Bar modelling as a powerful maths tool

Bar modelling is an often-misunderstood tool in mathematics. Rather than a strategy to solve problems, it is more a tool to expose ‘the maths’. This is explored by Charlie Harber in her 2018 blog, KS2 Maths SATs Papers: the power of the bar model. 

The bar model, when used consistently, can support children in tackling complex problems by enabling them to identify the knowns, unknowns and calculations.

Bar models are pictorial representations used to draw out parts and wholes to support with visualisation of a maths problem or scenario. Once the structures of additive and multiplicative models are understood, they can make something that seems complicated feel simple!

Developing a secure understanding of bar models starts in the Early Years and throughout Key Stage 1 where the foundations of part-whole relationships are explored.

In Key Stage 1 and Lower Key Stage 2, children should be given opportunities to explore both additive and multiplicative models.

Additive models will include structures of:

• aggregation – where two parts are combined
• augmentation – where one quantity is increased by an amount
• take away – where one quantity is decreased by an amount
• comparison – finding the difference between amounts

Key multiplicative models will include: 

• multiplication – where the group size and group number are known
• sharing – where the product and number of groups is known
• grouping – where the product and group size is known

If these key models are explored and understood, children can combine structures when tackling more complex worded scenarios as exemplified in the following questions from reasoning paper 2 from the 2024 Year 6 SATs.

Of the Year 6 maths SATs papers, children often find the reasoning papers the most challenging, as their ability to unpick the maths and find a starting point is challenged, as well as their maths subject knowledge.

Exploring Year 6 maths questions – structuring thinking and building confidence

Below, I have explored bar modelling with three Year 6 maths questions from the 2024 Reasoning Paper 2, and the dialogue I would use with children to support them.

Note that the language of part and whole is used in all the questions and is a concept built upon from the Early Years. 

Question 11 – bar modelling to expose the need for multiplication and division: 

Solutions

The missing ‘number of children’ = 24

The missing ‘number of adults’ = 5

Question 17 – bar modelling to expose the need for addition and subtraction: 

Solution

The price of the butter is £2.50

In this case, the bar model also helps to identify that the weights provided are distractors and are not needed to solve the problem.

Question 19 – bar modelling to expose the need for addition and subtraction: 

Solution

330 seats were filled between 7pm and 8pm.

If you enjoyed this blog, keep an eye out for part 2, where my colleague will explore questions from reasoning paper 3 from the Year 6 SATs 2024.

Further professional development

Our popular training is back this year in a new format.

Across four half-day sessions, we unpick tools for supporting children in meeting age-related expectations in maths using a range of different strategies, including the bar model!

Statutory assessment outcomes from schools who participated in this training in 2023/24 show that children made, on average, 6 months more progress than their 2022/23 cohorts.

Resources you may be interested in

Bar modelling in maths progression

• Download sample pages 
• A combination of photos of concrete resources, pictorial bar models and abstract calculations brought together to exemplify how to solve problems from across the maths curriculum.

Year 6 maths gap finder: SATS preparation toolkit

• Diagnostic summative assessment papers for early identification of gaps
• Resources to support teaching and rehearsal of arithmetic and reasoning

Year 6 SATS analysis toolkit

• a suite of resources and analysis tools to identify specific areas of learning strength and development for pupils and classes when using any past SATs papers as practice (2016-2024).

References

Contains material developed by the Standards and Testing Agency for 2024 national curriculum assessments and licensed under Open Government Licence v3.0.

Can you believe the first Phonics Screening Check was administered to Year One children in 2012? Those pupils are now around 18 years old. I wonder how those who didn’t meet the threshold that first year are now doing…? We are all a little older, wiser and perhaps greyer. But what have we learnt about:

• what comes up most frequently in phonics checks?
• what do we prioritise for teaching the next cohort of year one children?
• what reliable information does this snapshot test give us?
• how can we best close gaps for any children who have not yet met the minimum threshold in the check?
• what’s the best, most inclusive, form of acceleration for preventing gaps, especially for disadvantaged and vulnerable children (including those with SEND)? 

Those are indeed some thought-provoking questions, but ones we know we all need to probe. The link has sadly been proven time and time again between economically disadvantaged and vulnerable groups and lack of fluent word-level skills, ie accurate and speedy recall of the grapheme-phoneme correspondences, exception word knowledge and blending to automaticity… and especially so in terms of taking their spelling to fluency when writing. It does seem to be an almost accepted truth – if you are economically disadvantaged and/or vulnerable, or have SEND, then you are bound to struggle with spelling. And yes, English spelling has many, shall we say, foibles…historical footprints, if you will. But…our ‘hardest-to-reach’ children desperately need us to focus our attentions on areas of curriculum that are the hardest-to-teach. If we focus first on the hard parts, with our freshest energies and sharpest thinking, the rest tends to fall into place. So, the same can be said for spelling (encoding) and reading (decoding). Research (see reference list below) shows that the greatest impact on both word reading and spelling comes from a sharp focus on spelling.   In the words of a 7-year-old boy attracting PPG I met a few years ago, when I asked why he was practising some spelling during reading time: 

“If I can spell the word then of course I can read it!”

I was firmly put back in my box.

And blown away.

So, here are my thoughts and yes, a few dips into some of those hard-to-teach areas and hard-to-ask questions.

What does the Year One Phonics Screening Check tell us about children who have scored 32 (minimum pass-mark)? 

They may only be secure with Year 1 Term 1 learning (as measured against majority of the DfE Validated Phonics Schemes). It’s completely possible to pick up 32 marks with only Y1T1 learning secured, and remember this is not even a test of fluency, application amongst increased cognitive load such as reading a book, exception word-reading or even spelling. This is purely a health check of (a) can this child retain a core of survivability GPCs and (b) can they, in their own time – and possibly via heavy self-scaffolding such as sound-buttoning – blend the sounds together accurately. It’s effectively looking to see whether a child has learnt a Reception year’s worth of phonics over TWO YEARS.

Did children need to be secure with split vowel digraphs to pass? 

No. It was entirely possible to pick up 32 marks without them. This should ideally form the first block of teaching in Year 2, through the multisensory medium of spelling, to ensure no one is left behind. Split vowel digraphs are notoriously tricky for many children with difficulties in literacy, as are the following items in the table. The most reliable and memorable way to teach these for reading is to do so through teaching of spelling. Manipulation of concrete resources and graphemes (Sermier, 2021, as cited in DfE’s Reading Framework 2023, pp77, re children with ‘intellectual disability’; Labat et al., 2015; Lemons et al., 2018), writing by hand that helps light up the reading bits of the brain (Van der Weel, 2024; Ihara, 2021; Askvik, 2020) and a healthy dollop of self-regulation and metacognition (EEF) in the form of proofreading and self-correcting, all help create the sturdiest possible orthographic representations in the memory. This has now been shown, in just the past few years, to also be true of those with autism and Down syndrome, which indicates how rapidly the research in this area is evolving.

What other challenges may have tripped up the children who struggled most?

Words that look visually similar to another known word (key to notation below: what the child might offer / what the target word actually was > followed by examples of next steps):

• nob/nop; blink/blenk; scrub/scrup > possibly a speech and language issue; use mirrors together with multisensory approach to GPC revision; use a sound phone that amplifies own enunciation into own ear (cf ‘explicit instruction’, EEF’s SEND 5-a-day)
• my/yim; lid/ild; isn’t/sint; spilt/split > needs support with consistent left-to-right eye tracking; overlearning, making eyes, finger and voice all match at same time (eg ‘stroke the word’)   
• shed/sheb; drop/brop; dew/bew > deal with b/d separately! Don’t teach confusable words / GPCs together – teach for difference. Overlearn and become truly fluent with one confusable, get it really automatic and sturdy in the memory, then brain is able to rule an option out and narrow down possibilities. Get one ‘really there’, then deal with the ‘nearly there’.

Were there any shifts in trends of test design in 2024?

• Rise in occurrence of ‘ball and stick’ letters (ie b, d, p, q) – security in these has a correlation with mileage of ‘just right’, or fluency level, text reading. This could be a way of checking for this. Or just a shift towards checking that prior learning is very much secure.
• Slight increase in consonant digraphs. Tiny shifts each year, but worth bearing in mind.
• BIG drop in occurrence of vowel digraphs, from 18 to 20 to 13 this year. This suggests a real shift back towards checking on children’s prior learning. This could also be supported by the most commonly occurring target item being ‘ball and stick’ letters and adjacent consonants, and until this year also vowel digraphs.
  
  
   

Use this analysis tool for diagnostic assessment to identify children’s gaps in learning and next steps.

For more guidance on acting on this year’s check results PLUS the whole back-catalogue of analysis grids for all past PSC papers, book onto our e-learning:

Year 1 Phonics Screening Check: next steps 

References

Blogs: 

A Tale of Two Biggies: the two A’s of Phonics (Assessment & Application)

Podcast:

‘Fight'n Words? The Dyslexia Debate Revisited’ in conversation with Julian Elliott
https://pod.link/1629058475/episode/6a94fa813f6eca3570286b53f5d75b59 

Articles cited:

Labat, H., Vallet, G., Magnan, A. and Écalle, J. (2015). Facilitating effect of multisensory letter encoding on reading and spelling in 5-year-old children. Applied Cognitive Psychology, 29 (2015), pp381–39. doi: 10.1002/acp.3116

Lemons, C., Allor, J., Al Otaiba, S. and LeJeune, L. (2018). 10 Research-based tips for enhancing literacy instruction for students with intellectual disability. Teaching Exceptional Children, 50 (4) (2018), pp. 220-232. doi: 10.1177/0040059918758162

Sermier Dessemontet, R., de Chambrier, A.-F., Martinet, C., Meuli, N., and Linder, A.-L. (2021). Effects of a phonics-based intervention on the reading skills of students with intellectual disability. Research in Developmental Disabilities, 111, 103883. https://doi.org/10.1016/j.ridd.2021.103883  

Van der Weel, F. and Van der Meer, A. (2024). Handwriting but not typewriting leads to widespread brain connectivity: a high-density EEG study with implications for the classroom. Frontiers in Psychology. 14:1219945. https://doi.org/10.3389/fpsyg.2023.1219945

Ihara, A., Nakajima, K., Kake, A., Ishimaru, K., Osugi, K. and Naruse, Y. (2021). Advantage of Handwriting Over Typing on Learning Words: Evidence From an N400 Event-Related Potential Index. Frontiers in Human Neuroscience. 15:679191. doi: 10.3389/fnhum.2021.679191

Ose Askvik, E., van der Weel, F. and van der Meer, A. (2020). The Importance of Cursive Handwriting Over Typewriting for Learning in the Classroom: A High-Density EEG Study of 12-Year-Old Children and Young Adults. Frontiers in Psychology. 11:1810. doi: 10.3389/fpsyg.2020.01810

In 2019 I started as the new CEO of a Church school trust.  It was one of the most fulfilling, rewarding, demanding and pivotal roles of my life.  When I left the trust to take a new role as CEO of HFL Education in 2023, the MAT had grown from seven schools to thirteen, and we had elevated those that joined the trust as ‘inadequate’, despite initially have histories of in-year deficits – and managing a pandemic along the way.  It was genuinely a joy to be part of the trust, and I still feel the pains of not being with my ‘family’ when I see them thriving and growing under new leadership.

As I look back, we went through a lot as a team.  There are things we were (and are) enormously proud of, and of course, things I would have done differently.  There always are.  But in hindsight, what were the key lessons I learned from the journey?

1.    Know why you are and who you are

Our trust was a Church school trust, so for them (and for me) values and vision came first.  What did we believe about life and learning?  That is the most important thing to keep returning to, in good times and bad.  You have to know what is different about your trust and what is the same, and why.  One minor example was branding and communication – you can be dazzled by other trust’s posters and social media, and end up thinking you need to write more, do more all the time.  Reach were clear that they are the cornerstone of schools in their trusts, and that local branding comes first.  Other trusts like Harris are more unified with umbrella newsletters and websites.  Some local trusts have their CEO visibly out and about in the community every day, with daily photos on social media and in local news.  You can’t be all things to all people – so knowing if you are all about empowering local schools and providing the quiet ‘wind beneath their wings’ or whether you are trying to build a ‘Specsavers’ type brand for schools to drop into is key.  What kind of trust are you, and how does that inform how you communicate, brand, teach, lead, write job descriptions?  Spending time with school leaders to work this out together is so helpful – you need to be united on this before you can do anything.  If your staff are expecting to be a branch manager, you need to say if that is not what you do, to both manage expectations, but also make sure you are all pulling in the same direction.  All trusts are so different, and finding your own trust’s place in the world is key.

2.    Have high standards, but not at the expense of your staff

It goes without saying that standards and outcomes for pupils are priorities for all trusts.  We all, as professionals, know that academic outcomes are one of the most important indicators of long-term health, wealth and happiness.  So, the need to drive up standards and improve practice is something all trusts focus on – even outstanding schools need to be ever reflective and improving.  However, it is also important to keep in your mind, as you push on standards, that your staff are your greatest asset.  Treat them well, with respect, with support in times of crisis and celebration in times of joy.  I am really proud that the turnaround in our schools’ fortunes did not come because we burnt things to the ground and started again.  I genuinely believe that school staff want to do well and make a difference, they need the leadership, framework, support and training to help them – just like children do.  We are all just big children.  So, a people strategy, including a clear understanding of how you will enable staff to be individuals that can flourish in a larger community, is key. It is tempting when you have a deficit to remove CPD budgets, but this is something that needs managing with care.  Get it wrong and you have your school winging down the standards tables and getting tricky OFSTED outcomes; your pupil numbers go through the floor, and you are in a worse deficit than before.  Some of the schools joined us as inadequate had endured this cycle – so ensuring that people were invested in and kept in touch with the wider educational agenda was something we knew we had to preserve (even if through cost efficient methods…)

3.    Get governance right; make it a priority

I think I learnt this one the hard way!  Governance is so important in all trusts.  When I started at the trust, we had regional governing boards and a trust board.  It sounded sensible on paper, but the regional governors were exhausted, could not know their local schools well enough, and were not located in the community.  It took me too long to do this, but we moved back to a trust board and local advisory board model in the end.  Local presence is both local insight and local marketing.  It is both local challenge and local support.  And our governors could be tough on us in the trust, but they were driven, passionate and engaged people.  And they were worth their weight in gold.  I would always rather deal with an adult that is hard for me to manage than an easily led, agreeing one who doesn’t care.   So, as soon as we could, we hired a governance officer who put the advice, training and support in that they needed and deserved – and this is one key role that you need someone brilliant in.  Luckily, we had someone who was able to drive local governance and build us a supportive army to drive up standards.  Trust Board members are vital too – they can be asked a lot and need to be given the tools and support to do their roles.  My Chairs were challenging, strong – and their fortnightly catch ups were key and immovable times in my diary.  So, a governance officer and local governance is, to my mind, a necessity not a luxury.

4.    Have growth in mind all the time, but don’t seek growth at all costs 

Growth is always talked about at MAT events, and I have had many peer-led conversations about how to grow.  Whilst it is important to know what your growth strategy is, achieving this is never about a sales strategy or sales activities.  Having a growth strategy based on your vision (phase? area? rationale? benefit?) is key, but the main thing is not to sell in order to grow.  Maintained schools are terrified that trusts will be somehow be trying to gobble them up, and so are always on the defensive because of rumours and stories they hear circulated.  The most powerful tools you ever have for growth are your staff, governors and parents (who are also staff and governors).  Treat them well, help them, support them – and word will spread.  Sometimes you need to remind staff to say the things they say to you to their friends, but it is a bit like OFSTED.  Things are not quite right if you get out of bed to do well in an OFSTED.  A good report should be the by-product of serving your community well.  Growth is a by-product of being a strong, supportive community.   Being honest about what you can and can’t do, and who you are and who you aren’t is actually far more attractive to schools than a glossy brochure and promising that ‘life will go on as before’. It won’t, it can’t – things do change.  But I found that honesty is attractive, and although we walked away from some schools, and schools walked away from us, many schools were appreciative of and engaged by our clarity and as a result, we found the schools that were happy with our way of working. 

5.    Protect your trust; do due diligence wisely

Linked to the above point is the need to be really, really careful about your trust as schools ask to join.   Every school added to a trust is a responsibility, and equally is the responsibility for all schools in your trust.  Knowing that, and making sure that your family can remain strong and stable is worth far more than the income from another school. That means that due diligence is very important – for you and for the prospective joinee.  You have to know where you and they will have to adapt to make it work, and genuinely know if this is a step too far for either of you at that time.  The DfE process for joining a trust is not due diligence, it is the DfE’s legal process – you absolutely have to do your own careful research yourselves before entering the DfE process.  I learnt that the hard way too, and the resultant new process made everything better for everyone.

6.    Be outward facing  

It is so, so important to get out and about.  We say that to school leaders all the time, but as a trust CEO it is even more important.  By ‘getting out’ I don’t mean just to visit people, but to have a structured focus visit – how do they manage their IT?  How do other trusts manage their estates?  Contrary to everything I read before I took up my role about how aggressive and competitive MAT CEOs are, I found that (almost) everyone I spoke to was honest, humble and willing to share.  Networking events can feel self-indulgent, but I never regretted going out when I did, as I always came back with a new perspective or idea.  I didn’t do this enough, and if I went back to a trust in another CEO role, I would make sure I spent planned, structured time to learn from others.  It is not a luxury; it is core to trust improvement.

7.    Know when to leave

I loved my job, but I do believe this is also important – I knew when it was time to go.  As with all leaders, I had strengths and weaknesses.  As with most recruitment processes, I think my strengths were what the trust was looking for at the time.  But as time passes, you build your strengths into a system, and they become the trust strengths.  Your blind spots equally get magnified over time as they mirror your leadership style.  As a trust leader, your job is about children and learning, and I do believe we have a moral responsibility to do the best for them.  There comes a time when you have to see that you have done all you can.  Moving on so they can find another mix of other strengths and weaknesses which mirror their current profile is healthy.  

My time as a CEO of a trust was an absolute honour and privilege, and I hope one day to return to a similar role.  But for now, I have another job I love, working with MAT CEOs, and my wonderful trust has the perfect leader for their next stage in their journey.  I wouldn’t want it any other way.

We are pleased to introduce our new termly newsletter, specifically crafted to meet the unique needs of Multi-Academy Trusts (MATs). Each MAT faces distinct challenges, such as navigating complex regulatory frameworks, managing diverse school environments, and striving for both compliance and educational excellence. Our newsletter aims to inform and educate, providing valuable insights, resources, and updates to support your trust in thriving.

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