Is maths a universal language?

If the above question is inserted into a ‘well known’ internet search engine the following answer is produced as a headline:

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No, mathematics is not a universal language. It is, however, the study of universal truths.

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The next available item to click produces the following:

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‘The quote has many forms, but is basically "mathematics is a universal language." My question is if this is true and we met aliens could we use mathematics to talk to them? The fact that we could show them we know what 10 is doesn't seem like much of help in saying "do you come in peace?"'

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This is beautifully answered on the next available click by an explanation of ‘Why Math is the Only True Universal Language’:

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‘To start with, mathematics does not have a clearly defined, universally accepted definition. However it is safe to say that anything that studies the interaction between quantities, variables, structure, and change, is mathematics. Mathematics is not a tangible thing, but actually an abstract concept. There are a great many ways of expressing mathematics; the one you are probably most familiar with is the base ten Arabic format that permeates science right now. The base, the symbols, the structure, and the methods used to express mathematics can all be radically different and yet, it is still mathematics. Other civilizations have made other ways of expressing mathematics, and if we ever run into alien intelligence, it is likely that they will use a different system than we do. But the system is not the thing.’

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This particularly resonated with me as on my travels as a TLA I consistently hear teaching staff referring to maths as a universal language, especially when they are discussing children new to this country. What is not discussed is the issue that although the base and the structure are the same ‘the symbols’ can be radically different. I have had the lucky experience of working in many different locations with a wide range of primary aged children who are new to this country after arriving from a variety of countries. I also used to view mathematics as a completely universal language until I carried out some Action Research with a group of girls of Polish heritage in Year Six who were both new to the country and to the English language. The research was prompted by my frustration in my not being able to ‘get through’ to the girls and teach them maths effectively.

As the school had several interpreters I adopted the ‘Lesson Study’ approach which involved both observing and interviewing the girls. It was the latter that provided the ‘light bulb’ moment that made me realise that the girls struggled with maths because some of the symbols in their country were different to those used in the UK. These were:

Ten divided by two is written as   10:2

Five multiplied by two is written as   5·2

Six point three is written as   6,3

This revelation, of course, changed the way that the girls were taught which included the ‘re-learning’ of mathematical symbols. Even between the girls there were also some differences because they came from different parts of the country. One girl, for example, used the ‘X’ for multiplication. From that moment the girls’ confidence pointedly increased, their whole demeanour became more positive and they made significant progress in solving calculations. They were all very competent but had been held back by their frustration at not understanding the symbols because they weren't used in the same way as they had previously experienced.

This led me into investigating mathematical symbols from around the world with the intention of generating a reference chart for others to use. Its completion became increasing problematic as different symbols are used in different parts of some countries and in others two or more are used in the same country. As examples though, some of the differences are shown below:

The character used as the thousands separator.
In the US, this character is a comma/separator (,). In Germany, it is a period/decimal point (.). Thus one thousand and twenty-five is displayed as 1,025 in the UK and 1.025 in Germany. In Sweden, the thousands separator is a space. In the UK either a space or a comma/separator is used.

The character used as the decimal separator.
In the UK, this character is a period/decimal point (.). In Germany, it is a comma/separator (,). Thus one thousand twenty-five and seven tenths is displayed as 1,025.7 in the UK and 1.025,7 in Germany.

The majority of European countries use the decimal comma. Among them are Spain, France, Norway, the Czech Republic, Denmark, and more. However, it’s important to note that the United Kingdom is an exception because they tend to follow the Imperial System, which uses the decimal point. Curiously, Switzerland and Liechtenstein are different, as they use a point as a decimal separator, and an apostrophe (‘) for thousands.

Digit grouping.
This refers to the number of digits contained between each separator for all digit groups that appear to the left of the decimal separator. For example, the 3-digit group is used predominantly: 123,456,789.00. However, notice that Hindi uses a 2-digit grouping, except for the 3-digit grouping for denoting hundreds: 12,34,56,789.00

The way commas/separators and period/decimal point are used in large numbers in Italian, is the reverse of what is done in English. In English we use commas/separators to divide the thousands e.g. 12,345 – There is a comma/separator after the number twelve. However in Italian, a period is used instead of a comma. So, the number already mentioned would be written 12.345 in Italian.

Consider the following:

9,876 (English) – 9.876 (Italian)

246,017 (English) – 246.017 (Italian)

1,3247,968 (English) -1.3247.968 (Italian)

8,554,631,902 (English) – 8.554.631.902 (Italian)

The reverse also happens with numbers less than one (decimal numbers). Instead of using a decimal point as in English, a comma/separator is used instead.

35.8 (English) – 35,8 (Italian) 9,876.3 (English) – 9.876,3 (Italian) $1.75 (English) – $1,75 (Italian)

In Switzerland: There are two cases. 1'234'567.89 is used for currency values. An apostrophe as thousands separator along with a "." as decimal symbol. For other values the SI style 1 234 567,89 is used with a "," as decimal symbol. When handwriting, a straight apostrophe is often used as the thousands separator for non-currency values: 1'234'567,89.

Currency formatting might also need to be taken into consideration these following elements including negative-amount display:

 

Percentage sign: In Arabic, the percent sign follows the number; as Arabic is written from right to left, this means that the percent sign is to the left of the number, usually without a space: %48

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As mentioned, all of the above is not intended to be a definitive list. It is more to highlight the issue and to raise awareness of the difference in symbols in mathematics in different countries (and within countries). The knowledge of these symbols can be revealed by talking to the children before any maths teaching takes place in an ‘assessment of any previous understanding before learning takes place’ scenario.

If mathematics is a universal language (in this case with the same base and structure but with different symbols) then understanding some of the different symbols used may help us to provide assistance to those children who are new to the country. 

In strategic partnership with Matrix Maths Hub, the HFL Education maths team lead the Mastery Readiness stage of the National Centre for Excellence in the Teaching of Mathematics (NCETM) Teaching for Mastery Programme. Since 2015, almost a third of primary schools in England have been involved in this national programme.

A question that we are often asked is, ‘What is the difference between Mastery Readiness and the teaching for mastery programme?’

Our answer?

The Mastery Readiness year allows schools to slow down, take stock and then move into the developing year from a place of genuine understanding of their needs and priorities on the road to teaching using a mastery approach.

During the year, Mastery Readiness leads work with 1 or 2 teachers from each school (the project teams) as well as the Headteachers. The project team participate in half-termly workgroups and collaborate with teams from other schools as well as receiving half termly school visits.

The school visits are personalised to the needs of each individual school and support is based around the five areas of Mastery Readiness, also called the Catalysts of Change.

These catalysts are:

• vision and shared culture
• mathematical mindsets
• subject expertise
• systems
• and arithmetical proficiency

 

 

 

In this blog, Nicola, Doug and Laura reflect upon and share some of the things schools this year have done to drive improvement and really prepare for the next step in their journey to teaching maths using a mastery approach.

Vision and shared culture

Although there is no strict hierarchy within the five catalysts for change, our first workgroup is always based around vision and shared culture as this acts as the driver which then leads into all of the future work that the schools go on to do.

School visions this year included:

Delivering an engaging and challenging curriculum, embedded with practical and fun activities which enable all learners to achieve and be successful.

For all children to have the secure mathematical knowledge for their next stage in learning and life. To instil a love of maths for all pupils.

To achieve teaching and learning where the whole class can work together regardless of age gaps. Teachers understand how to teach the curriculum across more than one year group in a practical and enjoyable way.

Each school in the work group brings its own unique context and this is reflected in their vision. However, there are many commonalities and one of the most common elements has always been that desire for all to enjoy maths and achieve their full potential.

Mathematical mindsets

The intended impact of exploring Mathematical Mindsets is to instil a ‘We are all in this together’ scenario where the profile of maths is significantly raised and is valued by all stakeholders across the school. This positivity increases the aspirations of all concerned and creates an atmosphere where success is recognised in many forms and equally celebrated. Working in this atmosphere allows for a sense of purpose and belonging. Growth becomes the customary central focus that is recognised as being achievable by everyone involved.

Starting points for developing this type of mindset were based on the outcomes of the Mastery Readiness self-evaluation that was carried out by each school in the first workshop.

Considerations included:

• Do all staff understand that classes contain previously high / mid / low attainers, but they do not limit the achievement of all pupils through labels such as ‘most able / less able’, ‘good / no good at maths’ and ‘can / can’t do maths’?
• Do all staff proactively promote a ‘can do’ attitude to mathematics for all pupils through a set of ‘positive norms’ for the mathematics classroom, including the use of ‘yet’, depth of understanding before speed and valuing learning by mistakes?
• Do all staff believe all pupils can and will achieve in mathematics?
• Do all staff encourage an inclusive ‘learning together’ culture? Do all pupils feel a sense of belonging in the mathematics classroom?

The answers to these questions were graded between 0 and 3 on a scale from being ‘currently not a feature of practice’ to ‘a central feature of practice’.

This provided key personalised information for each school and prioritised actions could be formulated based on specific evidence explored with the project team.

In Tewin Cowper C of E Primary School, this reflection led to the realisation that a change to mathematical mindsets would be a key aspect in the school’s journey towards teaching using a mastery approach. Both the Headteacher and the subject leader for maths articulated that the school had a vision and set of aims but this subject specific aspect had not been considered previously. Developing positive mathematical mindsets would form a sound foundation for the enhancement of pedagogy throughout the school. 

When the Mastery Readiness Lead discussed the next course of action with the subject leader, it was realised that, from a leadership point of view, it would be beneficial to include all of the staff in the process and an effective starting point would be to create a Mission Statement. This would be the ‘driver’ for the rationale of a positive mathematical mindset and would guide the attitudes and consequent actions of all staff through the realisation of its potential impact, especially on the children.

The project team held a staff meeting to discuss and analyse the idea in depth and a whole school Mission Statement was created. This was deemed to be,

“Inspiring the Mathematician in everyone by championing enquiring, investigative minds.”

The content of the statement was developed through analysis of specific words and statements that staff agreed as being important.

The next step was to utilise the Mission Statement as the driver for the school’s actions moving forward.

It was decided that an explorative approach in lessons should be introduced to staff. This would begin using the ‘Distributive Leadership’ model with the subject leader using the approach in her classroom and feeding back to teachers regarding the impact in a staff meeting. Staff then trialled the approach themselves and evaluated the process.

The staff meeting also included the exploration of a ‘low threshold entry – high ceiling’ approach to keep the class together and provide opportunities for constant formative assessment by the teacher. This was delivered with a focus on the use of the Concrete Pictorial Abstract (CPA) approach as scaffolding for learning and specific exploration by the children with a focus on mathematical talk and the use of specific language. See this blog to read more about the CPA approach.

 

The project team learned that there were:

Opportunities for children to learn as a whole class

There is ‘space not pace’ for learning and going slow meant children were able to make connections to previous learning. This meant that children began new learning from a more level ‘playing field.’ Children were demonstrating that they could use alternative / their own methods and sharing verbal reasoning. ‘Low threshold/high ceiling’ meant all children are able to access the level at their own level and self-challenge, thus less requirement for adult instruction. Discussions mean that children are using rich mathematical vocabulary.

Opportunities for formative assessment

All teachers reported that there was bountiful opportunity for assessment with one teacher commenting, “Amazing assessment –they knew more than I thought!” Teachers reported that the freedom of it allowed them to make informed judgements of when to move children on within the lesson. Lots more discussion and exploration meant teachers could move around and challenge verbally. Discussion revealed children’s level of understanding of concepts and mathematical vocab. This formative assessment also allowed for teachers to use day by day planning and be flexible.

Opportunities to change mindset

Lots of teachers reflected that children were sometimes confused by the expectation that they take a more active role. Children were starting to take more ownership of the learning. Things that appeared initially to be obstacles to the teaching e.g. “Children are used to digital clocks, not analogue” provided rich discussion.

At the end of the Mastery Readiness year, the project team revisited the self-evaluation to re-grade the statements.

It was clear that significant progress had been made and there had been a definite positive shift throughout the school. This was evident not only in classrooms but also from the general atmosphere in and around the school. The project team asserted that clear foundations had been laid, which were based on deep understanding of the catalysts for change to enhance mathematical mindsets through the ‘driver’ of the Mission Statement and the main theme of Vision and Shared Culture.

Systems

Within the ‘systems’ element, schools consider their overall curriculum provision for mathematics.

• Is maths taught daily?
• Are medium term plans sequenced coherently?
• Are systems in place to support the cycle of both formative and summative assessment?

Schools were supported by their Mastery Readiness leads to explore these areas and to prioritise actions. The project team, with the support of the Headteacher, then led and managed any changes made so that ownership was fully with each school.

A key priority for Bedmond Primary Academy was to consider how to structure teaching in small steps, incorporating the CPA approach to develop secure understanding of mathematical structures and processes.

To do this, they began by visiting a local school to share strategies. Following this, they decided to purchase a set of planning materials that would provide small step sequences of teaching across the school and modelled examples throughout to ensure consistency across the school and continue to develop subject knowledge and expertise.

The project team then trialled the resources in their classrooms before enrolling staff on training for the autumn term. Teachers reflected that by participating in the training, they felt more familiar and confident with using the resource in their classrooms.  As part of the implementation process, potential barriers were also considered, with the project team discussing lack of previous exposure to some of the models and representations that children would explore as well as heightened language expectations.

The careful consideration to the implementation process helped give the school a clear vision and plan moving forward this year to develop their planning and teaching, as well as the use of purposeful summative assessments.

Diagnostic assessments have been introduced this year and early in the summer term, additional time was given to teachers to thoroughly analyse their outcomes. This knowledge was then used to inform and tweak medium term plans for the remainder of the year. The process also lead to successful and informative transition meetings enabling teachers to plan for a flying start from September. The project team noted how useful this process had been and leaders have included the use of diagnostics, and time to complete detailed analysis, as part of their assessment cycle moving forward.

For more information: ESSENTIALmaths planning and diagnostic materials

Arithmetical proficiency 

In order to successfully reason mathematically and solve problems, it is important that children are also proficient with their arithmetical skills and this is why two workgroups this year were dedicated to this catalyst; the first focusing on additive knowledge and the second on multiplicative.

Three characteristics that underpin arithmetic proficiency are accuracy, efficiency and flexibility.

Accuracy involves pupils having secure knowledge of their number facts, making meaningful recordings, applying their knowledge of relationships between numbers and double checking their results.

If efficient, children will be fluent with a range of calculation strategies with the ability to choose their method based on the calculation they are tackling.

Pupils with flexibility are able to move smoothly between different strands of maths. They have more than one approach and are able to choose and carry out an appropriate strategy.

This was something that Cassiobury Junior School were keen to explore as, although many of their pupils were competent when using formal written methods, there was less flexibility and efficiency in terms of the methods chosen, with pupils often over-reliant on those written strategies.

Another area for development that the project team had identified was that pupils were less confident at recalling prior learning when needing to apply it to real life problems and scenarios.

The school decided that in order to provide pupils with additional opportunities to revisit taught knowledge and skills, and to encourage discussion about multiple strategies, that they would introduce fluency sessions.

To support staff with planning and resourcing these sessions, the school purchased the HfL fluency session slides which revisit taught concepts and are then used on a regular basis to establish familiarity and understanding and increase confidence for all.

The project team trialled the sessions in their classes and reported that, even after only a week, pupils were participating more confidently using precise mathematical vocabulary and discussing their reasoning with peers. And pupils reported how much they enjoyed the sessions! The project team planned to introduce the sessions and resources in a staff meeting during the summer term ready to fully implement them into their weekly timetables in the autumn term.

To further support the school’s priorities around times tables recall and multiplication strategies, teachers will include a slide with a multiplication focus and a mental strategy element in each of their sessions so that key number facts are regularly rehearsed.

Therfield First School also focused on developing fluency sessions in their mixed-age school as part of their Mastery Readiness year. Read more about their journey here: Children are seeing themselves as mathematicians – the impact of CPA and fluency sessions in my mixed age class

Subject expertise

Last, but by no means least, comes subject expertise. Secure subject expertise and pedagogy for both teachers and teaching assistants is essential in developing mathematics. Knowing the age-related expectations and the necessary building blocks to get there is key to all children keeping up. It is important that children explore and learn about mathematical structures through concrete and pictorial representations. For children who find it difficult to articulate their thinking, this will support them to develop that and also allow teachers to pick up on any errors or misconceptions.

This year, with the disruption to schooling, it was more crucial than ever to be able to identify the starting points of the children in each concept and to do this, teachers needed to be aware of each small step in the sequence of learning.

Tannery Drift First School found gaming to be a valuable tool for identifying these starting points following the national lockdown in spring 2021.

Knowing how important it is for children to have a variety of calculation strategies in their toolkit, the Year 2 teacher wanted to check which strategies were being commonly used for addition and subtraction and whether or not the children were recalling number facts automatically in the process. Information gathered from observing and interacting with children playing games was then fed into teaching plans going forward.

The Year 4 teacher thoroughly enjoyed the designated gaming sessions and the children did too. As the games were played more than once, things could be dripped in over time and opportunities could be taken to support development of language or stretch thinking based on what was being observed. This was particularly beneficial for the teacher in the parallel Year 4 class who was new to the school as it provided opportunity to get to know the children in a very low-threat and fun environment whilst also highlighting areas of strength and areas that would require further teaching.

The children not only enjoyed playing the games provided, they began to design their own. For example, in Dodge the Crocodile, they thought about how to ‘trick’ players by including common misconceptions and errors on their path. Enabling the children to do this relied on the teacher’s strong subject expertise.

Collaborative planning is an approach the school have been using to plan the progression across the school in the teaching and rehearsal of times tables. Following workshop 3 around multiplicative reasoning, there has been a whole school focus on the recall of known multiplication facts and strategies to work out unknown facts.

Exploring the structure of multiplication with the children, along with modelling the use of accurate language, has enabled them to articulate their strategies for working out unknown facts. Both project teachers have noticed a big difference in the way their children play with numbers to reach unknown facts.

Year 4 have been working on the 12x table and focussing on understanding of mathematical structures.

• Can we partition it?
• Could we use double the 6x fact?
• Could we use 10x and 2 groups more?

Teaching happens as a whole class to secure understanding and then game choices are provided for rehearsal.

Thank you to those schools who shared their journeys so far in this blog.

For more information about Mastery Readiness and to apply, please contact primarymaths@hfleducation.org.

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Blog authored by Laura Dell, Nicola Adams and Doug Harmer.

After running a recent bar modelling course, I was asked by a year 6 teacher “How would this look in the past SAT papers?” She wasn’t questioning the relevance of bar modelling, rather she was seeking support to bring this method to pupils. Further discussion revealed that she was worried that she would model it wrong for the pupils. I did understand what she meant – she was new to the method and wasn’t yet confident in the models that she presented to the children and wanted to make sure that she wasn’t missing anything. So this is my attempt to show how bar modelling could have been used in the past SAT KS2 papers – I am not guaranteeing that I am not ‘missing something’, but this is how I see it (if you see it differently, brilliant – let the debate begin)…

Throughout these examples I'd ask you to remember that the purpose of bar modelling is to support the pupils in identifying the mathematical relationships within the questions and support identification of operation, not to provide the numerical answer. Pupils still need to be efficient in calculation. Also, even though I have presented one model for each question, there are many more models possible – it comes down to the pupils’ ability to reason and explain their models.  Embrace the differences!

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Reasoning Paper 2 2017

 

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Reasoning Paper 3 2017

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References

STA, Reasoning Papers 2 and 3 (2017)

“Yeah, they're great to use in maths, but you want every child to be curious, don't you? Across every single subject, because that's how you learn."

Ashleigh Calver, Assistant Headteacher, Maths subject lead and Year 6 teacher, Longlands Primary School, Hertfordshire

"We want our children to be curious in all subjects; we want them to make connections across the curriculum".

Suzanna Neate, Assistant Headteacher, Maths subject lead and Year 6 teacher, St Margaret of Scotland Primary, Bedfordshire

So what exactly are these leaders talking about? They are discussing with us the Greater Depth Maths materials which were developed to respond to the requests that we, as a team of maths advisers, we're always hearing. “Got any ideas for my greater depth?” You can read our earlier blog about this where we consider, in detail, what Greater Depth in maths could actually mean and the importance of having this discussion within your schools.

What we ended up developing moved wildly beyond our original thoughts; it became about the development of the whole child - skills and behaviours that could be used across the whole curriculum but nurtured in a maths classroom - and about the inclusion of ALL children.

This blog aims to share some of the incredible outcomes for children and teachers who have already engaged with the resource.

How schools have started

Longlands Primary School (Broxbourne, Herts) and St Margaret of Scotland Catholic Primary School (Luton, Beds) trialled the materials in their schools to see the impact on children and teachers. Each school chose a different deployment pathway; however, they both started with a launch staff meeting led by one of the authors of the books and HfL Primary maths advisers.

During the launch staff meeting, teachers had a chance to explore both the books and the online resources and got stuck in to the 6C cycle detailed in this training slide: 

 

 

 

The training

Suzanna, Year 6 teacher, St Margaret of Scotland

 

As Suzanna expresses, one of the strengths of the resource is its flexibility: "You can do it however you like." It can be led by your current school development priorities. You can choose to focus just on a couple of the Cs or complete the whole cycle.

Following the staff meetings, leaders supported their staff to choose how they wanted to use the resources. For Longlands, this meant choosing a task and then exploring the curious prompts, encouraging children to ask mathematical questions which they could then explore through the connect section (where they solve the problem).

St Margaret of Scotland decided to take a task and go through the full 6C cycle, spending between 3 and 5 lessons exploring the task fully.

 

 

Getting started

Siobhan, Year 5 teacher, St Margaret of Scotland

 

 

 

Teachers were surprised

Suzanna, Maths subject lead, St Margaret of Scotland

 

Focused development of behaviour

Longlands decided to focus on developing children’s curiosity. One of the impacts we have seen of essential remote teaching is that children have returned to our classrooms more passively.

Curiosity, to me, is the internal desire to acquire new information. Our bodies reward us when we have satisfied curiosity with a shot of dopamine, so we feel good. Curiosity has been shown to benefit both learning and memory (Gruber, Valji and Ranganath, 2019). If we can make children curious, they will be self-motivated and take control of their own learning.

However, allowing children to develop their own lines of enquiry and feel that they are pursuing their own investigations can make some primary teachers anxious. The Greater Depth Maths resources enable teachers to stay in control by using the 'covert' strategies provided, meaning the children believe they have self-generated their own questions. Yet, they have in fact been operating within tight parameters.

Progression through the complete cycle

As mentioned, St Margaret of Scotland decided to undertake a complete cycle; they wanted to assess the children's current abilities across all the 6Cs. To do this, they made use of both the Stumble Support and the assessment rubric which are key elements included within the resources.

 

 

Suzanna, Maths subject lead, St Margaret of Scotland

Enabling access for all – Stumble Support and assessment rubrics

 

All the resources required to enable teachers to accurately identify ALL children's strengths and development requirements are provided across all the behaviours and skills. For example, Suzanna discusses how the teachers identified that some of their higher attaining children in maths needed to enhance their communication and collaboration skills and so this became a focus.

Stumble Support is provided throughout the resource to help teachers see how to get children moving forward again in their learning if they become stuck. This comes in many different forms; teacher questions, additional PowerPoint slides, alternative concrete models, speaking frames, word banks etc. In this video clip, Suzanna speaks about how the unique Stumble Support and the assessment rubrics work in tandem.

Impact on teachers

One of the outcomes we repeatedly encountered on our trips back into the project schools was the sheer joy and enthusiasm that radiated from the teachers. They enjoyed teaching the tasks. They expressed that they felt freedom in their teaching.

“I really enjoyed using it.”

Ashleigh, Year 6 teacher, Longlands

“I enjoyed letting the children lead it.”

Steph, Year 1 teacher, Longlands

Gavin, the Year 4 teacher, spoke with such enthusiasm to us about the task that he had explored with his children. He told us how both he and the children thoroughly enjoyed the task and how when he told the children that they were moving on to a new concept, this was meet with an ‘Oh!’ from his year 4s as they were so keen to continue. Gavin has decided that he will revisit the task at a later point to allow the children to move through some more of the 6Cs.

 

 

Developing skills beyond knowledge

Pauline, Year 6 teacher, Longlands

 

Skills beyond knowledge

Both schools have kindly given us permission to share some of the children’s work created during their exploration of the tasks.

 

 

Here, Year 5 pupils were tackling a problem involving fractions at the Connect stage. This child chose to use playdoh alongside their own pictorial representations.

Ashleigh, the maths subject leader, commented that using the manipulatives supported children with their use of language and in drawing connections to their knowledge of equivalence.

She also noted that one child, who often lacks confidence to join in class discussion, was able to demonstrate what a resilient learner she was; she and her partner developed their own way of recording and tackling the problem.

Teachers also made sure to capture what the children were saying using the sentence stems on the curious prompts. The prompts helped the children to articulate their initial thoughts in full sentences before exploring the prompt further and heading into the ‘Develop and Deepen’ element.

 

 

Steph, the Year 1 teacher at Longlands, described to us how she revisited the prompt in short 10-minute sessions across a week where the children begun by investigating the task and then by the final session, were exploring their own representations.

 

 

The children were excited to revisit the task and like Gavin, Steph plans to come back to the task in the summer term to explore more of the cycle.

Teachers at St Margaret of Scotland use ‘floor books’ across the curriculum to capture collaborative work from their classrooms and decided that this was how they wanted to record children’s responses to the tasks. Each floor book was created along with the children, who take ownership and much pride in them, as we are sure that you will see from these snapshots.

 

 

Task: Crossing the River

Key Stage 1

 

 

 

Task: Triple Trick

Lower Key Stage 2

 

It has been an absolute privilege for us to work with both Longlands and St Margaret of Scotland: to set them off on their journey with the greater depth materials and to later visit them to see the excitement generated and impact of their work so far. We can’t wait to see where they take it next!

You can also download some free samples.

 

 

The Herts for Learning primary maths team are also able to offer bespoke in-school training.

Please contact us for further information and details of pricing on 01438 544464 or email info@hertsforlearning.co.uk.

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References

Gruber, M.J., Valji, A. and Ranganath, C., 2019. Curiosity and learning: a neuroscientific perspective.

 

Blog authored by Charlie Harber and Laura Dell.

Ofsted looming can feel overwhelming, and we know all our colleagues are feeling the anxiety too. As the leader of a core subject, like mathematics, none of us want to get it wrong and we want to know we have reflected our school in the best possible light.

Ed Farrell, the maths subject leader from Cowley Hill Primary School in Borehamwood felt sure this was the year for their turn and started to make plans for September. Here he shares his experience which, to his surprise, he found very valuable and almost “enjoyed”!

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Last academic year, in preparation for learning in September, we took note of the DfE guidance “Teaching a broad and balanced curriculum for education recovery” (DfE July 2021):

When deciding what to teach to support education recovery most effectively, leaders can help all pupils by focusing on making sure they are fluent and confident in the facts and methods that they most frequently need in order to be successful with further study.

In the context of missed education, it remains crucial to take the time to practise, rather than moving through the curriculum content too quickly. What pupils already know is key. Progressing to teaching new content when pupils are not secure with earlier content limits their chances of making good progress later.

The sequence of teaching mathematical content is also very important: gaps need to be filled before new content is taught.

We had already been using HfL Back on Track resources. Included with these was a complete set of diagnostics and so we were able to be precise about key concepts for each year group and the gaps in children’s learning.

You may be interested in a recent blog by a Year 5 teacher who explains how he used it with his class: Getting back on track in primary maths – tracking back to build up.

We knew that to support the progress of pupils, the acknowledgment of potential gaps in learning and the subsequent careful navigation forwards would be key, alongside being confident in assessing when pupils were ready to progress.

The identification of gaps was (and continues to be) a team effort. Leaders and class teachers work collaboratively to agree which strands of maths may, or may not, have been secured; in its infancy, this process began as identifying which strands teachers would revisit with each year group if time was not an obstacle. Yet with the pandemic, it evolved into a robust process, pulling the teaching team together to collaborate in support of a curriculum that considers and acknowledges gaps in learning and allows time for action to close them.

The most recent version of this process involved acknowledging that content taught remotely, at any point in the last three academic years, had to be carefully weighed up as the reality of suggesting that something had been ‘secured’ remotely, was not something that could be done with great confidence.

Further to this, content taught face-to-face had its own disruptions (attendance (of both pupils and staff), restructuring of the day to accommodate bubbles amongst other measures and timings etc.) and therefore also had to be considered carefully.

Stemming from this analysis, we identified and categorised the outcomes as follows:

• those where the majority of pupils had secured the outcome statements and were ready to progress;
• those where further intervention (through fluency sessions and main teaching) was necessary to support progress to ensure that students were ready to progress; and finally,
• those where the exploration was deemed to have been too limited and would likely be required to be taught as new learning rather than a progression of skill or reactivation.

Using the identified gaps, the priorities for each year group were sequenced, whilst simultaneously weighing up missed opportunities as well as future opportunities. Future opportunities identified and the need and manner for expansion to accommodate the missed prior opportunities was agreed and logged. This was conducted whilst also keeping a clear focus and awareness upon the age-related outcomes, as this in turn would, and has continued to, challenge and raise the bar for lower attainers.

 

 

With the curriculum mapped, and a new course for the current academic year plotted, staff were then supported and led to identify (using the maps) areas where reactivation of prior learning would be required as pre-steps before the expected age-related sequence of learning could commence.

For example, our current year 6 cohort had limited exposure and opportunity to develop and strengthen their understanding around area and perimeter, including the relationships within. To support this identified gap, early intervention was implemented with the intention being to re-activate prior knowledge. In some cases, this meant recalling teaching from the year 4 curriculum.

Using fluency sessions, short, sharp, yet frequent opportunities encouraged the accurate use of previously taught vocabulary. The beauty of these short sessions lays in the ease of repeating a skill through simple and fast editing of previously used materials. The following examples firstly demonstrate the initial session where the formulae of area was revisited and reactivated for fluency in recall; the calculation was reasonably straightforward (namely to support the promotion of mental fluency).

 

 

 

With the process now re-explored, in the next revisit, the task has not varied; focus is still upon the concept of calculating area and differentiating between the examples. The placement of the shapes varied which fed conversation around the length/width despite the orientation of the shapes. The unit of measure was also varied to include decimal notation – another area where children had limited security at the end of the previous academic year, as noted within the curriculum mapping outlined previously.

Moving ahead, with the pre-teaching conducted through the fluency sessions, the sequence of learning around the relationships between area and perimeter began with an additional opportunity for pupils to demonstrate their current level of understanding.

Making the best use of time

The benefits of our spiral curriculum had been compromised to some extent, where some outcomes and even entire strands had not been previously explored. Yet this is where the fluency sessions come to aid. Our ongoing fluency foci allow for pre-teaching to be embedded in support of overcoming the barriers and some of the excessive gaps presented in the pupils’ learning. Short, sharp and frequent exploration allows for reactivation of prior learning, some of which had not been called upon for durations in excess of a single academic year (an example would be the exploration and securing of telling the time and confidence and fluency over the units of measure involved; first explored in KS1, the year 4 cohort had received very limited opportunities).

Highlighting links between strands allowed staff to exploit them in a manner that benefits the pupils. Despite the initial (and momentary) increase in workload, the necessity to amend the maths curriculum was just that, and progress in knowledge and understanding of pupils in all cohorts would have been considerably hindered if we hadn’t done it. This, in turn, would have subsequently created an even greater increase in teacher workload stemming from what could have been an endless cycle of backpedalling and the burning of time we simply do not possess. Staff are encouraged to make the best use of time for we cannot create more, yet we can use it creatively. We ensure that the pace of teaching is not increased in a manner that could create a greater number of ‘stragglers’; instead, our ethos has remained clear and crucial:

deeper learning takes longer yet lasts longer and creates a far more secure understanding for problem solving.

Short inspection – 12 July 2016

Our school’s previous Ofsted report noted that the next steps and ongoing focus should include attention to ensuring that ‘higher-attaining pupils are consistently challenged by all adults, so that they achieve the very best they can in writing and mathematics’.

The level of challenge provided, whilst initially focussed upon the higher-attaining pupils, has since evolved into a robust curriculum where all children are consistently challenged in all year groups. Substantial time was placed into the development of reasoning across the entire curriculum; pupils are challenged to describe, explain, convince, justify and prove their findings.

 

 

School inspection – 05 and 06 October 2021

With the pandemic, the current intentions include the necessity to support the curriculum recovery.

Our reasoning thread has been a key factor of doing so and was positively highlighted during our recent inspection. Children in a range of year groups were observed being asked to convince or justify their findings, as opposed to being directed to check in response to an error – in doing so, pupils self-corrected and subsequently demonstrated the depth of their understanding, providing valuable input for teachers’ ongoing assessment for learning.

The implementation of our spiral curriculum was praised, namely where adjustments were made and justified with outcomes supporting the decisions. A thorough dive into the curriculum mapping process previously detailed noted ‘where pupils are coming from, currently at, and heading to’ as being vividly clear to leaders and teachers alike.

My actions as a leader were placed under the spotlight when it came to effectiveness of leadership; the key to success here was honesty around strengths and areas for development. To ensure consistent, highly effective practice, ongoing monitoring is a part of our school life. I have adopted a democratic style in which my leadership is participative; it is far from finger pointing and rule reader.

Instead, throughout ongoing monitoring, areas for development are highlighted and discussed with staff in need. Following this, agreed actions and support provided are logged within a proforma where a two-week cycle is stated. The impact of agreed actions is then reviewed and discussed further. The outcome of this was highlighted within the inspection, under the impact and awareness of leaders within the school and complimented by some members’ recounts of the support for professional development provided. Whilst there may be a temptation to refrain from detailing areas for development within your school, I cannot stress the need to avoid doing so enough. Leaders lead; the inspector was keen to see how and identifying need for development is a major part of that responsibility.

The ‘doom and gloom of Ofsted’ is not something that was experienced during the recent inspection. Maintaining confidence in the actions taken, clear articulation of said actions, alongside an explicit vision for what is next, made the duration of the inspection a positive experience, rather than one many would label as undesirable.

The inspector’s questions were fair and reasonable, far from the hearsay of trying to ‘catch us out’. My conduct, effort and application as a leader, as well as those of the teaching staff, are all done for a common goal, the children.

Inspections can feel like a personal investigation into action taken (or not taken) although knowing why, how, and when these were taken proved to be valuable. On reflection, the entire experience was a recital of the consistent effort made in school to provide what we feel is a strong, purposeful curriculum that evokes free-thinking, able, confident problem solving.

If writing a ‘survival guide for inspections’, I would condense my experience into three key points:

 

Confidence

Decisions made are meaningful and discussed in a collaborative manner prior to implementing them. Acknowledging what has not necessarily been as successful as envisioned was respected and noted by the inspector as much as what had. They were keen to hear how decisions were made, why and (most importantly) what the impact of these was. Knowing the story of how we arrived at the point we were at during the inspection removed any potential weight from my shoulders. Maintaining confidence and conviction whilst articulating action taken relaxed not only myself, but the inspector.

Whilst on the topic of confidence, it would also be worth mentioning the necessity to be able to honestly highlight the strengths and areas for development in your setting. Noting areas of previous concern and action taken reaffirmed the effectiveness of leaders in school, as well as the commitment to development from teachers; ultimately, the pupils are at the forefront of all decisions made and this is what the inspectors need to be sold on.

As with many other inspections I have heard about, inspectors state when meeting the staff prior to commencing the inspection that the team should ‘continue as normal’ and I wholeheartedly agree. Deviating from usual practice stands out from a great distance – confidence is essential for all members of the team whether being interviewed and/or observed.

Articulation

Being able to clearly articulate was another key element. Practise using likely/common questions with your team – question your teachers and support them in being as clear as possible in their responses; have others question you also.

We held conference as a staff in the months prior to the inspection where questions were put out and we conferred on our responses – we were all telling the same story, yet some responses were buried amongst verbiage that ultimately led to deviation from the facts and the impact our efforts have had and are having. Leaders or not, we are all accountable and ensuring that all staff members are aware of this is not to create a sense of anxiety, but to encourage an environment where expectations and standards are consistently high for all.

It is also worth highlighting the need for honesty here again – inspectors are not looking to trip you up, nor catch you out; they are looking for confidence in your practice and vigilance in decision making.

Awareness

As a leader, this was the largest and most important part, yet largely stood upon the pillars of confidence and articulation.

Why have you made the decisions you have? What is/was the impact of these? Why is your curriculum in its current state – what has led you to this point? What are your next steps? What are the strengths of not only your curriculum, but your teaching staff? (Be honest!)

Overall, in my experience, the inspection was an enjoyable and considerably invaluable experience that was far from the labels of pressure and anxiety it has come to be so widely associated with. Ultimately, it became a platform to celebrate the commitment, achievements and dedication of the staff who collaborate to create and deliver the curriculum which develops the minds of tomorrow.

Subject leader toolkit for mathematics

Blog authored by Ed Farrell, Maths Subject Leader at Cowley Hill Primary School.

Last year, I had the opportunity of working with two colleagues to write materials to support schools in developing effective fluency sessions. Their development began in a Hertfordshire school when a colleague was working with a subject leader who felt that mental strategies for addition and subtraction just weren’t getting enough ongoing rehearsal time and so the children were not developing a bank of strategies from which to choose when needed. From there… they grew… and grew… and grew! Since launching the resources following our first training day in June, I have also had the privilege to visit and work with schools who are using them.

But what are fluency sessions? Fluency sessions, or Maths Meetings, are short, snappy sessions, preferably daily, which allow children to retrieve previously taught knowledge in order to rehearse concepts. Brown, Roediger III and McDaniel note that although the “brain is not a muscle that gets stronger with exercise”, when taught concepts are practised over time, the knowledge of that concept will become stronger. “Retrieval strengthens the memory and interrupts forgetting.” (Brown et al 2014). You will notice that I am using the term ‘taught concepts’, as opposed to ‘learnt concepts’ as just because we have taught something, does not mean the children have confidently learnt it!

Consider the amount of knowledge that needs to be retrieved to answer this KS2 SATs question from summer 2019:

 

 

What do the children need to think about?

• What do I understand about perimeter?
• Do I know what a regular shape is and how this affects the properties of said shape?
• Can I calculate the total of 6 sides with a length of 8cm each?
• If I don’t know 6 x 8, what facts do I know that could help me?
• If the perimeter of both shapes is 48cm, how does this help me find the area of the square?
• What is area?!

Less than 50% of our last year’s Year 6 cohorts achieved 2 marks on this question. The calculations required aren’t tricky – even if 6 x 8 is not a known fact, pupils could use their knowledge of 5 x 8 and one more group of 8, or double 6 x 4 etc. We would expect most Year 6 pupils to tackle 48 ÷ 4 with confidence before finding the product of 12 x 12. The numbers involved aren’t challenging; it is the pupils’ ability to link the mathematical domains fluently that is needed here.

It is often my experience that when I work with teachers and fluency is discussed, fluency is seen as being confident and quick at completing arithmetic type questions – mainly based on the four operations and calculating with fractions. It is at this point that I will say that when I first started teaching Year 6, once a week – normally a Friday – I would reteach particular aspects of the KS2 curriculum, such as adding fractions with the same and then different denominators, before practising an arithmetic paper and then going through the answers. Now don’t get me wrong – there is definitely a need for children to practise procedures and formal methods and build up the stamina to complete the arithmetic paper – but it was in my second year of teaching Year 6 that I started using mini revision type sessions straight after lunch, 3 times a week, to allow children to practise concepts that we had covered in our daily maths lessons. For example, I may have put on the board a number such as 3,600 and some percentages that I knew that my class were confident with finding, such as 100%, 1% and 10%, and asked the children to find some more interesting percentages of 3,600. Children would discuss and share their methods – both efficient and inefficient were welcome! The next day, a new number would be in the middle of the board and children would complete the same task but be encouraged to think about strategies used the day before and how they might like to use them. These sessions lasted about 10 minutes and were by no means perfect but the children became increasingly confident at talking about their maths.

The fluency resources that we have developed aim to support teachers to revisit and rehearse concepts. Each year group has 3 PowerPoints (one for each term) and within each PowerPoint are 6-7 concept slides. Here is an example slide from Year 4’s spring term:

 

 

It is advised that this slide is used to firstly rehearse using a pictorial representation (part whole diagram in this case) to multiply a 2-digit by a 1-digit number. This is revision from Year 3. Vocabulary and speaking frames are provided to support the children with their explanations. Approximately 2 minutes are spent with the children discussing how they could find each part before combining to find the total product. 3-4 other slides are then used during the 10-15 minute session to rehearse other learning – perhaps including place value with decimal amounts and estimating the value of a mystery number on a number line when given the start and end number.

The following day, the same slides are used again but with small tweaks.  Instead of the multiplier being 6, it could be 7. Instead of the start number being 34 (leading to the base facts of 3 x 6 and 4 x 6), more tricky base facts could be required, such as 37 (7 x 6 might be a less confident known fact for the class). When the children have had sufficient rehearsal of this slide then a more complex adaptation might be made the following week.

 

 

Initially children could talk about potential strategies and work out their starting points such as recognising that 294 has been regrouped into 14 and something. 14 less than 294 is 280 so 40 x something is 280. The slide could then be repeated the following day with no changes at all and instead the children could use what they have discussed yesterday to complete the blanks. To encourage that rich strategy talk, perhaps the question at the top of the slide should read ‘How could you complete the blanks’. Here the children are not only rehearsing their multiplication facts – I know that 4 x 7 is 28 so 40 x 7 is 280’, they are also reasoning about the relationships between base facts, finding an appropriate starting point and drawing out what they know. Fluency isn’t just about that fast recall of multiplication facts but being able to reason and articulate their mathematical thinking. But that’s another blog!  

I had the pleasure of delivering the initial launch training during a staff meeting at Oxhey Wood Primary School and Kerry Kent, Assistant Headteacher and maths subject leader, is leading the implementation of fluency sessions across the school. Already, Kerry reports that she is seeing a difference. Kerry noted that all children, regardless of prior attainment, are included within the sessions and that the small daily adaptations support less confident children because they have seen similar the day before. Precise vocabulary is being developed with the use of the word lists, sentence stems and speaking frames. Kerry also commented on how rapid graspers are being challenged by being encouraged to use all the given vocabulary or to think of multiple strategies.

James McKenzie, Year 4 teacher and joint maths subject leader, is leading the development of fluency sessions at Abbotts Langley Primary School. James attended our first training day last June and shared the rationale and resources back at school. As well as the use of fluency sessions, James is also developing the use of pre-teaching as a strategy to support pupils who are working towards age related expectations. Currently, James uses these 15 minute sessions 2-3 times a week with pupils across the Year 4 cohort to allow children to rehearse a concept, tracking back to a point where the children are secure, before looking at possible misconceptions and gaps in knowledge that may restrict the access to what will be explored in whole class teaching. Key vocabulary is explored and a new method may be modelled to support the children’s confidence for the upcoming lesson. James reflected that: “These sessions are also of great benefit to the teacher, as they provide a practice run for the main lesson. I am able to discard things that may not have worked or add new questions that have come from the ‘pre-teach group’. There will be times when a start to the lesson can be used to provide a whole class pre-teaching opportunity, to cue pupils back in to prior learning, reactivating and rehearsing this before building on this in the main focus.”

Fluency sessions could prove to be a very useful tool for when pupils return to school. These short sessions based on previously taught concepts will help to reactivate learning, encouraging pupils to discuss what they remember, what they know, and to highlight what they may have forgotten. These sessions will also provide low threat assessment opportunities and can be used in readiness for future teaching and learning.

The suite of resources, which include 3 PowerPoints per year group, a start-up guide for leaders (including 3 videos) and a staff development launch session (staff meeting resource) are now available to purchase.

 

Professional development opportunity

Join us for face-to-face training at the Hertfordshire Development Centre to:

• consider the rationale for developing fluency sessions. Why do pupils need to be ‘fluent’ in maths and what do they need to be fluent with?
• explore possible structures and ways of organising sessions
• consider the appropriate content for fluency sessions, including what is appropriate for the year group and time in the academic year
• take away resources for a staff meeting to use with staff and a full set of (revised and updated) fluency session materials to use and adapt with each year group

Developing effective maths fluency sessions

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References:

Brown P, Roediger III H, McDaniel M,  (2014) Make it Stick: The Science of Successful Learning

STA (2019) Key stage 2, Mathematics Paper 2: Reasoning: Crown publishing

 

Siân Muggridge is a Specialist Advisory teacher for Vision Impairment (VI).  Over the past few terms, Gill Shearsby-Fox and Siobhan King from the Herts for Learning Mathematics team have been lucky enough to learn from and with Siân to continue to develop understanding of how to best support pupils with vision impairment in their mathematics learning.  We have been able to discuss priorities and curriculum adaptations that support success and Siân’s practical adaptations of resources and learning opportunities know no bounds!  It is wonderful when you meet someone who is similarly passionate about successful mathematics learning for all children, and we are delighted that she has agreed to share some of her expert insight and tips in this guest blog.  

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How many numbers do you think you have read today?  Your clock as you got up? The expiry date on the milk carton? The numbers on the remote control as you turned the telly on? Three examples and we’ve yet to leave the house.

Now imagine if you never saw numbers in the environment, that you never notice the numbers on doors or how they go up in twos, or that you can’t tell how many apples are left in the fruit bowl without going and physically touching each one.  This is the experience for many children who have a vision impairment.

My name is Siân and I have been working as a Specialist Advisory teacher for Vision Impairment (VI) for seven years.  I first became interested in the relationship between vision impairment and maths when I met Nabeel (not his real name) in 2015; a Year 4 pupil with a moderate VI and a physical impairment.  Nabeel was articulate and doing well in English but, despite having a high level of support, adapting his resources, and using a special magnifier to see the board, he seemed to be really struggling with maths – even simple activities such as adding two numbers appeared to be beyond him. 

When we dug a little deeper, it appeared that Nabeel had serious gaps in his knowledge: he wasn’t secure with place value, he didn’t realise that numbers had a set order (ordinality) and he struggled with manipulating shapes and understanding pattern (for example he struggled to put 2 Numicon pieces together).  It was evident that there were gaps in Nabeel’s maths concepts that needed to be plugged if he was going to make progress. 

We therefore set some challenges that Nabeel could do each morning independently, focussing on developing his understanding of ordinality, place value and using manipulatives effectively to solve problems (see below).  It helped to plug some of the gaps in Nabeel’s understanding and, importantly, his confidence in maths grew.

 

 

As I continued working with more pupils with VI, I found several of them struggled with maths and, those who had little or no vision and were learning through touch (braille), often found maths the hardest subject to master.  Why should this be? 

Well, it’s not a difficult question to answer when we think about it – maths is a highly visual subject that relies on the ability to sequence and organise ideas visually (ICEVI 2005) and most children will often begin this journey before they start school. 

Children with VI will often lack these incidental learning opportunities (like the ones mentioned at the start) that their typically sighted peers have accessed.  Indeed, it has been argued that concepts such as subitivity, cardinality and ordinality can develop before children receive any maths instruction (Van Der Heyden 2011).  This is not to say that early maths concepts cannot be ‘caught up’ but that we need to know where the gaps are.  For instance, it’s hard to teach cardinality if you don’t yet know what the concept of same and different is. 

Indeed, it took me 3 months attempting to teach one-to-one correspondence with a little girl who had no vision by counting objects from one pot into another.  We could not get beyond counting 2.  Why two?  She had two hands and could deal with the concept of two. However, as soon as she put that object down, she didn’t really understand that it still existed and therefore started counting again.  She needed to develop the concept of object permanence (that objects still exist even if you can’t perceive them) before she could develop counting past 2.  Once she understood that concept however, she absolutely flew and could count as many beads or bricks as I could fit in the pot.

But why am I going on about this? 

Vision impairment is a low incidence condition, which means you can go through your whole teaching career without meeting a child with VI – particularly one who uses braille.  However, there will be plenty of children who may have, or had when younger, difficulties processing visual information and there will be some children who have missed those early incidental opportunities for different reasons.  To support mathematical success for all pupils, I have identified some helpful tips which could help to make maths a bit more multi-sensory and, dare I say it, a little bit less visually complex. 

Tips for nursery/reception

Children’s vision continues to develop until about the age of 7 so it’s not unreasonable to think that some children in Reception will be less adept at processing visual information or using their distance vision.  You may also find some children are being patched to try and strengthen the vision in their ‘weaker’ eye – obviously these pupils’ vision is reduced while they are being patched.

• consider the environment: a patterned tablecloth can be tricky when trying to complete a visually complex task.  You can stick a bit of black sugar paper or a mat down to provide a clear, high contrast background
• sometimes, providing a lipped tray to ensure that resources stay in the right place can help.  The one below (Picture 1) has red grip tape so that objects stay in place, and it provides a high contrast background

 

 

• a lap tray can be a very useful resource for a pupil on the carpet. The one pictured below is £9 from Ikea

 

 

• there are lots of incidental learning activities in play – some children may need more directing to them than others.  If you’re singing 5 little ducks, have you got the ducks in front of the pupil who needs a more multi-sensory way of learning?
• concept books can be useful for reinforcing ideas that pupils haven’t quite grasped.  In the examples below (which I promise didn’t take very long to make), the key factor is that concepts have been exemplified, enabling generalisations to be made.  A circle can be a coin, bottle top, a badge.  It can be big or small.  Two things can look the same or different.  They can feel the same or different or they can sound the same or different

 

 

 

• be careful when reviewing concepts such as big and small, that you are not confusing with other concepts such as colour or shape.  I once watched a lovely session with a TA and two pupils sorting leaves into big and small – she picked a big brown leaf and said big, then a small green leaf and said small.  Her third leaf was a big green leaf.  She asked the pupils whether the leaf was big or small.  They both said small!
• one-to-one correspondence: counting images on a sheet or objects in a line can be tricky for some children.  If you find pupils struggling, try counting from one pot to another; even better if the object makes a sound when dropped into the next pot
• it can also be helpful to add some tactile markers for counting and addition.  This can give some sensory feedback when counting.  Drawing a high contrast line around images/gems (Picture 6) or using a high contrast background (Picture 7) can also make them easier to discriminate

 

 

 

Tips for Key Stage 1

Of course, all the suggestions for Early Years will work as well for pupils in Key Stage 1, but it might be more evident that some pupils are struggling with maths concepts.  One of the major areas of challenge that we have found for pupils who are learning through touch (with braille and tactile diagrams) is how many activities focus on subitising (recognising an amount by looking at it).

• physically explore part and whole to see how numbers live within numbers. If you learn through touch, then you are always working from the part to the whole and you can never just ‘know’ how many objects or images are in the set until you have counted each one individually.  One way of getting around this is to focus on using fixed patterns such as Numicon.  Pupils learn to recognise the Numicon shapes and can compare sizes, learn about odd and even numbers, order numbers and begin to explore number bonds
• many pupils struggled with manipulating the Numicon shapes and fitting them on the white board as they would fall off. Children with difficulties with binocular vision or fine motor skills may also find this tricky so a colleague made a special board from MDF and dowls (see picture below) that allowed them to explore number bonds to 10 with Numicon.  Having the longer dowls meant that they could also be used with beads without them falling off

 

 

• tens frames: there are many ways to represent number bonds or amounts on a tens frame.  Magnets, washers, and a bit of card can make a handy reusable tens frame.

 

 

• having found that some pupils find it tricky to use number beads, I found two simple adjustments that seemed to really help.  Rather than using an abacus, Rekenrek or bead string, where the beads could easily slip back and forth, I found that this handy device (picture below) worked well as the beads stayed in place.  I had to look it up and it’s a golf stroke counter and, of course, you can make your own – just google it!

 

 

• another adaptation was to add washers in between the 10 beads on a hundred bead string (picture below).  This way, rather than counting each bead individually, the pupils were able to identify the tens quicky and count on from there

 

 

• younger pupils may need larger print size when they are becoming familiar with numbers and operation signs.  Have a look at early reading books and you’ll see most of them are in font size 28.  We found this large 100 square (picture below) had a variety of uses, for example, ordering numbers, exploring number patterns or as a physical number line

 

 

• it’s worth being aware that some “helpful” visual representations might not be that helpful for all the children in your class.  It might be necessary to reduce and simplify the visual information – changing pictures into shapes or even putting a thick black line around pictures can often just make them easier to identify

Tips for Key Stage 2

I have found that Key Stage 2 can often be more challenging as the use of manipulatives tends to be reduced and the organisation of mathematical ideas becomes more set (e.g., place value, column addition, formal division methods).  However, there are still things that I have found useful:

• keep manipulatives going for a bit longer.  It’s very hard to understand concepts such as regrouping (exchanging) if you can’t see the physical exchange process.  I was teaching a pupil, who uses braille for learning, how to do column addition; we had to first understand the concept of a column and then what was meant by exchanging a ten for ten ones.  We practised this multiple times before she was confident to do this ‘in her head’
• think representational rather than pictorial.  I’m sure everyone is familiar with the concrete, pictorial, abstract (CPA) approach to maths but what happens if the pupil you’re teaching can’t see the pictures, doesn’t process visual information very well or can’t easily draw their own images?  I would suggest the need to reconsider the pictorial and think more representational – for example, instead of using a worksheet that used different coloured place value counters to represent 1000s, 100s, 100s and 1s, I created some different shaped magnets that could represent the counters (picture below).  It saved huge amounts of time adapting resources and allowed the pupil with VI to work alongside her typically sighted peers

 

 

• reduce the visual processing.  I once observed a maths lesson focussing on place value where the pupil with VI meticulously copied out the place value chart from the board but unfortunately, she took so long, she had missed the explanation and didn’t understand what to do.  Having a ready-made chart that could be written straight onto can be very helpful for pupils who take more time to copy from the board.  I made this one (picture below) for the student I had observed as she found it very difficult to recall which way digits moved when they were multiplied / divided by 10 or 100

 

 

• be aware that pupils who rely heavily on their hearing and do not have the incidental opportunities to see numbers will often write numbers how they are said.  For example, 136 may be recorded as 100306.  These pupils need more opportunities to read and write numbers
• ensure that font size is not too small and there is enough space for working out calculations.  Unlike printed text, numbers and operations signs have no context to draw on and, if you make a simple error such as mistaking a 3 for an 8 or a + for a ÷ then your answer will be completely out!  There seems to be a tendency (especially for homework) to provide teeny weeny printed calculations.  While this might be beneficial for the photocopying budget, it may not be so good for the pupil who struggles to differentiate numbers and symbols
• highlight the operation signs.  Even if print is large enough, some pupils may struggle to differentiate between operation signs so use different colours to highlight the different operation signs and to make them easier to identify
• beware working walls!  (picture below) They can be a fantastic way to record lots of different methods of solving a maths problem but can be very confusing for any pupil who finds it hard to process visual information.  For some pupils, it might be helpful to have methods laminated on cards or magnetic boards in front of them that can be referred to

 

 

It might also be useful to focus on 2 or 3 preferred methods rather than attempting to use all of them – some methods may be less helpful because they are visually confusing (picture below).

 

 

Remember a pupil doesn’t need to have a VI to find things visually confusing and sometimes recognising this can be the difference between them understanding or getting left behind in maths.

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Guest blog authored by Siân Muggridge.

Recently, I have been working a great deal with teachers on developing their skills at scaffolding learning rather than setting different levels of challenge (also referred to as ‘Tiered Learning’). Let me start by clarifying one thing. I am not opposed to presenting different levels of challenge. There are times when it is absolutely the most appropriate form of differentiation for the learners. For example, pupils who have significant gaps in learning or cognitive difficulties. However, it is the dominance of this approach within our primary maths classrooms that concerns me.

What is differentiation?

At its most simple level, differentiation is the teacher’s response to the variation amongst learners in a classroom. However, differentiation is a vast field, with multiple aspects and opportunities that need to be considered when deciding on the most appropriate adjustments to facilitate maximum learning. There is not a ‘one size fits all’ approach; nor is there a magic bullet. It relies on our professional knowledge of our pupils and consideration of a few key questions:

1) What is the long term learning outcome that is required?

2) What is the learning goal of the individual lesson?

3) How do we get our diverse classrooms to the desired outcomes? In essence, how do we go from Q1 to Q2 and back again?

Tiered Learning

This is the dominant form of differentiation that I encounter not only in classrooms, but also in published (hard copy or digital) resources. Fundamentally, teachers provide different tasks/activities to meet the needs of different learners. This approach has been used in UK classrooms for a number of years because it does have benefits. It allows pupils access to learning that they can (or should be able to) complete independently. In examples of best practice, this is a short term change in the learning goal to support pupils to acquire the knowledge, skill or experience to allow them to move to the next stage of learning. It is successful when a pupil has:

i) a cognitive difficultly and is working on a detailed personal curriculum that reflects their individual learning needs

ii) a significant gap in conceptual understanding that must be addressed to ensure that they are able to continue to develop their schema with clarity. A short term drop to accelerate rapidly back up to the levels of their peers.

I am not going to discuss those pupils with cognitive difficulties in this blog – it would be far too long. Please refer to our previous blogs here and here for focused conversation if you would like to explore this further.

I ran, and will continue to run, classrooms where self differentiation of tiered learning will be part of my teaching style. Warning: this form of differentiation needs very close monitoring.

Let us consider the ultimate outcomes if some of our pupils constantly access less complex or less challenging learning than their peers (either through teacher direction or self-differentiation).  When do they get to reach the age-related expectations (ARE) pitch? If in Year 1 they are constantly exposed to learning experiences that are just below ARE, what happens to them in Year 2? Year 3? The gap continues to get bigger. Many Year 6 teachers feel the eventual outcomes of this – a pupil who was just off ARE in year 1 is now significantly off by Year 6. The pupil has missed many of the key learning experiences and their schema is fragmented, limiting their ability to translate learning across the curriculum. Used constantly and without consideration, children could be placed on metaphorical escalators that have different pre-set outcomes, and once you are on one it is incredibly difficult to jump to the escalator above as the gap increases throughout a school career.

Figure 1: Some of the strategies which can be deployed to support learning

It can be successful though, if carefully managed through clear gap diagnosis, focused intervention, careful small step management, and most importantly, that those expectations are firmly attached to ‘getting back on track’. As such, tiered learning is both very time and energy demanding. I would always ask, is there another way to achieve the desired learning outcome?

Scaffolding

But what about these other ways? What else should be considered? This is where the list is almost endless. Figure 2 summarises the more common strategies that a teacher can deploy.

Do bear in mind that other forms of enabling access and tiered learning are not always mutually exclusive. They can have a symbiotic relationship.

For the purpose of this blog, I am going to focus upon scaffolding.

‘[a scaffold is] that [which] enables a child or novice to solve a task or achieve a goal that would be beyond his unassisted effort’ Wood et al 1976, p90

Scaffolding works in a similar way to its physical counterpart – it allows pupils who are ready to learn the concept but with certain barriers, access to learning that they would not be able, or willing, to reach without it. A key outcome of this approach is that they access the same thinking / learning experiences and continue to develop their understanding on a par with their peers.

When working with teachers, consideration of scaffolding opportunities has shifted their thinking about how to approach the needs of their learners. It makes classrooms more manageable, as through successful scaffolding we can all grapple with the same learning, collective power to conquer together. Then on to the next… together.

So what would this look like in the classroom?

Beware of cognitive overload, try to keep it in mind. Stay focused on the key learning outcome, what exactly do you want them to learn in this sequence or step? For example, if the desired learning outcome is to be able to use the language of multiple and factor accurately, and pupils don’t know their multiplication facts, then providing them with a multiplication chart could be the scaffold (after making sure that they how to use it). If they don’t know their facts, this is going to be a barrier to understanding and accurately applying the language. By being given access to a simple chart, they can explore, secure and feel confident in the use of the key vocabulary language, and they will be able to apply it rapidly when they acquire more facts. The focus of the learning is not to be able to recall multiplication facts (that’s a different much longer journey).

1) Are there any concrete manipulative or printed resources that would ease cognitive load? Do they have access to the same resources that were modelled? Pupils should have access to allow them to model out, explore and confirm for themselves as and when they need to. Some pupils may need a manipulative mat in order to contain/structure their resources – or they might get ‘lost in the sea of Dienes’.

2) Speaking frames support pupils to structure responses in full sentences and in how to use new vocabulary appropriately. These have been a hugely successful part of ESSENTIAL maths, as the development of verbal reasoning is closely linked with the ability to construct personal schemas and knowledge into long term memory. In some cases, pupils are not able to track from the modelled speaking frame and need their own personal copy, so that they can track with their fingers etc.

3) Recording frames work as organisers, supporting pupils to stay focused on key learning and enable some pupils to have more practice. Let’s take these two key functions separately.

4) Check-it stations support the development of independence and self-correction at the point of learning. It has been proven that to find and identify errors in your own learning is vastly more powerful than having them identified for you. Recently in a classroom, I asked one boy who had completed a set of calculations how he knew if he had them all correct. He shrugged his shoulders and said Mr X would mark his book. How much more powerful would it have been for that pupil to have checked some of his responses as he was working through them? Check-it stations can vary in content and level of support, but the default setting is simply just some of the answers. Then the responsibility lies with the pupil to identify where and what the error is and how the correct solution is reached. This prevents the proliferation of error within a lesson, has the added bonuses of more confident, self-checking pupils and frees the teacher to work with guided groups. Many schools who employ this now have editing pens at their check-it stations as well. A quick word from the experienced check-it station user – have at least 2 in your classroom to avoid the bun fight as everyone heads at the same time.

Staying focused on learning

A maths classroom is full of distractions. A recording frame can support a pupil in internalising a thought process, supporting development of efficient and clear recording, learning to contain their pictorial representations and symbols to ensure that they can follow their own chain of thought. They can also be adapted to enable greater depth.

Figures 2 and 3 Examples of speaking frames from ESSENTIALmaths

 

Figure 5 year 1 recording frame with challenge, ‘Can you complete the rest of the recording frame to make this true?’

 

Imagine you are a child who doesn’t write as quickly as your peers. Perhaps you get lost half way through a sentence as you are having to work harder on spelling and letter formation. You get through less practice. You might never get to the ‘tricky’ questions (this is why misconceptions must be exposed early in a practice sequence, but that’s for another time). What is the impact on your learning? Is this because you couldn’t ‘do the maths’? No, it is possibly because your fine motor skills aren’t as well developed as your peers or that your working memory is getting overloaded with other stuff. But you will eventually fall behind and the gap will grow – this isn’t fair.  This is further explored in this blog here. 

We need to get as many pupils as possible to become fluent and confident mathematicians. This is not for school league tables, this is because maths is a key life skill required to function successfully in our society. The longer children stay on parallel curriculums, through constant tiered learning, the harder it is to move them off. Are we content to have different expectations for pupils? Is it okay to leave children trapped on metaphorical learning escalators?

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In this blog, I have only outlined a few of the different differentiation strategies that a teacher can deploy. If you would like to explore this further and gain a wider portfolio of strategies, please join us on our central training.

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Wood, D., Bruner, J., & Ross, G. (1976). The role of tutoring in problem solving. Journal of Child Psychology and Child Psychiatry, 17, 89−100

Berk, L., & Winsler, A. (1995). Scaffolding children's learning: Vygotsky and early childhood learning. Washington, DC: National Association for Education of Young Children

My name is Vicki O’Brien and I am the maths subject leader at Therfield First School – a small, village school. There are 52 children from Reception to Year 4 in mixed-age classes and I have been teaching here for 8 years. Teaching in a small school has many benefits – we know our children and their families really well and there is a real sense of community. In class, we are able to target our teaching based on our understanding of each child’s starting point in each concept as we move through the year. However, with maths teaching, our mixed year group classes do present a challenge for us. Firstly, because of the specific curriculum requirements for each year group and secondly, the varied individual starting points for children. But one thing I know about our school is that we always rise to a challenge.

Last academic year, I engaged with Herts for Learning and the Matrix Maths Hub in the NCETM Mastery Readiness programme and this journey to develop mastery teaching across the school continues for us this year. As a school, we have always been passionate about growth mindset but for some of our children, maths is a subject that has historically caused a level of anxiety for a variety of reasons.

I wanted to write this blog to share how considering our school vision for maths and implementing fluency sessions has led to my class becoming confident mathematicians who enjoy exploring, sharing and bouncing ideas off of each other and how I manage this logistically in a mixed-age class.

Why did I decide to add fluency sessions to our timetable?

We have tried a number of mixed-age planning materials and have now found one that works for our school in terms of concept mapping. However, having used these materials for the past 3 years, we felt that there were gaps in the teaching plans in terms of developing fluency and arithmetic proficiency. This is something that we explored through the Mastery Readiness programme and I felt that this was an area we could develop further.

The children were not always making connections between what they already knew and new learning, the steps were sometimes not small enough for all children to secure understanding and I felt there needed to be more time for practice. This was having a particular impact in our mixed age classes because it felt like we were going from one thing to another and it felt rushed.

In addition, we also felt that the materials we were using to flash back to prior learning weren’t enabling development of language or providing a low-threshold, high ceiling approach. The materials were designed for specific year groups, the questions were quite closed and didn’t explicitly involve giving any explanation beyond the ‘answer’. The children weren’t excited about them.

When we returned from the national lockdown in spring 2021, we needed something that would be accessible for all to help us find and fill any gaps as quickly as possible to allow teaching of the maths curriculum to continue with understanding. The children had engaged well with home learning but when back in class, were finding it difficult to articulate their thinking or express how they had worked things out. So in April 2021, we purchased the Herts for Learning fluency slides and have been using these in Key Stage 1 and Lower Key Stage 2 since Easter.

The launch

Just after the Easter break, I led an INSET for teachers and TAs to explore how use of these materials could benefit our children. We considered our shared vision and culture around the importance of maths and our belief that every child can achieve. We looked at our timetables to ensure that we could give this proper time to support the development of mathematical fluency.

What I did and what I’ve noticed

As I said, I started using the fluency materials with my Year 3 & 4 class in spring. I chose to do 3 slides that I felt could be used to explore both Year 3 and 4 content. My main aim was to get children to be confident and vocal about what they were learning – talking, proving and investigating.

In my class, I have children working at Y2, Y3 and Y4 level with some who could make links beyond that. To start with, I took something they had already learned to develop confidence with the structure of the sessions and so everyone felt comfortable to participate.

Over time, we explored the 3 concepts at great depth and made connections to things they had learned this year (and in previous years) using sentence structures such as, “If I know ‘this’ then I know ‘that’”. I noticed how engaged they all were – including those who have previously been reluctant to contribute to whole class discussion.

Multiplicative fact recall and reasoning

When we first started, the children had manipulatives on their tables to build arrays. I showed them one on the Interactive Whiteboard and they built it themselves with double sided counters and explored it using the language I provided.

 

 

For each array, children talked to each other about what they noticed and I modelled to them how we would describe it in full sentences, e.g. I can see 4 sixes. This is 4 times 6. The children then repeated this back, also in full sentences to support the language development.

 

 

For this array, the children flipped over the top row of their double sided counters. The overall array hadn’t changed but now we were describing the equal parts differently. We talked about ‘what’s the same?’ and ‘what’s different?’

Everybody has the manipulatives. The children don’t perceive others as ‘better’ or ‘worse’ at maths than them. This approach proved valuable for those ‘fast finishers’ as they were having to model and explain using accurate language. For all children, their multiplicative understanding was deepening.

Over time, they were able to consider how they would split arrays to make calculation easier. This depended on each child’s known facts. For example, the example below shows 6 sevens split into 3 sevens and 3 sevens. For some of my children, this helped with calculating. But for some who were not yet confident with their 3 x table, it was easier for them to split it into 5 sevens and 1 seven, allowing them to use a fact from their 5x table and then add on one more group of 7. This was allowed… and encouraged!

 

 

After four weeks of manipulating arrays and having these conversations, the children were increasing in confidence at working out unknown products and were thinking much more flexibly. We then continued without the manipulatives with larger numbers.

 

 

 

They were now starting to consider benchmarks more naturally and consider numbers that were easy to multiply. So below, spotting that they could calculate 10 x 7 + 5 x 7 to find the overall number of dots.

 

 

Geometry

 

 

The second concept we have focused on is geometry – properties of shape. For my mixed-age class, this was perfect to get some rich discussion going, developing accurate mathematical language at the same time. All children are able to contribute and over time, their knowledge became more secure and they are much more able to describe what they notice.

The focus might be on triangles one week and then quadrilaterals the next. Each day, it’s very similar… but different. This keeps the children thinking but allows them to continue to make connections each time and build up what they know.

 

 

I keep a bank of images to choose from to adapt the slides each time so it doesn’t take long to change it up to build upon the previous day. Having the language down the sides takes the stress away from the teacher and it’s there for everybody to use – adults included!

Place value… and more

When we first started a place value slide, it would just involve talking about the simple place value – thousands, hundreds, tens and ones. Now we talk about greater than / less than, patterns, rounding, doubling, halving, odd, even… Before I even say it, the children are adding or subtracting the numbers automatically. They are not constricted but allowed time to explore.

Sometimes we discuss a number as a whole class, bouncing ideas off of each other. Other times, children work in pairs to record their thoughts and to link the number to other areas of maths. If I notice an error or potential misconception, we can talk about it there and then in the session.

 

 

 

The children are growing in confidence to try things out and then share them with the class. It doesn’t matter whether they are in Year 3 or Year 4, they are bringing their ideas together and they have no fear of failing as it’s all a discussion. They pick up on things I hadn’t even thought about. When we learn something new in class, they bring this to the table too. So for example, after we had learned about measurement, they began making connections to the place value and thinking about how the numbers would be affected by multiplying or dividing by powers of 10.

Headteacher, Tara McGovern, says of these fluency sessions, “Children suddenly immersed themselves in the maths and were proud about what they know. They were exploring numbers, bringing what they knew to the table. We have found previously that children have stated that they find maths hard or said they can’t do it. They’ve put themselves in a box. But in these sessions, the children are enthusiastic and the collaboration between the children is a joy to see.

In one session for example, the children were talking about related facts. One child explained what would be ten times greater. Another child then suggested an inverse operation and it ping ponged around the class, accelerating from something that could have been simple fact recall.

These short bursts of fluency are buzzy sessions that move across the curriculum. Participation is high for all children and they are developing more independence and stamina as time goes on.

Confidence has been brought back into the classroom for the upcoming maths lessons and children are recognising themselves as mathematicians.”

As the class teacher, in the first half term, I noticed a big difference in what the children could do. Fluency slides have given the children a voice. Even those who may struggle in maths lessons can share what they know and learn from others.

 

Blog authored by Vicki O’Brien, Therfield First School, Hertfordshire.

 

Professional development opportunity

Join us for face-to-face training at the Hertfordshire Development Centre to:

• consider the rationale for developing fluency sessions. Why do pupils need to be ‘fluent’ in maths and what do they need to be fluent with?
• explore possible structures and ways of organising sessions
• consider the appropriate content for fluency sessions, including what is appropriate for the year group and time in the academic year
• take away resources for a staff meeting to use with staff and a full set of (revised and updated) fluency session materials to use and adapt with each year group

Developing effective maths fluency sessions

In nearly every class across the country the Autumn term’s maths learning starts with number: place value, magnitude – ordering and comparing number. A very good place to start if you ask me, as it is arguably the ‘key stone’ to all maths learning. I would say that without a secure understanding of number most other maths learning, especially numeracy, becomes a series of memorised processes using digits rather than numbers.

What do I mean by Numeracy?

I was fortunate to hear Mahesh Sharma speak at the British Dyslexia Association’s Maths and dyscalculia day and in his provided notes he defines numeracy as:

“…the ability to execute standard whole number operations/algorithms correctly, consistently, and fluently with understanding and estimate, calculate accurately and efficiently, both mentally and on paper using a range of calculation strategies and means.” (2019, pg 19)

In summary I think pupils need to be able to:

• identify the value of the numbers in the calculation
• choose the best strategy to solve the calculation
• understand what is happening to the values whilst carrying out the calculation
• complete the calculation accurately
• repeat this process many times

I would argue that some pupils can do the last two bullet points but can’t do the first three. These pupils have learnt a process with digits rather than becoming numerate. I think this is proved over and over by pupils who automatically use standard written methods for any calculation with what seems like very little consideration for the values in the calculations. For example: 245 + 99, £10 - £3.99, 3682 ÷ 1 and 816 x 10 would be answered -

The calculations have been answered accurately and pupils do repeat this many times but pupils have not: identified the value of the numbers in the calculations, chosen the best strategy or applied their understanding of number to help them understand what is happening to the values. They are not secure with number. In these examples the pupils have been accurate but a lack of understanding of number can lead to many errors being caused or misconceptions being repeated over and over again.

In these addition and subtraction examples try and work out what has gone wrong.

 

a. not included the additional ten regrouped when adding the ones

b. included all 15 ones in the answer, hasn’t regrouped

c. recognised that there are 12 ones but put the 1 ten in the answer and carried the 2

d. added together the four digits in the calculation 2+5+3+1

e. taken the biggest digit away from the smallest digit in the ones column

f. recognised the need to regroup to be able to take away 4 but not shown that there is now 8 tens in 92

g. recognised the need to regroup but unsure about how to regroup with a no tens

h. recognised that 5 ones is less than 6 ones so shown borrowing a ten but not changes the 5 ones to 15 ones and then taken the biggest digit from the smallest.

 

In all cases the errors are caused by a lack of understanding of place value.

What does the intervention need to be to help stop these errors?

In my experience what pupils who makes these errors is definitely DO NOT is need more practice of column addition and subtraction. In most cases pupils need to secure their understanding of what happens to tens and ones when counting up and down – regrouping ten ones for a ten when counting up and regrouping ten ones for ten when counting down. Once secure with this they need to be able to regroup numbers into hundreds, tens, ones in a number of ways.

I have found that one of the best ways to support pupils in securing this understanding is to play a simple game sometimes it is called banker, other times it is called race to 50/100/zero.

To play the game the pupils need a place value chart with either hundreds, tens and ones or just tens and ones depending on your start and end points and possibly the age of the pupils. In either case it is important that there are two tens frames embedded in the ones column. 

The embedded tens frames enable the pupil to recognise by looking when they have more than ten ones so knows that regrouping needs to happen. The easiest way to generate the number being added each time is by rolling a dice (a ten sided dice works best but an ordinary dotty dice works too). When using the game to count up to a given number the pupil rolls the dice and adds the same number of base-10 ones cubes to the chart as the number rolled. So if 7 is rolled seven cubes are added. At this point encourage the pupil to subitise how many more to ten – the tens frame supports this. Once more than ten ones cubes are in the chart the pupil will need to regroup. The pupil say the total and how it is represented, identify what is going to be regrouped and then restate the total after the regrouping and how it is now represented.

For example:

When the pupil groups ten ones for a ten ensure that they understand that this is the same value. The base-10 works really well for this as ten ones cubes is exactly the same size as a tens rod. This regrouping and explanation of what is happening needs to happen every time there is more than ten ones.  

 

This game needs to be play repeatedly because as well as developing understanding of regrouping they will also be developing their visualisation of the number and conservation of number recognising that as regrouping is done the total amount doesn’t change.

If you think back to the errors made in the addition calculations (a-d) all of those errors were caused by a lack of understanding of regrouping the ones (a-c) or not understanding the value of the digits (d). To bridge the gap between the improved understanding of regrouping from the game to the written strategy expanded column addition will help to make the place value explicit, initially the calculation can be done using the base-10, then represented pictorially and finally written.

For example for 57 + 25:

To support the understanding of how regrouping is needed in subtraction again an adaptation of the ‘Banker’, ‘Race to…’ games are the best starting point. The same place value chart is used but this time the pupils start with an amount and need to count down. This time when a decade boundary is crossed a ten will need to be regrouped for ten ones when there isn’t enough ones to take away. As before the explanation of the regrouping needs to be articulated each time it takes place.

As with addition all the errors made with in the subtraction calculations above were caused by insecure place value understanding. The most common mistake I see with subtraction is like in calculation e) when pupils take the smaller digit from the biggest, practice of regrouping using the ‘banker’ game will help reduce this common mistake. Then using the CPA (concrete, pictorial and abstract) approach and an expanded written method the link between the regrouping when counting back and written subtraction can be made.

For example 73 - 46:

It is the practical element of the game using explicit place value equipment (base-10) physically regrouping the tens and ones alongside the verbal explanation of what is being done that helps to secure the understanding of what is happening to the numbers as they pass through decades. When this is then linked to a calculation strategy pupils make the connections and are able to carry out the process of column addition or subtraction with a much better understanding both of the process and the value of the numbers within them.

They have become more numerate.

Thinking back to Sharma’s definition of numeracy what this blog hasn’t covered is whether column addition is the best strategy or whether another strategy would be better. Considering the original examples:

 

Arguably all of these would all be better answered using a mental strategy but that’s another blog.

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To find out more please book onto the following course

Mathematics intervention using a place value diagnostic assessment and teaching programme resource

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References

Sharma, M. (2019). Dyscalculia and other mathematic difficulties.