what effective peer discussion can look like and help them to understand the purpose of successful collaborative dialogue 
• Getting them reading their writing aloud to their friend
• Maintaining agency so learners see feedback as providing ‘suggestions’ for correction or improvement
• Providing question prompts that allow their peers to elaborate on their thinking and justify their choices
• Using speaking frames to guide their conversations: focus on specific praise and a specific area for improvement (linking back to the effect on the reader)
• Being positive and specific!
• Allowing the writer themselves to make any necessary amendments – there’s no scarier feeling than seeing someone else’s red pen heading towards your precious writing
• Making use of resources such as sticky notes if written feedback plays a part in peer responses
• Trying working in groups of 3 or 4 instead of sticking to pairs
• Providing ample opportunity for reflection and evaluation.

It is important to note that peer feedback is not a replacement for teacher conversations. Feedback should be inclusive and immersive and when children know that it is a supportive component to improvement, this will hopefully remove the fear factor and put them at ease having these discussions.

Explore the EEF’s teaching and learning toolkit to read more about how other high quality, carefully planned interactions between pupils can contribute to success.

Read about how this Year 6 class sought and received feedback from a range of esteemed experts. (The Writing for Pleasure Centre, 2022)

Pupil voice 

A good place to begin with developing feedback for writing in your school could be by conducting a pupil voice questionnaire. Ask your young writers, “what do your friends think about your writing?” and “how do you know?” or “tell me about a piece of writing that your friend wrote.” This will give you a real sense of whether peer feedback is actually happening, if it is happening effectively within an authentic environment and whether children see themselves as real writers.

Showcase their work

Celebrate the achievements of young writers by showcasing their work. Whether it's through classroom displays, school publications, or sharing with their readers, giving young writers a platform to share their creations boosts their confidence and reinforces the value of their efforts.

In the world of writing instruction, feedback is not about correction - it's a conversation that shapes the next chapter of a student's literary journey. By celebrating strengths, providing specific guidance, and fostering a growth mindset, teachers can inspire a love for writing and empower their students to become confident and skilled authors.

This is the first blog in our new series, ‘Inspire Me!’

In this series, we will take a manipulative and exemplify different mathematical concepts that you could explore using the concrete resource in your classroom.

The first resource that we will be focusing on is Cuisenaire Rods.

Not used Cuisenaire Rods before?

Have used them before and want to expand your repertoire?

Keen to challenge your confident learners through exploration of an alternative manipulative?

Invented in 1945 by a Belgian teacher, Georges Cuisenaire, there are 10 rods of different lengths and colours.

The smallest rod (the white rod) is 1cm in length and the longest rod (the orange rod) is 10cm in length. Other coloured rods measure between 2cm and 9cm.

Let them play!

We would suggest that when introducing your children to Cuisenaire Rods for the first time, you let the children ‘play’ and build with them.

Ask them to build a picture with them. Then give them some lines of enquiry.

If the white rod represents 1, what is the picture worth?

If the white rod is worth 2, what is the picture worth?

What if the white rod was worth 4?

Already we are exploring scaling and multiplicative relationships!

Cuisenaire Rods are a hugely versatile manipulative so let us inspire you!

Why use manipulatives?

Before I delve into how to use Cuisenaire Rods to explore the concept of algebra, let’s revisit the fundamentals for teaching concepts in mathematics.

In their ‘Improving Mathematics’ guidance reports, the EEF recognise that manipulatives and mathematical representations play an important role in developing mathematical understanding from Early Years to KS3 (1,2).

When selecting a manipulative, it’s important to ask yourself:

• How is this tool supporting conceptual understanding?
• Is this the most appropriate tool to scaffold learning?
• How will I enable students to work independently of the manipulatives over time?

Inspire me! Introduction to algebra

Fundamentally, algebra involves manipulating number to identify patterns and is described in the oxford dictionary as: “a type of mathematics in which letters and symbols are used to represent quantities”. 

Children’s experience of algebra begins in the early years where they begin describing and noticing patterns. Pre-requisite understanding builds through part-whole exploration in key stage 1 and beyond.

Step 1: Play with the Cuisenaire!

With no numbers involved (yet), allow children time to revisit and secure their part-whole understanding before gradually making connections with the abstract form of expressions.

Provide children with a set of coloured rods and ask them to find combinations which are the same length as the brown rod.

For example: 

When modelling the discovery of different combinations, share your internal monologue.

For example: “I know that two purple rods is equal to one brown rod. Purple + Purple = Brown.”

Provide children with time to explore other equalities with the rods using the following speaking frames to support them in making connections:

Step 2: make connections

Use the part-whole relationships the children have discovered between the colours to make visual connections with numeric values and represent ‘unknowns’ with letters.

Revisit complements to 10 or 100 with the rods. Give the whole a value.

For example:

Model how to represent the ‘unknowns’ as letters rather than use their colour names.

p + p = 10 or 2p = 10

g + y = 100

The children can use the same, familiar speaking frame to support them to make connections.

At this stage, children are still exploring and looking at representing the unknown rather than finding it.

Step 3:

Once the children have grown confident and familiar with the Cuisenaire model, we can begin supporting them in making links to a familiar pictorial representation - the bar model.

For example:

Which bar models could be correct? Which could not?

Extend the speaking frame from step 1:

Explore this further using expressions with an unknown, for example,  x + 30 = 100

Step 4:

Provide children with an equation with one unknown.

For example:

x + 24 = 100

Children could represent this in a bar model.

Ask the children: If x + 24 = 100, what else do you know?

Explore creating a fact family with Cuisenaire Rods and bar models, labelling each part and the whole.

Application

Challenge children to apply what they know to a problem like the one below from the 2019 KS2 SATs paper 3 (reasoning):

Keep an eye out for more blogs from the Inspire Me! series this term. To subscribe to our blogs: Get our blogs straight to your inbox

Join the HFL Education Primary Maths Team for upcoming training and events

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References

1 – Improving Mathematics in the Early Years and Key Stage 1, https://educationendowmentfoundation.org.uk/education-evidence/guidance…

2 - Improving Mathematics in Key Stages 2 and 3

https://educationendowmentfoundation.org.uk/education-evidence/guidance…

3 - 2019 Key Stage 2 mathematics paper 3: reasoning

Contains material developed by the Standards and Testing Agency for 2019 national curriculum assessments and licensed under Open Government Licence v3.0.

In many of my recent visits, I have worked with practitioners to discuss the development of communication and language: the need to improve serve and return interactions, the importance of developing opportunities for children to talk and to be heard and the growing need for speech and language therapist support. Leading the Terrific Talkers School Project for two years now, I find that many of my conversations look towards the development of language, however, there are times when these ‘developmental practice procedures’ are not specific enough in accelerating progress for children where speech is impacted. 

For this blog, I will use definitions from the Oxford Dictionary:

Speech: Speech is the expression of ideas and thoughts by means of articulate vocal sounds, or the faculty of thus expressing ideas and thoughts.

Language: the principal method of human communication, consisting of words used in a structured and conventional way and conveyed by speech, writing, or gesture. 

In a recent progress and attainment discussion with a group of teachers, we identified less than expected progress for a child experiencing communication and language as a barrier to learning. Practitioner knowledge, built on observation and interactions, indicated that pronunciation of some speech sounds, specifically ‘s’ was impacting the child in communicating his knowledge. The child’s key person explained that in a familiar context with specific resources the child could be understood and their understanding of serve and return interactions secure, however, when observed later that day, in the outdoor area immersed in the awe and wonder of his imagination the language used by the child was much harder to understand. Referrals to SALT services have been made and the child is one of many on a waiting list to be seen- but what can we do in the meantime? How can we best support the child to make progress? 

Doing something is always better than doing nothing

Considering what we do in the meantime depends on the child’s identified needs.

There are so many complex folds to the development of communication and language, including:   

Exploring which aspect is presenting a barrier is the key to identifying your next steps.

• Have you considered using Speech and Language UK's Progress checker? 

This can help you identify whether there is a speech concern and best practice would suggest completing this with the child’s parents/carers to get a holistic understanding of the child’s speech development. Completion of this tool can then be used to support the identification of specific barriers.

• You might also use the Oxford NHS speech sound development chart for further guidance. Completion of this tool can then be used to support identification of specific barriers.

So, I have this knowledge now what?  

Do you have any communication and language programmes running in your school or setting and if so, are they targeted to meet the identified need, in this case, to support pronunciation of the speech sound s? 

If you have experience in speech and language therapy, you might quickly recognise lots of opportunities provided for repetitive practice and games. Here are a few you might like to try in your provision. Identify which sounds the child requires support with and collect a bank of objects and pictures to play these games:

• Sound spotter game -Using a few objects/images beginning with different initial sounds. Say a range of sounds and ask children to identify and say the word beginning with the identified sound (s is for sock)
• Hidden sounds- use a feely bag with a few objects beginning with an identified sound. Ask the child to choose an object from the bag for the adult to say, progressing to the child saying the chosen objects

Sock sounds – place a selection of pictures/objects on the floor. Using a rolled-up sock/beanbag ask the child to throw the sock at one of the objects and say the word

Accessible websites for further guidance:

• The Oxford NHS speech sound development chart provides further guidance to support practitioners working with children. 
• Speech and language kids to provide advice on the correct teeth and mouth placement to support the production of speech sounds: One example from this website is: “For the S sound, have the child put his tongue in the same place as /t/ (use the placement cues from /t/) but then blow air out.  We call this the snake sound!  The child should blow air over the top of the tongue.  The sides of the tongue should be pushed against the upper back teeth.”            

For more information about the Terrific Talkers project or support for developing a communication friendly environment please get in touch. 

In the summer term of 2021, the Herts for Learning maths team had the pleasure of working with over 130 schools across Hertfordshire and beyond as part of our Summer Success in Maths Project.

Diagnostic assessments were used to identify gaps in learning, informing how teachers would plan to teach and secure specific prerequisite skills and knowledge to prepare children for their new year group in the autumn term.

Data shared within the project provided a good oversight of strengths and weaknesses across the participating schools. More details about the findings from the project are shared in the blog, ‘Coming back stronger in mathematics’.

An interesting aspect stood out for us when analysing the data from Year 5 and Year 6 children – the children’s use of division strategies. Of course, this blog is not aimed solely at these year groups – strategies for division begin before UKS2.

So… to division.

Division is a concept which often stirs maths anxiety into both adults and children. Yet, as Rachel Rayner points out in her blog, ‘Divide and conquer: Do we share or do we group?’ , division is perhaps given less space in the maths curriculum than addition, subtraction and multiplication.

Let me put you on the spot (sorry!). How would you solve the following division calculations? If appropriate, how might the children in your class tackle them?

• 84 ÷ 6
• 1872 ÷ 6

I wonder if you used formal calculation strategies as below, where short division has been used to find the quotient in each calculation.

Or perhaps you used an informal method by using mental methods and perhaps jottings?

In a previous blog, ‘Old maths vs new maths – the balancing act’, we explored how using ‘newer’ methods may make calculations easier and more manageable – and we will explore this idea again here.

Let’s take the first calculation – 84 divided by 6.

Is using short division the most efficient method for this calculation? If a Year 4, Year 5 or Year 6 pupil used this method, it would be worth considering how secure pupils are with connecting their knowledge of multiplication and multiples to division.

You might consider the following:

• do pupils look closely at the numbers involved before deciding on a calculation strategy?
• how confident are pupils with regrouping numbers flexibly and can they see the usefulness of this strategy for this calculation?
• are pupils really secure with the value of the digits in the number or are they following a procedure to solve the calculation?

It could be argued that the most efficient way to tackle this calculation is to look for the multiples of 6 within 84. We might be hoping to hear or see the following from pupils:

• I know that it is greater than 10 groups of 6 as 10 groups of 6 is 60. 60 and 24 is equal to 84
• we could regroup 84 into useful multiples of 6
• 60 is a multiple of 6 and so is 24

Within the Year 5 and Year 6 diagnostics, we wanted to assess whether pupils understood that regrouping numbers to divide is a useful strategy. The results made for interesting reading.

In the early summer term Year 5 paper, pupils were given the following question:

When dividing 98 by 7, which would be the most helpful way to regroup 98 to aid the calculation?

Can you guess which answer might have been the most popular choice?

Only 30% identified that the most useful regroup to divide 98 by 7 would be 70 and 28 – 70 divided by 7 and 28 divided by 7.

42%, however, chose response A.

20% chose C.

4% chose D.

4% did not respond.

This was out of 1,253 pupils across Hertfordshire schools.

Of course, all the part whole models show regroups of 98, and although answer C is regrouped into multiples of 7, they are arguably less helpful than the regroups of 70 and 28 which are likely to allow known facts to be used.

In the end of term Year 6 paper, pupils tackled a similar question:

Nadia is dividing by 9 using a mental method.

Which of these regroups for 297 would be most useful?

Only 29% of children (of the 566 in this data set) accurately identified that 270 and 27 were the most useful regroups of 297 in this case.

Again, the most popular response was answer A, with 36% of the children regrouping into hundreds and tens and ones without recognising that neither 200 nor 97 are multiples of 9.

Pupil scores showed more confidence with the following 2 questions:

72% of 1,253 Year 5s correctly identified 204 as the quotient.

Out of 1,054 year 6 pupils, 78% correctly identified 903 as the quotient.

It would be interesting to know how many children reverted to a formal written method to solve these questions rather than really looking at the numbers within each dividend.

Much like the calculation I posed towards the start of the blog, 1872 ÷ 6, the numbers within the calculation have been carefully chosen. Within these dividends are common multiples of the divisors. What we are aiming for here is securing understanding of regrouping as well as the use of base facts and place holders if required.

For example, for 816 ÷ 4:  

Again, a written method is not necessarily needed but may be the default method for many of our children and these can be prone to error if children lack true understanding of ‘the procedure’.

Regrouping for division may not be such a ‘new’ calculation method. Let’s consider the ‘chunking’ method of division – chunking off multiples of 10 and subtracting these from the dividend until a number less than a multiple of 10 was left. Really, this is what we are doing when we are regrouping but this is just much more efficient.

So how could we expose and encourage this strategy with pupils in KS2?

Here is a possible suggestion of small steps. The examples could be adapted for use in lower key stage 2.

• Begin by asking pupils to regroup a number in multiple ways using base-10 or place value counters. Pupils could be encouraged to represent regroups pictorially through part whole models.

• Following this exploration, the calculation 105 ÷ 3 is introduced. The multiple ways of regrouping 105 can then be revisited asking pupils which would be most useful for the calculation and why. Pupils may respond with the following:
  • ’90 and 15 are both multiples of 3.’
  • ‘ I know that 90 is a multiple of 3 because 9 is a multiple of 3. 90 is ten times greater than 9’
  • ’30, 30, 30 and 15 could be useful regroups as they are multiples of 3.’

All contributions are valuable and can then be further discussed.

• A recording scaffold such as the following example can then be used for pupils to explore calculations and question prompts such as:

• explore further examples with pupils choosing their most useful regroups
• useful regroups could be provided with pupils asked to use the given information to solve the calculation such as in this destination question:

The data collected in the summer doesn’t tell us that pupils cannot regroup numbers but perhaps it suggests that they are less confident with flexible regrouping (this learning starts in KS1) and with the knowledge of why we might want to use regrouping as a strategy for division – to make the calculation easier and more manageable.

Perhaps children can get a little hung up on the apparent sophistication of ‘complex’ methods when, why wouldn’t we want to make things easier for ourselves!

Moreover, by truly understanding and being able to manipulate the numbers in front of us - by creating useful regroups if appropriate for example – is actually perhaps a more sophisticated way of solving calculations.

References:

Questions and models from ESSENTIALmaths.