By the time this blog hits your screen, there will be approximately 26 teaching weeks until this year’s KS2 SATs tests. If you are teaching maths daily, that’s around 130 maths lessons to cover all the new year 6 learning and to revisit, retrieve and secure the rest of the KS2 curriculum. When you look at the numbers, time is tight.

Yet this precious time can get consumed by testing regimes in a bid for children to familiarise themselves with the test in the hope that their scores will improve with each new test they do. In the recent Ofsted maths subject report, the summary of the research review relevant to assessment stated:

As educators, we have likely all experienced the frustration of teaching something, feeling positive about the children ‘getting it’ to then be presented with blank faces when their understanding is tested or they are required to apply this knowledge in a problem solving context.

Teaching approaches to problem-solving is just as much a part of a strong curriculum as teaching the maths which problem solving calls on. In fact, it should be an integral part of the teaching and learning process.

According to Polya’s four-step model (1957), learners progress through four stages when faced with a problem: 

It is important that we model to the children the thought that gets us into a problem by slowing down the problem-solving approach and making the reasoning behind it explicit.

How can we teach the skills of reasoning and problem-solving?

Let’s consider how this may look in the classroom using a question from the 2023 key stage 2 mathematics Paper 3 (reasoning):

First, pupils should be encouraged to look at the models and information they have been given.

By removing the question, we are encouraging them to identify what they already know and to ask questions of what they have been shown.

Not only will this support them with noticing the maths and making connections, but it will also serve as a useful tool when they progress to stage 4 in Polya’s model and are reflecting on their solution. 
 

Next, we can start to refine the thinking of the learners by considering what vocabulary we would like them to be using when discussing the model provided. This signposts them to what they should begin to recognise.

The use of speaking frames supports the expectation of full sentences and analysis. At this point, you may ask pupils to consider what their peers have noticed, question it and build upon it: ‘If that’s true then…’.

By modelling to the children how we would filter the question and relate it to concepts, we begin to provide a tool for them to do this independently. 

Once the children have had the opportunity to find the maths and consider what they know from the image, we can reveal some more information for them to explore.

This is where they can begin moving into stage two of Polya’s model. They know what they know and what they don’t and now they can consider how to use this to devise a plan.

At this stage, you could ask the children to consider:

• How could we present this using a bar model?
• What questions do you have?
• What calculations could give you more information?
• What could the final question be?

This scaffolds the problem solving process and enables the pupils to have a far richer discussion around the question than simply solving it under test conditions. At each point in this process, we are reducing the number of possibilities for the solution which again supports the final stage of reflection. 

Finally, we reach the big reveal.

By the time we have shown them the problem in its entirity, children would have already unpicked what is required of them and feel more confident to tackle the calculation.

You may then simply ask them to solve the problem and could even extend their thinking by asking:

• How would you change the question to make it more challenging?
• Could you represent the maths using a concrete resource (e.g., Cuisenaire)?
• Could you represent the problem using two different models?

What does teaching offer that testing doesn’t?

When used on a regular basis (in place of continual testing), this approach will serve to familiarise the children with KS2 SATs test structure but also develop and enhance all children’s problem-solving skills in a collaborative way. This could be incorporated into fluency sessions, at the beginning of maths lessons or intertwined with lessons where the content is relevant.

Teacher modelling and rich discussion between peers will enable children to make links across the curriculum, support them to notice where connections can be made to what they know and develop their ability to decide what they need to find out and how. We want this to become part of their natural response when applying their knowledge and skills within a problem-solving context.

Upcoming training for Year 6 teachers

For practical, face-to-face training in Reaching the expected standard in year 6 maths, join us at the Hertfordshire Development Centre in Stevenage, Hertfordshire, on the 19th of October 2023. Use the link above to find out more. Booking is now open.

We can teach these problem-solving skills throughout children’s time at school

This strategy for exploring problems is useful not only in Year 6, but as a tool to support the development of these skills across key stage 1 and key stage 2.

By using images from real-life contexts, we can expose children to mathematical thinking and continue to build upon this as their schemata grow. 

Below, you will find an example of how the problem-solving approach could look in Year 1.

The HFL Education primary maths team have developed a set of these real-life reasoning slides. Look out for further information about them in our monthly newsletters.

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References

2023 Key Stage 2 mathematics paper 3: reasoning

Contains material developed by the Standards and Testing Agency for 2023 national curriculum assessments and licensed under Open Government Licence v3.0

Ofsted (2023) Coordinating mathematical success: the mathematics subject report available at: https://www.gov.uk/government/publications/subject-report-series-maths/coordinating-mathematical-success-the-mathematics-subject-report (Accessed: 19 September 2023)

September. Even more so than January, for those of us who work in education, the start of the autumn term is the fresh start, the time for plans and resolutions about how we want to work differently this year. The summer holidays have hopefully given some time and distance to the year that has passed, with a chance to reflect on our own professional WWWs and EBIs (‘what went wells’ and ‘even better ifs’). Now we’ve got past the initial busy first few days of term, it is a good time to reflect on plans for the coming year.

When we talk to SENCOs, their (academic) new year resolutions often include the ambition to be more strategic – to find a way to move beyond completing the never-ending paperwork and to increase their focus on leading inclusion across the whole school. We hear, ‘I want to get into classrooms more’ or ‘I want the chance to do more staff training.’ 
SENCOs understand the value of time spent advising, supporting, and training their colleagues and recognise that consistently high-quality teaching is one of the greatest levers to ensure good outcomes for learners with SEND.

Remember, the SENCO’s role comes with strategic responsibilities as well as managing the day to day: 

The recently published SEND and AP Improvement plan (March 2023) reinforces this expectation, describing the need for the SENCO role to be “whole-school, senior and strategic”

And yet, being strategic doesn’t happen automatically, and it can be hard to carve out time to focus on these high-impact activities. The reality is urgent tasks are visible, often important and every SENCO spends a significant part of every day reacting to demands on their time: a dysregulated child needs immediate support; a phone call interrupts your focus; staffing absence needs to be covered. 

This is not something to feel guilty about…. it is something to be curious about and to reflect on. The good news is that most strategic tasks often don’t take that long once you have found time for them.  The 80-20 rule is the widely regarded principle that if you focus on the right priorities you can get 80% of the total impact from just investing 20% of your time in the right place. (What is the 80-20 rule, and how to apply it in your life | Tony Robbins).  How might this work for you? For example, a one-hour staff meeting training about the early identification of need may have a positive impact that eventually saves you time minimising the number of ad hoc conversations with teachers throughout the year. Or, taking some time to research and introduce the latest approaches to supporting self-regulation, will hopefully mean that, over time, there are fewer incidents of dysregulated behaviour that urgently need reacting to.

So, if your new year intention is to be more strategic this year, here are five top tips for how you can increase the likelihood of success. 

1) Make a strategic plan: 

The first task is to be clear what leadership and strategic activities you want to complete. To maximise the impact, these need to be relevant to the context of your school, so a good starting point is to review your self- evaluation framework (such as the Hertfordshire SEND Benchmark and Planning tool) and your school’s development plan for the current priorities. Be selective, not comprehensive. From your settings’ priorities, decide only three or four strategic activities for the year that will add the most value.

The ‘Big Rocks’ concept might be useful here; the origin is unknown, but this story was made popular by Stephen Covey in his book The Seven Habits of Highly Effective People. Greg McKeown summarises it in his book, “Effortless”:  

In this metaphor, the big rocks represent the strategic monitoring and evaluation activities such as learning walks, training, policy review and development, pupil progress meetings, analysis of trends and patterns. The small pebbles are the operational tasks such as planned meetings and processes, referrals, administration. The sand is the day to day urgent and visible tasks that you need to react to.

Once you have chosen your ‘big rocks’ for the year, set SMART objectives for exactly how and when you are going to complete these and block out time for them in your diary. Research shows that scheduling when and where you will do something makes it much more likely that the task will get done. (How to Focus on What’s Important, Not Just What’s Urgent (hbr.org)). 

2) Find an accountability friend: sharing your plans increases the commitment to them

By publicly announcing an activity, for example that you will ‘launch a parent questionnaire in May 2024’ to your senior leadership team, to your teaching staff, or even to parents in a newsletter, there is less chance that this action will be ‘bumped’ by urgent firefighting tasks. 

Planning a joint activity with your SEN link governor is a great way to keep focus; whilst colleagues might interrupt you if working on your own to ask for help with an urgent issue, they are more likely to find another way if they can see you are in a scheduled meeting. To support you, we have provided templates for eleven strategic monitoring and evaluation activities you can complete alongside your SEN link governor in the Hertfordshire SEND toolkit. 

3) Do not try and do too much

If this way of working is new to you, then be realistic about what you can achieve within the time you have available. One good quality strategic monitoring or evaluation activity per half term might be an initially achievable aim. If you didn’t manage to fit in any learning walks last year, then it’s highly unlikely that you are going to be able to complete one a fortnight this year. So be kind to yourself and don’t set yourself up to fail.

4) Share the strategic tasks for SEND between SLT and middle leaders:

As Gary Aubin puts it, in ‘The Lone SENDCO’:

Can subject leaders include subject specific adaptations for learners with SEND in their CPD sessions for staff? Do all leaders review learners with SEND in their monitoring activities? Can whoever reviews attendance in your school provide you with the analysis for learners with SEND? Can your SEN link governor complete a website check for you?

Whenever you can work ‘with’ someone, you will also be adding that very important value– capacity building in others by sharing your experience, knowledge, and skills of SEND.

5) Identify the unimportant and non-urgent tasks that you keep doing

Find ways to either streamline these processes so they take less time, delegate them to someone else or give yourself strict time limits to complete them as quickly as possible. Also be aware of others who take your time; it can sometimes feel that your non-class time is swallowed up by whatever is needed; covering playground duty, putting up a stage, photocopying assessments. So be assertive enough to say the polite, “yes, but…” such as “I can help with lunch duty today, but that means I won’t be able to complete the attendance analysis I had planned.”  Once the requester is aware of the consequence of what they are asking you to do, often they will rethink and find a different option for solving their problem. 

Hold tight to your ambition to be more strategic – to spend more time in classrooms and in improving outcomes for learners. You will need to be adaptable and resilient as challenges on your time throughout the year are inevitable, but your ability to respond, learn from failures, and regroup will steer you through. 

And finally… don’t forget to acknowledge and celebrate the achievements and positive impact that result from your strategic leadership in the SENCO role. Good luck this year.
 

The technical details of the 2023 progress model for KS2 have now been published by the DfE. The model is broadly similar to the 2022 model, which I blogged about here. Every year there are minor adjustments to the ‘expected outcome’ for each Prior Attainment Group – these adjustments are down to slight changes in the national average outcomes each year. 

The more significant change affecting the model this year relates to changes in the way children were assessed at KS1 in 2019, compared to 2018. (The 2019 KS1 cohort are of course the children who have just completed KS2.) The differences at KS1 relate to how children were assessed if they were working below the ‘Working Towards’ standard. 2019 saw the introduction of the 'Pre Key Stage Standards’ (PK1 to PK4 at KS1) and the removal of P-scales from P5 to P8. In constructing a model for calculating progress from KS1 to KS2 in 2023, scores needed to be assigned to each of these pre-key stage standards so that a measure of ‘prior attainment’ could be produced.

The range of scores used then for converting KS1 attainment into a prior attainment score for calculating KS2 progress now looks like this:

For children with SEND assessed using P-scales, the points range from 0.25 (P1i) to 1.75 (P4).
As per previous years’ models, the prior attainment score is found by converting a child’s teacher assessments in reading, writing and maths into scores, from the above table, and then combining those three scores into one score as follows:

• calculate the average of their reading and writing scores, to give a score for English
• calculate the average of this English score and their maths score

Based on this ‘weighted average’ score, children are then allocated into a Prior Attainment Group (PAG) and this forms the input measure of the KS2 progress model (in all subjects). 
The full details are available in the DfE’s technical guidance, but here are two simple examples:

Example 1 – a child who was assessed at EXS in all three subject areas at KS1

Their combined prior attainment score is 8 points.
In 2023, the national average scaled scores for children with a prior attainment score of 8 points were as follows:

This means that, to achieve a better than average progress score, children with this prior attainment need to score above those figures. If they achieved a scaled score of, say, 107 in reading, their progress score would be +1.62 (the difference between 107 and 105.38). If they achieved a scaled score of 103, their progress score would be -2.38.

These average outcome figures are only marginally different from those in the 2022 model (effectively the same, to the nearest whole number).

Example 2 – a child who was assessed at GDS in all three subject areas at KS1

Their combined prior attainment score is 10 points.

In 2023, the national average scaled scores for children with a prior attainment score of 8 points were as follows:

In this example, to achieve a positive progress score, the child would need to achieve a scaled score of at least 114 in reading or 113 in maths.  These are both 1 scaled score higher than was the case in the 2022 progress model.

Understanding the way this works for writing is a bit more complicated, as I discussed in the blog about the 2022 progress model. 

I have produced a free downloadable attainment & progress calculator tool (scroll to the bottom of this page to download it). Using this tool you can easily see for any pupil, by entering their KS1 attainment and either their raw scores from the KS2 tests or their scaled scores, what the ‘expectation’ (i.e. national average outcome) is in the 2023 model. The tool then calculates each pupil’s progress score and produces a cohort-level average progress score. It will also indicate whether this score would be deemed to be ‘statistically significant’ (more on this below).

A cautionary word about analysing this data

It is always best to start from the view that data can raise interesting questions and form the starting point of a discussion, rather than jump directly to conclusions from the data. Data tends to raise more questions than it answers. 

The methodology used to calculate a progress model inevitably carries a bit of ‘statistical noise’ or margins of error, as it is based on what has happened ‘on average’ for different groups of pupils, but no child actually behaves in an ‘average’ way. Every child is unique and complex, and there are bound to be day-to-day differences in how they perform in an assessment activity. So, there may be children whose progress scores come as a surprise – maybe they performed better than you might have expected on the day of the test, or maybe the opposite. At cohort-level though, these variations (some positive, some negative) would hopefully mainly cancel each other out. It is when you find that a large proportion of learners within a cohort appear to have underperformed that we might begin to notice this as a pattern and ask questions about what happened.

This is where statistical significance can help us. As the size of the cohort increases, variations from the norm become more meaningful and might become more worthy of exploration.

It is important to be clear about what is meant by ‘significant’ in this context.

‘Significantly’ does not mean the same as ‘very much’ and ‘significant’ does not necessarily mean ‘important’

In common parlance we might use ‘significantly’ to mean ‘very much’ or ‘noticeably’, e.g. “it is significantly hotter today than yesterday”. In the language of statistics, ‘significantly’ has a more precise meaning than that – it is better to think of it as expressing a degree of confidence in the data.  If a progress score is statistically significantly above or below the national average, what that means is that we can say with a high degree of confidence that progress in that school is above or below the average.  It doesn’t necessarily mean that the progress score is a lot better or a lot worse than average, just that it is better or worse and that we can be pretty confident in that assertion. (95% confident, to be precise – although even a 95% certainty does still mean we will be wrong 5% of the time.)

Likewise, significance is not the same as importance. If something in the data is deemed to be statistically significant, that might mean it is important but, as the song goes, ‘it ain’t necessarily so’. (And something could be important even without being statistically significant.) 

It is for the users of the data to interpret and decide (taking into account all their other contextual knowledge about a school and that particular cohort of children) whether something is important or meaningful. For example, we might observe a significant drop in results from 2022 to 2023 – but that drop might have been entirely expected based on the two cohorts being very different, so we wouldn’t necessarily be concluding that some radical action was required based on that one observation. 

So, what about next year and beyond?

This 2023 data will be the last time we see KS2 progress scores for a few years. Next year, the children completing Year 6 will be the children who finished Year 2 in Summer 2020, when statutory assessments were cancelled due to Covid. The same applies to those children who complete KS2 in 2025. So, in those years there will only by KS2 attainment data – no progress measures at all (as has been confirmed by the DfE).

2026 and 2027 will see a return to progress scores based on the sort of methodology that we currently have in place, i.e. based on children’s progress from KS1. But after that it all changes. The children currently in Year 2 will not be statutorily assessed at the end of KS1 and their KS1 attainment will not be collected by the DfE. This cohort of children all took part in the Reception Baseline in the academic year 2021/22, and that will be the input measure used to calculate their progress when they complete KS2 in 2028. (This is, of course, assuming there are no changes to national policy between now and then.)

It is perhaps worth taking this opportunity to remind readers that progress is always more than just a number. The published progress scores serve to give us an indication of how well pupils have made progress compared to the national picture, but these statistics are always a proxy for what really matters, which is the actual lived experience of children in that school: how well have they made progress through a well-thought-out curriculum, how securely have they acquired the knowledge and skills expected of them, to equip them for the next phase of education? Even in the years when we will have no published progress scores, these will still be crucial questions to explore. I wrote more on this in a previous blog, What do we mean by ‘progress’ and how can we reassure ourselves that pupils are making it?
 

Question: What do Albert Einstein, Chris Packham, Jazmin Scarlett, and Sarah Rankin all have in common?

Answer: All of them were pupils with SEND who have gone on to have outstanding careers in science. 

• Albert Einstein, Physicist, dyslexia
• Chris Packham, Naturalist & tv presenter, autistic spectrum disorder
• Jazmin Scarlett, volcanologist, juvenile arthritis
• Sarah Rankin, Stem cell biologist, dyslexia & coordination disorder
• For more information see: The Royal Society – Celebrating Scientists with disabilities

A classroom teacher looking over their new class registers for September may feel daunted by the prospect of meeting the individual needs of a large number of pupils with SEND in their lab. However, with the right challenge and support there might just be the next Einstein in the classroom!

The number of pupils identified with SEND has increased by 87,000 since 2022 in English schools. (ONS 22nd June 2023) In secondary schools alone this number has increased by almost 24,000. Numbers have increased for the 6th consecutive year and there are now over 1.5 million pupils identified with SEND ranging from the most significant and complex to comparatively minor in English schools. (Ofsted Research & Analysis: Supporting SEND). Most of these pupils attend mainstream schools and are entitled to a broad and balanced curriculum, which includes challenge, high expectations and builds aspirations. It is important to ensure that the curriculum is not made easier but that students receive the right adaptations to the curriculum and high-quality teaching so that they are able to access new learning.

How can we ensure that our delivery of the Science curriculum enables our pupils with SEND to be successful?

Helping students to break down science knowledge and tasks is a key step in supporting them to be successful scientists. Rob Butler (ASE) describes sensible strategies to manage cognitive load for pupils with SEND during practical work. Adam Boxer (RSC) talks about the benefits of the slow practical. Whilst recognising the need for individual risk assessment for pupils with SEND, (CLEAPSS guidance, G077) there are many similarities between the two approaches which suggests that strategies which improve outcomes for pupils with SEND will also improve outcomes for all pupils. Teachers should feel reassured that high quality teaching for students with SEND does not come at a cost to other students. 

The EEF (Education Endowment Foundation) has said “to a great extent, good teaching for pupils with SEND is good teaching for all.” They have identified five specific approaches, which are particularly well evidenced, as having a positive impact on pupils with SEND.

Here we focus briefly on one aspect and discuss the importance of metacognition as an example. The EEF states that the use of ‘metacognitive strategies,’ which involves pupils thinking about their own learning, can be worth the equivalent of an additional +7 months’ progress when used well. 

A good starting point for developing metacognition in secondary science teaching is to focus on using the cyclical 'plan, monitor, evaluate and regulate' as a structure. Adapted here by John Spencer: Five ways to boost metacognition in the classroom. 
 

This can be used in science lessons alongside complex calculations, practical activities, and evaluations, allowing students to be systematic and strategic in problem solving. This supports the learning of disciplinary knowledge and working scientifically, which is embedded throughout the secondary science curriculum. 
Supporting pupils with SEND to use the metacognition cycle in their approach to scientific tasks enables them to break down the process and reduce cognitive overload.

A very brief overview of questioning to support pupils in working through the cycle is shown here:
 

Teachers explicitly modelling this thinking for the pupils and walking them through the cycle supports them in embedding the process and applying this questioning to further scientific tasks. Alongside this cycle the other EEF recommended strategies such as scaffolding and explicit instruction should take place. Watch out for future updates on this. 

This is just an example of how one of the recommended ‘five-a-day’ approach can be used within your science lessons to ensure that high-quality inclusive teaching is delivered. 

If you would like more information about, SEND, Secondary Science or the Secondary Science conference please contact:
Joanna.Conn@HfLeducation.org Secondary Science adviser
Anna.Mapley@HfLeducation.org Secondary Science adviser
Becky.Rothwell@HfLeducation.org SEND adviser.

In the first of this mini-series, (Making fluent and flexible calculators: why counting is not enough), I reflected on the importance of taking a deep look at how children solve simple addition and subtraction calculations. I considered why it wasn’t enough just to let them solve simple calculations through counting and the need to stop children from being caught in the counting trap.

This time, I have turned my attention to understanding why gaps in developing additive reasoning lead to difficulties later in children’s longer maths journey. I return to my crucial statement - 'Why getting the calculation correct is not enough’. But this time, I consider why we must blatantly pay attention to additive reasoning. If children in your class struggle with multiplication, division, fractions and/or percentages, they may not have already secured the strong foundations in additive reasoning that they need.

Shortly after publishing the first blog in this mini-series, Ofsted released ‘Coordinating mathematical success: the mathematics subject report’ (July 2023). As one of the main findings for primary schools, they highlighted:

“Pupils’ gaps in knowledge tend to be centred around, but not limited to, addition facts in younger year groups. This was for some, but not all, pupils. These early gaps in knowledge may not become apparent until a significant amount of time has elapsed. This is because it is possible, in the medium term, for pupils to understand what is being taught and then keep up with extra classroom support and slower calculation. However, this is at the expense of the later ability to access the curriculum.”

Additive reasoning is foundational to multiplicative reasoning

I have long questioned why some children seem able to move into KS2 maths easily and others struggle, particularly with multiplication.

Why am I talking about multiplicative reasoning when Making Fluent and Flexible Calculators explores, develops, and enhances children’s ability to solve addition and subtraction calculations?

It is because additive reasoning forms the foundations for developing multiplicative reasoning (see Figure 1 below). During the course of the project, we have repeatedly seen large proportions of pupils using counting to solve simple addition/subtraction calculations. They have yet to develop trust in a range of strategies and haven't developed the habit of looking for ways to simplify the calculations. Instead, they revert to counting or, slightly better, depend on a single strategy (for a more detailed discussion on the strategies children spontaneously use in the project, see the first blog in this series)

Figure 1: Reasoning progression phases from EYFS to KS4 and beyond

Each phase is progressive, building understanding and knowledge vital for success in the next phase. To be blunt, if a child doesn't secure the required learning in the additive phase, they will struggle with the multiplicative.

Too many children are trying to solve multiplication calculations with understanding and knowledge only from the counting phase. This is why some struggle to understand and accurately answer multiplication questions and find proportional reasoning, such as fractions, percentages, and ratio, ‘a bridge too far’.

To illustrate this, consider what happens when children solve 9 x 6 when this fact is not fluently recalled:

Child A

Child A is in the counting phase, so they solve the calculation by drawing an array; nine rows with 6 dots in each. Then they count each dot, starting from 1 (count all) or, slightly better, from 6 (count on).

Child B

Child B is in the additive phase and solves the calculation through skip counting. They now recognise that there are 9 groups, and each group is equal to 6. They no longer see each item (dot), but rather 9 repeated equal units of 6. They solve through applying repeated addition, so even though they are solving a multiplication calculation, they are employing strategies from either the counting or additive phase.

Child C

Child C’s solution is representative of the multiplicative phase. They have developed an understanding of the two different dimensions of multiplication - the number of groups and the size of groups. They can flexibly manipulate both dimensions to break the multiplication into groups to which they fluently know the answers. They must employ skills from both the multiplicative and additive phases to succeed at this stage.

In addition to the macro-progression of counting to additive to multiplicative, each phase has a micro-progression

Children start with unitary counting; counting all the dots, one to one. This strategy is not time efficient and has massive potential for error. They do not see the array as representative of multiplication or the relationships between two differing dimensions (number of groups/size of group) but rather just a series of single dots. The child’s ability to estimate if their answer is correct is minimal.

Rhythmic counting is how many children first solve multiplication calculations when they start to recognise groups; they emphasise the number pattern but are still counting in ones. When counting in sixes, they would whisper 1, 2, 3, 4, 5, and shout 6, whisper 7, 8, 9, 10, 11, shout 12 and so on. They recognise the repeated addition structure but still employ skills from the counting phase.

As children get more confident with rhythmic counting, they can often skip count some of the patterns but then revert to rhythmic counting to complete the calculation. This is called ‘hybrid counting’. For example, 6, 12, 18, 24, 30…whispering 31, 32, 33, 34, 35, call out 36; whispering 37, 38, 39, 40, 41, call out 42, etc.

The cognitive load is huge when using this method as children must keep track of numerous pieces of information all at once, including:
      a)    the number of groups of 6 that they have counted
      b)    which number they are currently on
      c)    which number comes next in the number sequence (particularly challenging if it crosses a decade boundary)
      d)    how many more they must count on to complete the next group of 6.

However, if children are confident in additive reasoning, then they can start to simplify the repeated addition calculation and compact the amount of information they are processing, releasing cognitive space. A child’s thinking might be:
I know 5 x 6 is 30, so 30 + 6 = 36, 36 + 6 = 42, 42 + 6 = 48, 48 + 6 = 54.

During this intermediate stage, children use additive reasoning strategies to solve the sub-calculations required to multiply 9 and 6:
 

Ability to use base facts 

In the first blog in this series, I discussed the importance of children developing mental addition and subtraction strategies to prevent them from being stuck in the counting trap.

But the ‘Fluent and Flexible’ project has also highlighted another issue. Thompson (1994) highlighted that children must understand that mathematics is about “working with relationships”.

The entry data from the project shows that many of the children aren't spontaneously using their base facts to help with other calculations (the National Curriculum refers to them as ‘derived facts’). Children do not see relationships between calculations and are not using what they already know.

For example, suppose pupils know that 9 + 4 is 13. It is then anticipated that they would use this information flexibly to solve other related calculations, such as 90 + 40, 19 + 14 and 13 - 9, but we found that many children involved in the project weren’t. Instead, they see all calculations in isolation. They aren’t in the habit of anticipating and seeking ways to make the calculations easier using known facts.

Not seeing associations between calculations has implications for their longer journey in maths. Children need to build the habit of seeking patterns and seeking ways to simplify calculations in preparation, not just for more complex additive calculations, but also multiplication (if I know 3 x 4 is 12, then 30 x 40 is 120, 13 x 4 is 10 x 4 + 3 x 4). Children need explicit instruction, time, and practice to seek and trust relationships.

Moving beyond known multiplication facts

The ability to use additive strategies is much more than just being more efficient within the times tables facts. As they progress through the curriculum, children will have to use additive reasoning to solve calculations beyond their known multiplication fact range.

Additive reasoning impacts multiplication success

To summarise, additive reasoning develops this kind of understanding and knowledge:
      a)    numbers can be broken into smaller numbers (composite units/part-whole models)
      b)    how to use base facts to seek derived facts – going beyond memorised facts
      c)    to select, with reasons, from a range of strategies to simplify the calculation.

In the 2023 Key Stage 2 mathematics paper 1: arithmetic, there were 15 questions that explicitly involved multiplication or division (excluding those that contained fractions). Of these, at least 9 can be simplified and solved using additive thinking strategies.

Examples include:

That there are many ways to solve each calculation is critical here. It is up to the children to decide. They have the agency so select the option which exploits their strengths.

We need to explicitly help them establish the habit of seeking patterns, build their understanding and trust in additive reasoning so that they will continue to look for relationships within and between numbers. This will help them to learn to seek out ways to simplify calculations rather than mechanically applying the same strategy over and over again.

This foundational flexibility enables them to build more sophisticated ways of solving calculations, preparing them to be successful with more complex calculations.

It is vitally important that we aren’t satisfied with children getting simple addition and subtraction calculations correct, but that we overtly focus on the strategies and the understanding that the child is applying. We should constantly question ‘are they building foundations to enable success for the next stage of their maths journey?’

Is this a key focus in your school?  

The HFL Education Primary Maths team can work with you in school to develop mathematical fluency through The Fluency Package or the teaching and learning of multiplication facts through The Multiplication Package.

Find out more about the Making Fluent and Flexible Calculators

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References

2023 Key Stage 2 mathematics paper 1: arithmetic
Contains material developed by the Standards and Testing Agency for 2023 national curriculum assessments and licensed under Open Government Licence v3.0

Ofsted (2023) Coordinating mathematical success: the mathematics subject report Available at: www.gov.uk/government/publications/subject-report-series-maths/coordinating-mathematical-success-the-mathematics-subject-report 
(Accessed: 31 August 2023)