When the new National Curriculum was first published a few years ago, I used to talk about the three ‘C’s’ of curriculum design with mathematics leaders. These being cohesion, coverage and consistency.

Cohesion referred to the design of the curriculum. The National Curriculum document simply presents a series of statements in domain content order for each year group. Leaders had to decide how best to connect this learning in a way that ensured the mathematics in the classroom flowed from lesson to lesson and helped pupils see the interconnectedness of concepts. Think, for a moment, about pupils you know who do not really have a sound understanding of proportionality. I wonder if this is a result of how they have experienced fractions. As part of the number system, as quantities and in the context of shapes – but separately. Developing cohesion is somewhat trickier than deciding which bits of learning need to be apportioned to which terms and weeks.

Leaders also had to ensure there was coverage. By this, I do not mean that pupils had to simply ‘experience’ all of the curriculum statements and deep learning would follow. More specifically, pupils experienced the breadth of learning required to help build their schema. The skilful curriculum design provided sufficient opportunities to revisit and deepen understanding.

And what about the three aims of the curriculum? Don’t forget fluency, reasoning and problem-solving. Not as bolts-ons but manifested in the way that the curriculum is delivered.

With regards to consistency, I meant maintaining an unwavering focus upon ensuring high quality teaching in all classrooms. Simple to say – much trickier to do!

Inherent in all of this was the pressure upon leaders to ensure that colleagues had sufficient subject knowledge and teaching expertise. By subject knowledge, I am referring to the expertise teachers need to have about the conceptual development in maths. And how this informs the selection of effective teaching approaches for a concept. This being across the phases teachers work in and not just the year group they currently teach. This is challenging in one subject let alone across all of the others that make up a primary school’s curriculum!

The challenge of this for leaders is the reason, I assume, most have reached out for support. By support, I mean resources.

Only a few years have passed and it would appear that another important organisation has become very interested in curriculum design as part of their quality of education judgement[1]. I would accept that my original three ‘C’s’ were very broad. They were presented as useful hooks for thinking. I am now happy to accept OFSTED’s three I’s: Intent, Implementation and Impact as the new lens and language that all leaders use.

It is not my intention with the rest of this blog to offer a critique. Let’s not start about how often I have heard that this is the year of reading – that’s perhaps for another time! Instead, I wish to use this as an opportunity to re-explore curriculum design for mathematics and offer some points of reflection and support. I think these new lenses are helpful. I also wish to consider sources of evidence that might be used in an inspection. This isn’t offered as a checklist to help leaders feel they have all bases covered if visitors arrive. Instead, as points of note to consider the reliability and accuracy of our own evaluation.

It was, and still remains, our key intention to help leaders ‘take control of their curriculum’. The National Curriculum is a broad framework. It needs to be adapted and made right for each school’s context. So what does that mean?

The inspection framework offer an insight.

“Inspectors will take a rounded view of the quality of education that a school provides to all its pupils, including the most disadvantaged pupils (see definition in paragraph 86), the most able pupils and pupils with SEND. Inspectors will consider the school’s curriculum, which is the substance of what is taught with a specific plan of what pupils need to know in total and in each subject.”

(paragraph 167)

Now to Oftsed’s three eyes, I mean I’s.

Intent is defined as the way a school’s curriculum sets out the knowledge and skills that pupils will gain at each stage. In essence, the curriculum plan. Implementation refers to the way in which it is taught and assessed. In Ofsted’s words, “to build their knowledge and to apply that knowledge as skills”.  I often describe this as the way in which the planned curriculum manifests itself in the classroom. Finally to impact. This is the outcomes pupils achieve as a result. It is important not to lose sight of this aspect as I have frequently and incorrectly heard leaders suggest that Ofsted are no longer interested in outcomes. Secure and deep learning is the reward for a rich maths curriculum, well planned and expertly executed. Whether that manifests itself in test outcomes is a slightly different point.

The handbook outlines a number of points that are worthy of further consideration. So let me take each of these in turn and offer some thoughts.

I referenced ‘cohesion’ earlier. In many respects, this is equivalent to intent. As such, it is reflected in the careful planning of the curriculum. The way that the knowledge and skills are identified and sequenced. Both in terms of conceptual development across the primary phase but also the conceptual development within a year group. It is seen in the clear end points that this is building to. For most schools, the national curriculum steer this.

The bullet point I am interested to ponder on in paragraph 170 is this one.

“The curriculum reflects the school’s local context by addressing typical gaps in pupils’ knowledge and skills”

I agree. The school’s context in this sense is about what your learners are like. Patterns in weaker areas of mathematical understanding and mathematical behaviour that inform curriculum design. So pause for a few moments and consider these questions.

---

Are there any areas of learning in your mathematicians that are not as strong as you would like?

Is this common across the school or isolated to classes or groups?

As a result of this what adjustments are needed to your curriculum?

---

Your response maybe to focus upon increasing time for certain key learning focuses. Sharpening fluency. Some schools have identified key skills in each year group. This is a useful step. What is required here though is to ensure that these are secured. Of course, these skills may change over time as learners will be different. Identifying and prioritising certain areas of learning is very helpful but the ensuring that pupils actually do secure these is the most important aspect. Clear intent. It is skilful implementation that leads to impact. Read more about this in Siobhan King’s excellent blog Detecting shaky learning and dealing with it .

The handbook suggests that the main source of evidence for this is discussion with senior and subject leaders. It is essential that you own your school’s mathematics curriculum. Most schools make use of resources to support this. It is crucial that leaders know how the resources supports them. That it was selected and adapted because it provided the key focuses that teachers and learners need. Does it help with the sequence and flow of learning? Does it support the effective use of representations that secure deeper conceptual understanding? Often referred to as ‘Concrete Pictorial Abstract (CPA)’. Does it support the use of the correct mathematical conceptual language and the language to work on the mathematics?

I move now to explore implementation. As I have previously stated, I think about this as the ‘reality in the classroom’. When you are next describing what you think is happening in each of your mathematics classrooms, pause and consider what your thoughts would be if this was the reply. “Let’s go and see this in action.” In some ways, this is the thread of the ‘deep dive’ approach to evaluation in the inspection framework. As leaders, it is fundamental that we have an evidenced view of the reality. Hoping that your colleagues are is not the same as knowing they are. This isn’t about engaging in increased monitoring exercises. Talk to any of the maths team and they will tell you that ‘weighing the pig doesn’t make it any fatter!’ This is related to how leaders actually lead on professional development in mathematics. How they improve teaching. And the final part of any improvement (or enhancement) cycle – knowing it is working as you want it to.

This is the essence of the bullet point that references teacher’s ‘expert knowledge’. I would suggest that this is the perennial challenge and core motivation for leaders. It may seem obvious but is nonetheless worth stating. With strong knowledge of conceptual development and a repertoire of effective teaching strategies, practitioners are then more likely to teach more effectively. Ensuring that learning is secured for all because they are very acutely aware of what misconceptions pupils face and so can teach to expose and address them. They are secure in what to look and listen for and so are the masters of assessment.

I have the frequent pleasure of supporting teachers so an example may help. It is from a Year 1 classroom a few weeks ago. The lesson focused upon developing pupils’ use of the 0, 5 and 10 benchmarks. A crucial skill that takes time to develop in Year 1.  Pupils were using a demarcated number line to place various single digit numbers. The aim here was for pupils to use the 0, 5 and 10 as anchor points to help position other numbers. Indeed, the pupils were able to place 6 quickly using the 5 benchmark. They also had ’0-20 beadstrings’. I asked many pupils to show me ‘6’. In each case, pupils counted the beads individually. They were still counting and so hadn’t secured the notion of helpful benchmarks in numbers. Not once did any pupil slide the 5 red beads and then 1 further white bead. You might say that this is where the teacher was going next.

As leaders, it is crucial to not be too presumptuous. In our roles as leaders and supporters of teachers we need to ask. I did. It was clear that the teacher hadn’t noticed. And as we explored the journey to date in year 1, it was also evident that many of the earlier key learning points were not quite secure either. The teacher, by their own admission, was not that clear about these essential components. So we unpicked this in a bit more detail and identified precisely what learning needed to be revisited and secured. Importantly, also identified what this would look like when it was happening.

I do not offer this vignette as criticism of teachers at all. I have a very privileged position in working with and supporting lots of teachers to develop their practice. More crucially, I offer this as an example of what leaders need to do. The teacher had planning support resources but needed to spend a bit more guidance to unpick the conceptual learning underpinning this sequence of learning.

I had a recent conversation with an Ofsted inspector and we talked about the need for leaders to ‘look at’ and not ‘look for’ in classrooms. It may be pedantic but the former provides the leader with a position of inquisitiveness. To look carefully and consider what the information is really revealing. Further questions really help. In the example above, the discussions with pupils in lessons really helped understand what they hadn’t quite developed yet. From moments like this, the skilful leader is able to identify the support and guidance that will have the greatest impact.

In the handbook, there is reference to the transfer of knowledge to the long-term memory. I am not quite sure of my own understanding of this neuroscience and so are more enamoured with the phrase “new knowledge and skills build on what has been taught before and pupils can...” use these in a range of contexts with fluency (that last bit is my addition). This relates again to the design of the curriculum. Its flow and connectedness that supports teachers to manage the learning journeys for pupils. I am looking forward to returning to work with the Year 1 teacher in a few weeks.

As with intent, the handbook identifies a range of sources of evidence. These are well known. Discussions with leaders about the programme of study and how pupils are progressing through it; observations of learning in classrooms;  pupils’ ‘work’; and discussions with pupils. Of note and I applaud it is the reference to the support for teachers. I refer to this as the professional development of teachers. I regularly talk about this essential ingredient with leaders. It can take a myriad of forms but is the only catalyst for even greater practice.

The framework also identifies how this can be evidenced through conversations with teachers. I note this last point as I rarely hear leaders talk about this and it can be a really rich source of feedback.

Impact, in the handbook, is aligned to “what pupils have learned”. This is hard to disagree with. Positive learning outcomes should and will be the result of a rich maths curriculum, which is well constructed and taught. National assessments are useful indicators but not necessarily the only measure. A view of what is happening in classrooms, in books, through interviews will add to that picture.

Aligned with the earlier point about ‘local context’ to address gaps in learning, there is reference to the learning success of disadvantaged pupils and pupils with SEND. In many senses, this will be the ultimate litmus test of your curriculum intent and implementation. The extent to which your curriculum design and delivery supports the success of your most vulnerable learners. By vulnerable learners, I do not necessarily preclude this to the ‘defined’ groups. I simply state that any pupil who is in danger of not securing the learning is vulnerable.

To support subject leaders, I have generated a short set of questions. These are not intended to be exhaustive or used as a definitive script. Instead, they are offered as further prompts for reflection and evaluation. They have been created from many conversations with school leaders and listening keenly to those involved in school inspections or curriculum evaluation.

---

Subject leaders

How do you decide what to teach in each year group? Why this? Why now?

How do you support the development of subject expertise?

With regards to the sequence ... what learning is next? ... what came before?

What is your approach to revisiting key learning?

How do you check pupils are making progress through the curriculum?

How well do pupils secure key learning?

How are your pedagogical approaches matched to the learning taking place?

What’s the reality in the classroom? Let’s see that in action.
 

---

Teachers

Why did you teach that...? Where is the learning heading? What came before? How has this shaped your decisions?

Why have you decided to teach it like that?

How did the lesson content and activities ensure the learning was secured?

How are you supported to develop your understanding of the key components pupils need to learn?

How are you supported to develop the best ways to teach these?

What has been the impact of that?

---

I believe that the three lenses offered to us by Ofsted are useful. I would, however, urge readers to return to the crux. This is not about the need now to write out a lengthy statement for three I’s. A quick reflection to ascertain salient points would be helpful. Not swathes of time dedicated to create a set of rhetorical soundbytes. The quality of your mathematics curriculum and how well it is served up in classrooms is what you need to invest energies into. This will be evident in what you say about how you have designed it for your school, but ultimately about what happens in classrooms and the secure learning pupils build as a direct result.

---

References

Handbook for inspecting schools in England under section 5 of the Education Act 2005, The Office for Standards in Education, Children's Services and Skills (Ofsted), May 2019

This blog has been written for teachers but can be shared by schools with their parents as the ideas within it are useful for the learning of multiplication tables at home.

To get in touch with the HFL Education Primary Maths Team about our blogs, resources and services, email us at primarymaths@hfleducation.org.

Much of the key learning of multiplication tables happens across Year 2, Year 3 and Year 4. It is fair to say that children who know their multiplication tables up to 12 x 12 (with a good amount of understanding as well as recall) cope better with the demands of the maths curriculum in many areas, such as formal written division, equivalent fractions, percentages and ratio and proportion.

So much of the mathematics curriculum in Upper Key Stage 2 is built upon a good understanding of multiplication and division and recall of the multiplication tables. Where a child has not yet remembered the necessary facts or understood their connections to each other enough, they often end up using what becomes an inefficient ‘counting up from 0’ strategy to figure them out. Not having facts at fingertips or fast strategies to get them will slow down the bigger calculations they are trying to solve and place additional pressure on working memory when problem solving (as they are adding in additional steps to work out multiplication facts rather than recalling them).

Many schools ask parents to help at home with the learning of multiplication tables. We think of it, like regular reading, as an area where parents’ support is really beneficial. What we need to do is be more specific about ‘how’ to support this at home. This blog aims to support the ‘how’ to learn a multiplication table (from the beginning) and then also how to rehearse and maintain it.

First, to be clear about the expectations within the National Curriculum from Year 2 onwards, there is specific reference to the recall of multiplication and division facts. It should be acknowledged that some pupils find learning their tables harder than others and so this may not be a smooth path but the general expectations are:

Year 1 – Count in multiples of twos, fives and tens

Year 2 – Recall and use multiplication and division facts for the 2, 5 and 10 multiplication tables

Year 3 – Recall and use multiplication and division facts for the 3, 4 and 8 multiplication tables

Year 4 – Recall multiplication and division facts for multiplication tables up to 12 x 12

 

Starting from the beginning:

If a child is learning, for example, the 3 x table for the first time, we need to set out a path. This should include:

 

 

A lot of what is thought of as practice for multiplication tables, e.g. games using speed of recall, in reality actually falls under the rehearse and recall phases, rather than the initial learning, as they rely on children already having some base knowledge to draw upon. When children begin learning a new multiplication table, it is important to give time for exploration and building a picture of what is happening, allowing the opportunity to physically make the facts and then rehearse them in a range of ways before focusing on the memory and retrieval.

At the ‘learning’ phase, children benefit from seeing the multiplication table build up from the beginning, looking first at one group of the amount (e.g. 1 group / row of 3) and then building up by adding another group / row of 3 each time and seeing what the total becomes. This helps children to link multiplication to repeated addition, e.g. linking 4 x 3 (four rows of three) to 3 + 3 + 3 + 3   and knowing that both make 12.

Any small resource that there are lots of can be used for this. At school, this might be cubes but at home, small toys would work equally well – cars, beads or even a handful of dried pasta. For the purposes of this example, I have borrowed my son’s collection of dinosaurs.

A selection of the arrays (equal rows) are shown. The child would build each in turn, working up through 1x, 2x, 3x, 4x… and talk about each one, noticing how another group / row of 3 is added each time.

 

 

As a note here, it is also good at this stage to talk about no groups / no rows of 3. If there were no groups of 3 dinosaurs, there are 0. This reinforces the idea that 0 x 3 = 0.

Counting and tracking the number of groups on fingers is really important to help children understand one lot more or less as well as to build up a familiar pattern of multiples ready to learn facts. Avoid counting in multiples at the beginning if you want to build a good understanding of relationships between facts. Children generally begin by counting every object in a row at the beginning of skip counting. Help this by encouraging children to ‘whisper’ numbers that are not a times table fact, touching each object and shouting the number of the object at the end of the rows. For example…

 

 

In this array, the child might point to each object starting top right, move along the row whispering 1, 2 and shouting 3, move to the next row whispering 4, 5 and shouting 6 etc. They repeat this on their fingers; tapping a finger and whispering 1, 2 and shouting 3.  In school, you could try with cubes on fingers as these teachers, working with the fabulous Professor Jenny Field to develop skip counting, demonstrate below. Over time, encourage children to miss out the ‘whisper numbers’ and skip count more.  Finger tracking at this stage is crucial to help children negotiate the number of groups. We can make it more efficient once children can skip count by asking… Do we need to count up from 0? If you can remember your 5th fact, could you count on from there?

 

 

A nice thing to do at this point (with arrays of small items still available) is to turn the facts into a set of cards with the ‘question’ on one side and the ‘answer’ on the back:

 

 

Once these cards have been made, there are lots of options for playing with them; first of all, in order to build some memory recall and then, once the child is starting to remember what is on the back, moving to playing with them out of order to further secure the learning. The point here is about taking time to build confidence and develop memory. Repeated rehearsal should strengthen the memory so don’t rush to reach the out of order and speed rounds.

 

 

Keep it light and manageable:

• The process needs to be broken up into several steps to maintain motivation and concentration and to build up rehearsal.
• Think about whether to stick to one multiplication table at a time or return to explore a small number of less well remembered facts. It is not a race to learn them all as fast as possible. Focusing on one family or set at a time will allow the child further opportunities to build confidence and develop a more lasting and meaningful memory of the facts.
• Play and create your own games with the cards; whether playing against a partner or children creating their own new rules for using the cards.
• Ask questions such as, “Which facts do you think you already remember?” “Which facts do you think are harder to remember? Why is that?” “How can we remember them?” “How could you use other facts you do remember to help?” This helps children to take more ownership in the learning of facts they see as ‘harder’ to remember (see Further Professional Development Opportunities below for more on this).
• Once you feel a multiplication table has been cracked, keep the cards safe and return to them after a break of a week or two. Revisiting tables to keep them fresh will also help to embed the learning.

Continue the rehearsal:

Once you feel a particular multiplication table is becoming secure, two further elements are needed: application of the knowledge (helping children see when they might need to use them) and also continued rehearsal. This helps to maintain the memory. Application might be about noticing how the knowledge can be used in everyday life or in other areas of their mathematics learning.

 

 

The ‘continued rehearsal’ could be where online games come in and here at Herts for Learning we love a card game. Look out for the games posted @Hertsmaths and our FaceBook group Herts for Learning: ESSENTIALmaths. 

The main messages:

Take each multiplication table one at a time. There is a logical order which usually works; 2s, 5s and 10s first (usually around Year 2), 3s, 4s and 8s next (usually around Year 3), then 11s, 6s, 9s, 12s and then 7s come later (usually around Year 4).

Take time to develop an understanding and then memory of each multiplication table rather than skipping to recall; this comes later.  

It takes time for most children to develop their memory recall of the full set to 12×12 (and some do struggle more than others to do this) so developing and personalising the approach is important.

Once a table has been rehearsed and is fairly secure, remember to revisit it again and again to keep if fresh.

---

Further professional development opportunities

Join our digital, on-demand training to explore effective teaching strategies for rehearsal and recall of multiplication facts. The modules are full of practical ideas and resources to take away to enable pupils to learn multiplication facts with understanding and recall. 

Supporting pupils to learn multiplication facts – effective strategies for rehearsal and recall

Here is an article that schools could share with parents about the hardest to learn times table facts.

Here is an article that might be of interest to teachers looking for professional development around the teaching of times tables.(pdf)

 

To keep up to date: Join our Primary Subject Leaders’ mailing list

To subscribe to our blogs: Get our blogs straight to your inbox

 

---

Further blogs to read:

The beautiful array

This blog was first released in September 2020 and has been updated with the latest DfE guidance and additional resources.

---

From the 2021/22 academic year, the multiplication tables check (MTC) is statutory for all year 4 pupils registered at state-funded maintained schools, special schools or academies (including free schools) in England.

The administration guidance provides details on:

• how to use the MTC service (including the pupil try it out area)
• how to enable access arrangements
• how to administer the check

Multiplication tables check administration guidance

 

I want to develop the fluency and application of multiplication facts. Where do I start?

Many adults have gone through the trials and tribulations of learning to drive. Whilst some take to it immediately, others need more time and space to hone their skills. During our multiplication research project, Making Sense of ‘X’, we have similarly found that some pupils acquire new facts with relative ease, with others needing extra support to develop multiplicative fluency.

In the Herts for Learning maths team, we have been supporting schools and pupils by using this model to reflect on how curriculum time is spent. In our new online training programme, we have used the learning to drive analogy to break down the crucial stages:

 

 

Taking a mock driving test every week without any instruction or support between tests is unlikely to improve driving skills. Whilst assessment is an important element of any learning, we believe there are important stages which need to be prioritised before pupils are ready to be assessed in their multiplication tables. Just like focusing on a new manoeuvre, for the start of each multiplication table, pupils will need time to go back to learn and rehearse before mastering the new skills.

 

Learn: building, deconstructing, drawing and describing

When learning to drive, novice drivers need to begin with expert instruction. They need plenty of time dedicated to hands-on experience and discussion, with support to make connections to prior knowledge and skills. We believe it is the same for multiplication, with time needed to build, deconstruct, draw and talk about the structure behind the facts.

 

Throughout our ESSENTIALmaths plans, we use speaking frames such as this to support pupils to describe the structures they have built and drawn. We would want pupils to be able to move between different representations which could involve manipulatives such as counters, cubes, beadstrings and everyday objects. Our previous blog looked specifically at this phase in detail.

One way schools have incorporated this stage during home-based learning has been an array hunt where pupils can find, create and draw arrays inside and outside the home.

 

 

Developing rehearsal

Once you have been taught the basics of mirror, signal, manoeuvre, it is time to rehearse the different elements, starting with easier manoeuvres and building up stamina over time. There are countless ways of rehearsing multiplication tables but one way of providing daily opportunities is to use a counting stick for your focus times table. As well as building fluency in counting up, down and out of order, there is also the chance to build in reasoning.

 

 

This example shows possible questions and responses to describe a missing value. To begin with, pupils may produce simple responses such as “I know it is 28 + 4”. However, once they are familiar with the concept, pupils often come up with ever elaborate responses which show a level of much deeper understanding.

You can’t beat a traditional counting stick which can be handled and manipulated but when needed, this interactive online counting stick is a great alternative: Mathsbot: Counting Stick

 

Opportunity to recall

Once there has been plenty of time for varied rehearsal, it is time to for the learner driver to recall how to perform a range of manoeuvres in increasingly complex situations. Learners may need to go back to the ‘learn’ or ‘rehearse’ stage before progressing on. Part of the recall stage in terms of multiplication could involving low-threat gaming opportunities. This could start with a simple bingo board where the ‘caller’ reads out a product, with the players finding the corresponding calculation. Alternatively the players could be provided the product and the ‘caller’ calls out the calculation.

 

 

Other gaming opportunities could involve dice or playing cards, either for a focus times table or a mixed set of calculations. It is essential that pupils have the underlying understanding from the ‘learn’ and ‘rehearse’ stage and are not attempting to recall disconnected facts and, in some cases, inadvertently practising incorrect facts.

The Herts for Learning Primary Maths YouTube channel has provided gaming opportunities to embed different parts of the primary curriculum. This video provides a collection of three multiplication games which can be used to provide pupils with varied recall opportunities.

 

 

The use of assessment

Like teachers, driving instructors use on-going assessment to inform which further teaching, rehearsal or practice opportunities are needed before the final assessment of the driving test.

It is up to schools how they use purposeful assessment to support teaching and learning but we would encourage all teachers to use testing as a brief stocktake opportunity, with the bulk of teaching time being spent on the stages that precede ‘assess’. One aspect of the MTC which we are regularly asked about is the timed element of it. This is something that many pupils may need building up to, to ensure a focus on speed is not brought in too early; inadvertently impeding the learning.

By the end of the progression, we aim for pupils to be able to take the wheel and recall and apply their facts with independence and confidence. There are often bumps in the road, and obstacles to overcome, but it has been fantastic to see pupils across the schools we work in building their skills to then apply their increased understanding across the maths curriculum.

 

What next?

• Reflect on the ‘learn, rehearse, recall, assess’ model. What is already going well for you? Is there any aspect that could be enhanced to support your pupils?
• Are all year groups playing their part to ensure pupils have the pre-requisite skills to allow pupils to be fluent by the end of Year 4?
• What aspects of the ‘learn, rehearse, recall, assess’ model could be used for pupils in UKS2 who are yet to secure all the multiplication facts up to 12 x 12?

 

Is this a key focus in your school?  

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

Find out more about the HFL Education Curriculum Impact Packages. 

Alternatively, join our digital, on-demand training to explore effective teaching strategies for rehearsal and recall of multiplication facts. The modules are full of practical ideas and resources to take away to enable pupils to learn multiplication facts with understanding and recall. 

Supporting pupils to learn multiplication facts – effective strategies for rehearsal and recall

 

---

More information is available at:

• www.gov.uk/guidance/multiplication-tables-check-development-process
• www.gov.uk/government/publications/multiplication-tables-check-administration-guidance

 

Further related reading:

• Starting from the beginning: how to learn times tables

 

To keep up to date: Join our Primary Subject Leaders’ mailing list

To subscribe to our blogs: Get our blogs straight to your inbox

Parents are powerful! 

Whilst we know that work inside our schools is crucial in supporting our children’s learning, what happens at home plays a significant part too. I think that most teachers know this and it is acknowledged early in the EEF guidance report, “Working with parents to support children’s learning”. What can be challenging is ensuring that all parents know their importance and crucially, that the school – parent relationship is fully developed to enable all parents to support their children’s learning.

Perhaps it is worth noting here, that when I talk about parents, I include all parents, carers, grandparents, older siblings… in fact, anyone who has significant caring responsibilities for children in our schools. And of course, just as each child brings with them their own strengths and needs, parents too have varied contexts and starting points. Particularly in considering mathematics, parents can bring with them their own anxieties. So how can schools fully engage parents and enable them to support their children’s mathematical learning? 

Look at evidence carefully considering your school context

Whenever we look at evidence, it is crucial to ensure that we consider it critically. This is none more so the case than when aiming to identify what is effective in supporting parental engagement in children’s learning. Firstly, a school’s plan should meet the needs of their communities and be “informed by an understanding of families’ lives” (EEF, 2018, p10). Secondly, “strategies for parental engagement will be different for different age groups” (EEF, 2018, p13). And finally, even the EEF recognise the “limitations of the current evidence base” (EEF, 2018, p11). 

In knowing this, the priorities must then be to make sure we understand the lives of our families through listening to and working with them; to establish the aims of parental engagement based on our understanding of pupil needs at different developmental points and to ensure that whatever we chose to put in place, we continually evaluate how effective it is being in meeting those aims.

Consider what effective parental engagement looks like within your context

To establish the next steps in developing parental engagement within your context, it is best to reflect on what already happens, what is working and where there are things that you feel could be improved. Be clear about the development that you are trying to implement – where is it needed? For whom? What would it look like after successful implementation?

I would urge caution here and ask that you consider the eyes that you are looking at this development through. What a school views as effective parental engagement may not be the same as the views of parents within it! Step one then remains to establish true understanding of the lives of families in our schools – their needs and the opportunities they present.

There are however many aims that schools may consider to ultimately support children’s mathematical development, and these might fall under four broad themes:

• opportunities – working with parents to understand where there are relevant links to learning and opportunities to develop maths understanding
• recognition – enabling parents to notice and value their child’s mathematical working and achievements
• interactions – considering together what makes an effective interaction and supporting parents to be able to work through mathematical activities successfully with their children
• modelling – developing parental understanding of how teaching has already happened in school and being able to reinforce this or simply ask relevant questions about it at home

 

 

Ensure that support is practical and linked to learning

Whichever focus is selected, what matters is how able parents are as a result to take specific actions to support mathematical learning. So, sharing information about curriculum content may provide context, but what will support improvement is parent understanding of their importance and practical strategies and resources to enable them to unleash their power. 

This was a key point we considered when designing the HfL training, “Enabling parents to support primary mathematics at home”. For example, as part of the training, one pre-recorded session and associated resources is heavily focused on helping parents to see the importance of talk and understanding how they can notice and use everyday opportunities and their child’s own interests to develop mathematical thinking and articulation. Schools are provided with a template and supported in designing a bespoke parent session. Through practical and accessible examples and supportive sentence stems, parents are then encouraged to identify actions they can take immediately which will support mathematical learning.

For anyone who follows the Herts for Learning maths team on social media or YouTube, you will know that we are huge advocates for the use of games to support learning rehearsal. Time spent showing parents how to play and understand the purpose of well-chosen games can support continued rehearsal and reinforcement at home. Ensuring that parents have the resources to play the game at home once shown can be overlooked but will make the difference in terms of impact on pupil learning.

Consider how to maximise the impact of homework

Homework is another potential opportunity to support parents and to make links to learning. The EEF guidance identified a few points relevant to homework in primary schools and these are outlined below.

• at primary level, the evidence is strongest for short and focused homework projects.
• the quality of the homework completed is more important that the absolute quantity.
• schools can improve the quality of homework by ensuring that homework tasks are tightly tied to main class teaching.
• parents can have a positive effect on homework completion and help children to develop effective learning habits. (EEF, 2018, p16)

It was through considering these key points, requests from schools and consideration of how to support parents, that the HfL home learning tasks were written. These tasks are shared (all 118 of them) as resources accompanying a second pre-recorded session in the training. The resources have been developed to support parent understanding of pupil mathematical learning with key features that will enable parents to ask relevant questions, understand taught models and to support learning reinforcement.  

 

 

Plan for successful implementation

Successful implementation will be key in establishing impact and this should be considered while gathering evidence and planning new developments. It is so important to develop your vision, share it with stakeholders and to plan carefully to make it a reality. 

If you are currently reflecting on your vision and considering how to engage parents in supporting children’s mathematical learning at home, join the primary maths team for our on-demand Enabling parents to support primary mathematics at home training.

Following the training launch, teachers said:

“I am going to incorporate the Maths Everywhere examples into the KS1 and KS2 parent workshops next week.”

“I liked having the key parent session template slides ready to adapt and also the focus on language and times tables.”

“I loved the idea of sharing videos of playing games. I plan to talk to our EYFS and KS1 teams about filming short videos to share on our school website for our youngest children and families to engage with at home”.

“The home learning tasks will really help with teacher workload.”

“The home learning resources are great. I love how they link in with the same methods and equipment that the children are used to seeing in class.”

 

 

Enabling parents to support primary mathematics at home

---

References and further reading:

Working with parents to support children’s learning – Guidance Report, EEF, December 2018, p.16

www.educationendowmentfoundation.org.uk/education-evidence/guidance-reports/supporting-parents

Why mixed-age maths?

As a 1.5 form entry school, our journey to mixed age teaching in Maths began in September 2017. For a few years our Maths data had identified that the lower and prior middle attainers were not making the desired progress, therefore after carrying out lots of research on Maths sets verses mixed attainment/age teaching and after considering the aims of the National Curriculum, we decided to take the plunge and try mixed-age teaching. Since then we have never looked back!

Parent presentation

At the start of the year, we invited all of the parents to a presentation on why we were moving to mixed age maths as we predicted some parents may be apprehensive about the move away from streaming. The only concern voiced by a minority of parents was the possible negative impact on the highest attaining pupils. Here we presented them with the benefits of mixed-age maths verses streaming for pupils from all starting points, showing them the kinds of challenges that would be available to them. This engagement with our stakeholders was a key piece of work in order to get the parents to put their trust in what we were planning to do.

Phase 1

The first phase of mixed age maths teaching was to introduce it to Key Stage 1 and Lower Key Stage 2. We sought support from the Herts for Learning Maths Team and the Teaching and Learning advisor spent time with each team, unpicking the first sequence of the HfL mixed-age ESSENTIALmaths planning and supporting teachers with a range activities and assessment opportunities.

In addition to this, we explored a range of maths resources to support the learning process. This in turn meant that each class was provided with five maths toolkit boxes with all of the key resources to support the teaching of the sequences.

Differentiation

Having two different year groups in one class meant that teachers needed to think carefully about differentiating the learning. Throughout the year, teachers explored many different teaching styles. When teaching the whole class, they would often provide the children with low threshold, high ceiling tasks from the HfL ESSENTIALmaths mixed-age planning. This meant that all of the children within the class could access the learning and it could be extended further through extension opportunities.

Independent learning

Children during independent learning were provided with three challenges**. They were guided to select appropriate tasks and use their maths toolkit boxes to not only support them but to deepen and reason about their mathematical learning.

This approach evolved further and through assessment for learning, the teachers would sometimes teach in smaller focused groups. This ensured that misconceptions were quickly addressed, children were provided with the relevant support and that pupils who needed challenging further were challenged.

Teaching strategies

One of the strategies we embedded as a school was the episodic approach, teachers would often start by teaching the whole class using the HfL Essentials mixed age planning and would then ‘peel off’ throughout the lesson, deepening the learning.

There were times at the beginning of the lesson that the current higher attaining learners would start on their challenge independently while the teacher taught and consolidated learning with the rest of the class, once these pupils had consolidated their learning the teacher would teach the higher attainers and extend their learning further.

As well as the teachers exploring the different teaching approaches, they also began to think about the seating arrangements within their classroom. During the input, the children were often taught in mixed attainment pairs, this encouraged the children to work collaboratively, using the toolkits and encouraged them to explain their reasoning.

Whilst sampling the pupils' books, the journey they were securing demonstrated excellent progress compared to starting points. They were using clear pictorial representations to demonstrate their understanding and their reasoning was much improved.

When speaking to the children, about their learning and the mixed-aged classes, they were extremely positive. They liked the different structures of the lessons, they liked being provided with the three tasks and they all felt that they were being challenged.

Impact

The impact of mixed-age teaching has been very positive. Over the course of the year, I had the opportunity to meet with teachers to find out how they were finding teaching in this way, I was able to observe lessons across the school, sample pupil’s work and meet with the children to find out how they found working in mixed-age classes.

Class teachers were extremely positive about the shift to mixed-age classes, they were enjoying using the HfL ESSENTIALmaths mixed-age planning and could see the benefits in mixed-age teaching. The lessons observed demonstrated that the children were working collaboratively and were developing their reasoning and problem solving skills together.

The children were more regularly seen to be working in mixed attainment pairs, where they were developing their understanding of a concept and reasoning together. This enhanced the children’s mathematical vocabulary and understanding, as well as further developing their ability to reason. The HfL ESSENTIALmaths speaking frames were excellent scaffolds to support with this. Teachers were also using a range of manipulatives to enhance their teaching and the children were using the maths toolkits to support and deepen their learning.

Our maths data also showed that the prior higher attaining pupils had not been disadvantaged.

Due to this we decided to implement mixed age maths into Year 5/6 at the beginning of September 2018.

Across the school all of our classes are now mixed-age and this is continuing to have a really positive impact for all of our children.

Where next?

Not only could the impact of mixed-age maths be seen within the teaching of lessons, books and when talking to the children, it was also reflected in the data.

Our Year 2 data for 2018, rose to 85% at ARE in comparison to the previous year, which was 80% and we had 38% of pupils assessed as working at greater depth in comparison to the previous year at 22%.

The data for the end of Years 3 and 4 also saw a rise in the amount of pupils at ARE and working at greater depth.

**These were set up to match year group expectations and as further deepening tasks.

---

Amanda Taylor Assistant Head, maths subject leader and Year 3/4 teacher at Eastbury Farm Primary School, Northwood.

---

ESSENTIALmaths mixed-age planning resources are available in the ESSENTIALMATHS subscription.

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:

---

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

---

The next available item to click produces the following:

---

‘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?"'

---

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

---

‘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.’

---

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

---

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.

---

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!

---

Reasoning Paper 2 2017

 

---

 

 

---

---

---

 

---

 

---

---

---

---

 

---

Reasoning Paper 3 2017

---

---

 

---

 

---

 

---

 

---

---

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.

---

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”!

---

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.