Strong foundations: why primary to secondary transition is so important

Thursday, 27 May 2021

Back in 2015, Ofsted published a paper titled ‘Key Stage 3: The Wasted Years?’ Those of us teaching in the period that followed might recall the onslaught of meetings declaring that Key Stage 3 was now our priority. ‘What about year 11?’ we might have asked. Yes, they remained our priority. ‘And Key Stage 5?’ Also our priority.

The reality facing secondary schools is that exam classes will never not be the priority — and it would take a bold headteacher indeed to direct staff to focus their interventions and marking time on S1 or year 7 at the expense of their exam groups. In an ideal world, we would offer our best to every single pupil who steps foot in our classroom without burning ourselves out or surrendering our own right to a life outside school. In such a world the milk in the staffroom fridge never runs out, the photocopier never jams, and the IT works best on days when you are expecting visitors or an observation.

Truthfully, teachers must regularly draw a line and accept that everything that falls beneath it will receive whatever we have left to give, even if some days all that’s left is an empty tank. Non-exam classes often find themselves falling under this line, not because they are seen as less important but simply because with finite time and resources, something has to give. May often marks a turning point — with exams receding into the distance behind us, we are free to turn our attention back to those groups who might have missed out.

As we move towards the final half term of the year, the opportunity to capitalise on the freed-up time created by the submission of CAGs and the departure of exam groups is more important than ever. The question persists: which remaining priority to prioritise first?

The Transition Challenge

Even in a ‘normal’ year, primary to secondary transition can often find itself near the bottom of the list, yet it is perhaps one of the most influential periods in a pupil’s life. In the absence of taster lessons and transition days, secondary schools head towards September knowing less than ever about their new intake. Vast swathes of information — both academic and pastoral — are lost every year in the handover between primary and secondary, compounded now by two years of disruption to education. The feedback from #MathsConf makes it clear that any division between primary and secondary is a “false dichotomy” — colleagues want to know what their pupils have experienced earlier on in their schooling, because it forms the basis of what they will study next. It is simply the case that in an increasingly loud and demanding list of priorities, something has to slip down the list.

It isn’t just knowledge of students that is lost when teachers in different phases don’t have time to talk. There is a reason why primary and secondary teachers are trained separately: the contexts and methods in which they operate differ vastly. Primary teachers are there from the very beginning, introducing pupils to the world of education through which they will navigate at least until adulthood. Every superhero has an origin story — and Marvel turns these into films because audiences understand the vital role our past plays in shaping our future. To grasp why our pupils embrace certain concepts but struggle with others — at its core, to grasp how they think about problems — we need to know how they learned mathematics in the first place. Their primary teachers are the only ones who can tell us.

Teachers as Builders

Imagine you inherit a house, partly built, with instructions to complete the job as you see fit. You have planning permission to build as many storeys as you like, to add extensions, perhaps even to extend down and build a basement (or wine cellar, if I indulge my own Grand Designs ambitions for a moment). What would you do first?

You’d probably want to know what’s happened already; what materials were used to build the walls behind the veneer, and the location of hidden cavities beneath load bearing structures. You’d want to see the initial blueprints, hear from the previous owners how things had gone before. You’d want time to walk around the entire site, imagining what it could become in the future, how long it would take, where you might encounter difficulties. You would want all this to happen as thoroughly as possible, to prevent problems occurring further down the line.

What would change if you were suddenly told you had only one year to complete your project? Would you scale back your ambitions to ensure you had time to prepare fully, or would you cut back on the planning and race to get as much done as possible?

I’m sure we’ve seen enough Channel 4 documentaries about cowboy builders to know the dangers of the latter — yet this is exactly what the school system seems to push us towards when we take this metaphor and apply it in the classroom. If your new architect presented you with kitchen designs in which the cabinets and countertops didn’t actually fit into the space, you would rightly demand that he or she think again. When it comes to schools, however, it can feel as though getting through the curriculum is more important than getting to grips with it — even if that means some pupils reach the end of their formal schooling with huge, unaddressed knowledge gaps that can’t be disguised with a tall plant or a strategically-placed rug. For the child who begins secondary school lagging behind their peers, they face at least five more years in education trying to catch up, knowing that those ahead won’t be slowing down to wait for them.

Such pupils find themselves further disadvantaged if their understanding of core concepts is itself too weak to carry them through the next stage of their education. Ofsted’s latest subject report in mathematics highlights this, stating that, “When planning curriculum content, teachers also need to prioritise ‘forward-facing’ knowledge. [This] includes the mathematical methods that pupils will take with them on their journey.” We might illustrate this with an example as simple as the equals sign. Let’s imagine Pupil A understands ‘=’ to mean ‘the answer is’, while Pupil B understands ‘=’ to mean ‘is equal to’. The two may be fairly indistinguishable in the early stages of their education — certainly both might achieve correct answers on the same questions. But later on, when asked to balance equations, Pupil A might suddenly find themselves struggling. This simple, but influential, misconception is tricky to diagnose when disguised behind a much more complex problem — and this isn’t the only issue. Where in the secondary curriculum is there space to loop all the way back to this core knowledge, the meaning of different mathematical symbols?

And this is where both teachers and builders find themselves in full agreement: the key to success is a secure foundation. With this in place, we can happily add and knock down walls at our leisure; without it, we are simply kicking trouble down the road.

Securing the Foundations

Primary teachers know exactly where the gaps are — which topics are secure, and which won’t stand up to a strong wind. But without enough time to share this information in full, secondary teachers then find themselves covering old ground looking for weaknesses. This is part of what Ofsted criticised in their 2015 report: an acceptance among departments that the first year of secondary school would involve a degree of repetition.

How does one prevent this?

We’ve been puzzling this over at Complete Maths, and it’s why we created our ‘Secondary Ready in Mathematics’ course. With six weeks of content covering the key fundamental concepts of mathematics, it is designed to act as a de facto building inspector — identifying and plugging knowledge gaps so that all pupils begin secondary school ready to tackle more complex ideas. More than this, it also signposts what’s left to build: as a pupil works through the course, their teacher can track their assessment results from each idea studied.

No course can replace the deep, personal knowledge primary teachers hold about each child in their class — but ours can create a degree of consistency, perhaps even predictability, which allows transition conversations between colleagues to focus on other topics. Some of our customer schools have gone so far as to purchase access to the course for all their feeder primaries so that teachers can feel confident that every student will reach them with the same foundational knowledge in place. More importantly, it lets pupils who’ve slipped behind catch up without their classmates having to slow down and wait.

To wake up each morning and watch the sunrise from your bed, an architect needs to place the master bedroom on the east side of the house; a builder must ensure the window sits at just the right height that you don’t need to get up to see the sky; and an electrician should install plug sockets in such a way that you can position your bed opposite the window without sacrificing a bedside lamp — in short, a series of experts need to cooperate in planning and delivering a shared vision. This is precisely how successful transition should feel.

Just as Ofsted pointed out, this means pursuing a ‘forward-facing’ education. When presented in the right way, even young children are capable of engaging with the type of conceptual knowledge that will serve them well later on. Certainly there is space for teachers to explore different methods of solving the same problem, but ultimately the method we encourage pupils to use should whenever possible be the one they can carry with them for life. EYFS teachers rarely sit down with their post-16 counterparts to share teaching methods, but if we want to equip pupils with the best tools from the start, shouldn’t they? The exchange of knowledge between primary and secondary should go both ways: not just what our pupils learned but how they learned it, and how they need to learn it to be successful later on.

We want, through our transition-focused courses, to facilitate the type of conversations between teachers that lead, in the end, to the outcomes we hope one day our pupils will wake up to.

Planning for the Future

We have so little time with our pupils before we must, once again, send them into an exam hall to have their progress assessed, and their knowledge and understanding quantified. At this pace, there is no time to waste time. We cannot escape the paradoxical notion that everything is a priority, but we can find our way through these competing demands by asking ourselves what actions might offer the best pay off in the future. After all, the final owner of our metaphorical house is the pupil themselves — it is they who will shape what it eventually becomes, and they who will inhabit it, so we owe it to them to provide a secure foundation on which to build.

The Power of 1%

Friday, 23 April 2021

If you're seeking to improve teaching, the sports field might not be the first place you look for inspiration. But, Hannah Gillott argues, the lessons learned by professional athletes can be just as applicable in the classroom.

Nowadays, British Cycling is synonymous with success. Such is the team’s domination on the world stage that from 2007-2017 British cyclists won 178 world championships, 66 Olympic and Paralympic gold medals, and five Tour de France victories. Its leading figures are household names and Sports Personality of the Year winners, despite the fact that most of us would struggle to name all of the individual events for which medals can be awarded.

It was not always so. Until 2004, British cyclists had won just one gold medal at the Olympic Games, and no British rider had ever won the Tour de France in the 110 years since it began. In ‘Mastermind: How Dave Brailsford Reinvented the Wheel’, Richard Moore describes how British cyclists were so poorly viewed that one leading bike manufacturer refused to sell bikes to the team for fear it might hurt sales if their product was seen to be used by Brits. But in 2003, British Cycling appointed Dave Brailsford as its new performance director and the transformation began. In 2004, Great Britain finished third in the cycling medals table at the Athens Olympics, and by 2008 the team were top, six gold medals ahead of second-placed France.

The team’s transformation under Brailsford has deservedly become part of sport lore, and his methods are the subject of numerous articles, books and videos. But what does any of this have to do with teaching?

The power of 1%

Brailsford believed in the power of marginal gains: the notion that a series of small, 1% improvements across numerous areas added up to one big improvement. As teachers, we know just how true this can be for our pupils — how many times have we optimistically requested a re-mark in pursuit of the one extra mark that might push a grade up, or watched as one minor mistake early in a response unravels an entire solution? Or, indeed, breathed a sigh of relief when a single adjustment to the seating plan transforms behaviour in the following lesson…?

The potential in marginal gains, however, should not be reserved for our pupils. As professionals, many of us long for more opportunities to discuss and plan as a department, rewrite curriculums and schemes of work, or spend unhurried periods watching one another teach. We can only imagine what we might achieve given unlimited time, but in its absence we might look to Brailsford for the alternative.

British Cycling’s upturn came about through a desire to seize every tiny opportunity to make improvements. Brailsford went so far as to commission custom-built mattresses for his cyclists to use every night on the Tour de France, each one bespoke to its user’s way of sleeping, in the belief that no one performs at their best after a less-than-optimal night’s rest. Headteachers reading this need not worry that I am about to suggest setting aside hard-fought budget for new desk chairs in every classroom. There are plenty more cost-effective ways for schools and departments to factor small, impactful changes into their daily routine.

Reconfiguring CPD

Often, the phrase ‘CPD’ conjures images of conference rooms, cover requests, and a day of checking your emails under the table to see how many of your Year 10 will need to be set detentions on your return. The high cost means only one or two members of your team will likely be able to attend and, despite returning with a thick ream of notes and the zeal to disseminate them, they will then need to distill a day of talks and workshops into a twenty minute feedback slot at the next department meeting. CPD days are often brimming with expertise and inspiration, but rarely can this be enjoyed in full by the whole team without an opportunity-cost of lost learning time and a huge dent in the budget.

At La Salle Education, we’ve tried to challenge this by running our #MathsConf on a Saturday, for a far lower price, and —when Covid-19 meant we moved online — sharing all the workshops afterwards on our Teacher CPD College, alongside our existing virtual courses. We believe this model makes CPD accessible for all teachers, not just those chosen to attend a mid-week, in-person event.

But let’s imagine for a second that we’re describing this model to Dave Brailsford. Teachers, we would tell him, receive a day of training, which they then go and implement in the classroom.

“What happens next?” he’d ask. “What happens every day? How are teachers supporting each other in identifying new targets for improvement? How are they ensuring they scan every area of their practice, continually, for the 1%?”

And he would be right. Improvement happens gradually over time — it’s rarely linear, and you can rarely pinpoint the exact moment it occurs, just as you can never see the grass grow. Yet with the right conditions in place, it happens. Our attitude to CPD should reflect this by turning the traditional model of ‘workshop leader in a room with delegates’ into a continual dialogue, whereby knowledge is not stored in a cupboard to be opened every now and then but instead shared out, in small doses, as often as possible to as many teachers as possible - and refreshed when needed.

Ten minutes per day

Finding the 1% shouldn’t mean hours and hours of effort —it resides instead in small, sustainable, everyday efforts that add up to something much bigger. Think, for instance, about the notion of morning meetings. These usually take place over ten to fifteen minutes, and are primarily reserved for admin: announcements, notices… in other words ‘things that could have been an email’.

What if we reclaimed them?

Imagine the new morning agenda. ‘Ten minutes to share something that worked really well yesterday, and you think others should try today.’ ‘Ten minutes to share a problem or puzzle you’re encountering with a class, and to hear others’ suggestions.’ ‘Ten minutes to discuss the most common misconceptions from the last assessment’. ‘Ten minutes to share key takeaways from last week’s CPD’. Suddenly, marginal gains aren’t just part of the agenda —they’re all of it.

This time does not, of course, need to be constrained to the morning —but identifying a regular slot, with a pre-determined focus, could be transformative. When a team commits not to a meeting but to a process, even a passing conversation in the corridor can be powerful.

Deliberate practice

In the last edition of ‘#AskMark’ , Mark McCourt expounded on the various stages of differentiation. He writes, “Purposeful practice keeps the pupil at the limit of their competence and, therefore, creates the cognitive conditions for learning to occur…Deliberate practice is also goal driven, but draws upon what is already known in a domain to improve performance.” Teaching something so that pupils understand it and can recite it back is one thing, but learning can’t truly be said to have taken place until pupils can apply their knowledge independently through practice. The same can, of course, apply to us as professionals —one might argue that a conference or webinar has no value until its takeaways have been applied in the classroom.

If we are to commit to the notion of marginal gains, then we need to challenge ourselves on a daily basis to push the boundaries of our teaching. This doesn’t just mean trying new things gleaned from CPD or morning meetings, but also seeking feedback on their effectiveness. Feedback can come from all directions —pupil voice, a drop-in observation by a colleague, or reflecting on pupils’ work at the end of the lesson. For it to be truly meaningful, it must also be linked to a goal or objective for the class in question. In this way, every teacher is continually developing and improving their pedagogy to best meet the needs of the pupils in front of them at any given time.

As long as time and space is created firstly to set goals, and later for feedback and reflection, then leaders can trust that their team will be improving, a little at a time. As Dave Brailsford’s example proves, lots of little adjustments can lead to huge overall improvements.

An empowered team

Brailsford didn’t just oversee a shift in the performance of individuals, but also in that of the team as a whole. He describes in one interview how over time every member, from the mechanics to the athletes themselves, felt empowered to share suggestions for improvement, regardless of whether it fell into their area of responsibility or not. Far from seeing this as criticism, team members instead viewed it as part of their collective responsibility to find ways of optimising performance.

Feedback in a school often flows from top to bottom, but if departments are serious about adopting a continuous 1% improvement model then every member needs to feel empowered to speak up if they see an opportunity to do things better. This requires an enormous level of trust, and a belief that even teachers early on in their careers have valuable insights to share. At La Salle this is one of our guiding principles, and explains why we offer the same platform to teachers with many different levels of experience at our #MathsConf. Truthfully, it might sting when someone new, or at the start of their career, sees something we hadn’t, but it doesn’t invalidate our experience or undermine our expertise. Accepting feedback, or even criticism, instead marks out a teacher filled with confidence and secure in the knowledge of their own progression.

‘What’ then ‘How’

Too often, especially in the field of EdTech, the solution is of more interest than the problem. Shipping customised mattresses to various hotels in France is a novel idea, but it only holds value if it works. Innovative technology, flashy graphics, or impressive conferences are of no use to schools unless they result in better teaching, and consequently better outcomes for both pupils and teachers. This might sound strange coming from an employee of an EdTech company, but it’s easy for that message to be lost — especially as we emerge from an intense period of innovation, driven by school closures.

There will be times when it’s better to rip the whole thing up and start again, but those times are few and far between, and even then often occur against a backdrop of time and budget constraints. A committed team, who share the same goal of becoming better practitioners for their students, can go a long way even without much time or money to spare if they embrace the power of 1%. That's precisely why La Salle exists: our company was founded by teachers wanting a better way to support their colleagues. Why is that so important to us? Because, put simply, when you empower educators they improve more than just their teaching — they improve the lives of the pupils in front of them, too.

We recently heard from a school using our Teacher CPD College to run a ‘CPD Book Club’ — watch out for a future blog exploring how they did it. We’d love to hear from other schools using our platforms to run similar initiatives. Tell us in the comments where you’re finding your 1%, and we’ll share your journey in future blogs.

Find out more

This article explores the idea of 1% marginal gains in more detail.

You can hear Dave Brailsford describe his approach first-hand in this video.

#AskMark Volume 4

Friday, 23 April 2021

Welcome to the fourth edition of #AskMark, a weekly series in which our founder Mark McCourt responds to your questions. This week, Mark answers questions on the challenges facing Primary teachers seeking to implement a mastery approach, and what to do with the student who always finishes early...

Don't forget you can submit your own questions too - simply tweet @LaSalleEd using the hashtag #AskMark .

Claire Rodger (@ClaireRodger6) asked:

What would you advise for primary schools who wish to implement a mastery approach? Classes are always mixed ability and attainment gaps are often very wide.

Thank you for your question, Claire. It is certainly the case that, in the UK and other Western jurisdictions, primary classes often contain pupils spanning a large attainment range. In year 4, say, it is common to find pupils who are operating at a mathematical level beyond what would typically be expected of a year 6 whilst also finding pupils who are operating at a mathematical level below what would typically be expected of a pre-school child. This is an enormous challenge for teachers to overcome.

There are approaches we can take to make such classrooms more effective, including in-class groupings, which change from subject to subject, day to day, for instance. Alternatively, when it is time for maths, we could use a common hook from which to spin off many different activities – so pupils are, in the face of it, working on the same problem, but at varying levels of complexity.

You ask about implementing a mastery approach in such a classroom. These two goals (having classrooms with very wide attainment gaps and running a mastery approach) are not compatible. A mastery cycle approach to teaching and learning relies on the group being closely enough aligned in terms of attainment to allow for the effective use of the elements of the model – that is, prerequisite quizzing and pre-teaching until all pupils are ready to progress with learning the new idea, whole class instruction, working on tasks specifically about the new learning goal (not spanning a large range of access points), ongoing formative assessment with immediate corrective teaching, extension tasks related specifically to the new idea, and testing at the specific level of difficulty of the new learning goal.

A mastery approach does not work with a large attainment gap.

This is why the main formulators of a mastery approach would often use non-grade settings – in other words, mixed aged classrooms – to enable groups to be a homogenous as possible in terms of their current attainment.

But this is not a very useful response to your question, is it? There’s little point in me just saying it can’t be done.

Primary schools can use a mastery approach. In fact, if a mastery approach is to work anywhere, then it is critical that primary schools use it. But it needs to be implemented from the very beginning.

Which perhaps brings us on to the next question...

Christopher Such (@Suchmo83) asked:

Mastery learning in mathematics relies on teachers addressing children's gaps in prerequisite knowledge before an idea is taught. It seems to be accepted, understandably, that when the gaps get too great, mastery approaches are a non-starter.

This being the case, I wonder about what happens at the start of education. In my (admittedly limited) experience of working in KS1, gaps in number sense, spatial awareness, attention span, etc are often vast, to the point where addressing them before a new idea can be learned is impossible without delaying the teaching of the new idea for a very long time. This is the case despite the excellent work undertaken in reception. (I'd go as far as to say that I found a mastery approach harder to implement in KS1 than in upper KS2, despite the greater absolute gap in maths attainment in the latter). I suspect that for a mastery approach to mathematics to be successful in KS1, the prime areas (and number sense)focused upon in reception would need to be prioritised for significantly longer than they are, with carefully judged focus on those children who need more support in these areas and a concomitant delay in formal instruction, perhaps until the latter stages of Y1. I wonder whether there are any other mastery advocates who would disagree with this assessment.

If so, how might the practical issues that I have described be addressed?

Thank you for your question, Christopher.

It is not unusual to find implementing a mastery approach somewhat easier in KS2 than KS1 in the current system (which has been a product ofmany years of a conveyor belt approach to curriculum). I’d suggest this isbecause KS2 pupils – even those at the lowest level of attainment – have abetter developed schema of knowledge than KS1 pupils. Making sense of new ideas is only possible through constructing that sense from ideas already understood – so the older pupils have more stories, metaphors and images on which to call. The vast majority of what children learn is beyond the content of the school curriculum, so it is often a happy surprise to find that older pupils can construct new meaning in unexpected ways.

There is a fear (and I don’t use that word lightly) amongst many primary teachers that there will be professional consequence to them if they, themselves rather than the pupils, do not ‘keep up’ with the curriculum. I’ve written many times about this conveyor belt approach problem, so won’t labour the point here, save to say that it is understandable that teachers in KS1 feel they are unable to work on an idea such as number sense or place value for a very prolonged period of time. Teachers know (and will tell you privately if they feel they are not at risk of admonishment) that moving on through the curriculum content when pupils have not yet gripped obviously prerequisite ideas is a reckless and idiotic thing to do. But they often feel they have no choice.

I think we make a huge error in England by starting formal mathematics early and racing towards a view of successful mathematics learning that holds written algorithms up as the way of identifying whether or not a 4-, 5-, 6- or 7-year-old child is doing well. I suggest we would perhaps have a great deal more success in our aim for all pupils to become mathematically literate if, instead of the current fetish for standardisation of written responses, we provided an early years (up to age 7) education that focussed on truly understanding numerosity, place value, proportional reasoning and relationships between quantities – none of which is best achieved through a worship of written algorithm.

A prolonged view of early years education (as is not uncommon around the world) would, in my view, give a much stronger foundation for all pupils to construct a successful understanding of mathematics as they continue through school.

All of this can be achieved through a mastery approach, but it requires a shift in policy that takes the aim of all pupils having secure foundational knowledge in place before embarking on a more formal process of using that knowledge to develop the ability to communicate through mathematical symbolism and convention. Sadly, there is little appetite forsuch an approach in the UK.

Nilam Patel (@NILAMPA04557349) asked:

What do you do when some students finish an independent task before others?

Thank you for your question, Nilam. It’s a perennial problem, isn’t it? We all know that moment when a pupil says they are finished long before we expected them to be. Even the most careful planning and the most diligently selected tasks that take into account everything we know about the pupils in order to get the level of difficulty just right can sometimes leave us surprised by the speed at which an individual grips and overcomes the problems. This is, of course, also a pretty lovely moment –it shows the pupil has really applied themselves and worked determinedly to nail whatever task they have been set. I think there are two important responses that should come next.

Firstly, we should appreciate that some pupils absolutely love to completetheir work quickly, but that this might not always align with completing their work carefully. So, teachers take the time to ensure they have presented their thinking elegantly and with mathematical precision – we should always be encouraging pupils to treat mathematical communication with the attention it requires.

Assuming they have indeed completed the task to the highest level of accuracy and mathematical sophistication they can, then, secondly, I think it is important that all teachers have up their sleeves a range of challenging prompts that extend the task, engage the pupil in serious thought, and keep them working at the limits of their comfort zone. Thereare lots of stock responses we can quickly use that have huge impact (perhaps it would be a good idea if teacher training included dozens of these stock responses, so we all know them before facing the situation you describe?)

Some of my favourites include:

  • Can you generalise your solution?
  • What if this approach was applied to *insert new scenario*?
  • What if the questions were posed in a different base?
  • Under what conditions would your solution break down? What are the boundary conditions of the idea we are learning about today?
  • Now that you have gripped this new idea, what ideas did you once hold to be true are now probably false?

I hope that is a useful starter for the list... but would love to hear other people’s favourites. Perhaps add in the comments below?

Andy Waters (@MrAJWaters) asked:

Do you think mixed attainment classes are essential to teach maths via mastery?

Thank you for your question, Andy. This has been addressed in the previous questions, so I’ll just briefly reiterate: absolutely not. Mixed attainment (where the gap is large) and a mastery approach are not compatible models.

We know that learning takes place at the boundaries of our comfort zone, so let’s take the education of all pupils seriously and ensure that the new ideas we are asking them to grip are at the right level for them.

Got your own question for Mark? Simply tweet @LaSalleEd using the hashtag #AskMark and he’ll answer as many of your questions as he can.

#AskMark Volume 3

Thursday, 15 April 2021

Welcome to the third in the series of our new weekly #AskMark, in which our founder Mark McCourt responds to your questions. This week, Mark answers questions on differentiation, and how parents of young children can best prepare them for beginning school.

Don't forget you can submit your own questions too - simply tweet @LaSalleEd using the hashtag #AskMark .

Mahnaz Siddiqui (@MahnazSiddiqui) asked:

What is differentiation in mathematics? What does it look like?

Thank you for your question, Mahnaz. Differentiation is a huge topic, which could easily fill a book on its own. I’ll try to do it some justice in a short response, but would love to hear your thoughts too – feel free to add to what I have written here in the comments below.

If we are working with more than one pupil, then it is always the case that there will be variation in the experiences the pupils have had to date, in their understanding of ideas, in the maturity of their knowledge schemata, in how quickly they can make sense of a new idea, and in how keen they are to do so.

Differentiation is simply a teacher’s response to all of the variations that exist within a class. Understanding that the class is made up of individual human beings with vastly different lives means that teachers can appreciate the burden upon them to ensure that all pupils have a successful experience of learning whatever new idea the teacher is planning for them to grip.

So, differentiation is just a way of saying how the teacher reacts to the pupils in front of them. This is a continual process and changes from class to class, idea to idea, even day to day. Perhaps it is helpful to consider the phases that a teacher and class progress through as they work together on a new idea.

To begin with, the teacher will seek to establish the pupils’ ‘readiness’ for learning a new idea – this could be through some sort of diagnostic activity or through discussion or through detailed prior knowledge of the pupils. Clearly, pupils will have differing levels of readiness – some will have forgotten things, some will have missed key moments, some will have independently prepared more than others, etc. The first stage in differentiation, then, is the actions the teacher takes based on an individual pupil’s readiness. For some pupils, the teacher may react by providing corrective instruction, working carefully to undo and overcome a misconception, for example. For other pupils, perhaps some pre-teaching will help them to connect partially forgotten ideas. Other pupils will be perfectly well equipped to proceed with new learning having demonstrated their readiness by mastery of the diagnostic activity – the teacher might react here by extending the pupil’s domain expertise in a prerequisite idea by asking them to work on an unfamiliar problem or they might simply allow the pupil to progress to the new learning. This will depend on the teacher’s plan for classroom management and whether or not they wish all pupils to receive the introduction to the new idea together.

When pupils are ready to learn a new idea, the next step is instruction. We know that understanding new ideas relies on understanding earlier, pre-requisite ideas. This is how we construct new knowledge – by linking it to what is already understood and using this understanding to ‘bridge’ to new meaning. The teacher can do this by using story-telling and metaphor. To enable metaphors to come to life and have mathematical meaning, the teacher uses models. The models are explored in examples and these examples form the way of narrating the instruction.

The second step in differentiation is, therefore, when teachers react to how readily (or not) pupils are making sense of the instruction. They do this by changing the examples, the models and the metaphors they are using to animate their instruction. The order in which these changes are made is really important. I have written about how to react during the instruction phase in this blog, Models, Metaphors, Examples and Instruction

All pupils (all people, in fact), grip new ideas at different speeds. The purpose of instructing pupils is to bridge from a mathematical idea in my head and understood by me to one that the pupil is able to make meaning of. Working out whether or not the individual pupils in front of us are making appropriate meaning is best achieved through dialogue – as we narrate an example, we then ask them to work on a similar problem and narrate back at us their thinking. In other words, we are using the to-and-fro of examples and problems as a conversation between teacher and pupil – the pupil is forced to articulate their meaning.

The next step in differentiation is, therefore, to react to the pupils in front of us by varying the number of examples they are asked to respond to until each individual is communicating the meaning you are aiming for. This is just a way of checking that the meaning is being received. We should not be fooled into thinking that their ability to articulate the correct meaning is an indication that any learning has taken place. At this stage, it hasn’t. But we do now know that we are able to ask the pupils to work independently on problems. We can now ask them to do some mathematics.

Doing mathematics is an absolutely vital step in learning mathematics – it is through doing that pupils begin to learn.

It is important that we do not stop at the point of them knowing – at the point they were able to give the correct articulation. Imagine a pupil learning to play the piano, for example. The teacher could tell them the keys that need to be pressed and in what order, with what pressure and at what pace in order to produce a certain tune. And the pupil could articulate back at the teacher the precise instruction – they know how to play the tune. But that doesn’t mean they can play the tune.

A teacher could explain, through the use of several examples and problems, how to multiply over a bracket, say, and a pupil can articulate back at the teacher the precise instructions – they know how to do it. But that does not mean they can multiply over a bracket. This is why we now give the pupils ample opportunity to actually do the mathematical skill. We want pupils to be so competent in doing the new mathematics that they achieve a fluency in doing so. That is to say, that they can perform without the need to give attention.

The next step in differentiation is clearly the amount of doing that we ask of individual pupils – they will all achieve fluency at different rates. Once the new skill is something pupils are comfortable with, it is time to start learning.

This might sound a trifle odd and some people might argue that surely, if the pupils are fluent, they have learnt what they need to. But this is just the first step. Learning only occurs at the boundary of our current ability. All pupils have pretty much unlimited potential, but they only continue towards expertise if they continue to operate at their limits. Automaticity is a poor aim for any lesson – it represents a pupil who is no longer learning.

To ensure that learning is maintained, we now ask the pupils to engage in practise.

Effective practise occurs in phases too. Firstly, teachers should create opportunities both in the classroom and beyond, for pupils to engage in purposeful practise – this type of practise is goal driven. Considering the mathematical skill that the pupil has been working on and now has automaticity with, teacher and pupil examine carefully the common errors that the pupil makes.

For instance, the pupil who can fluently multiply over a bracket may well forget to multiply the second term in the bracket two times in every, say, ten questions. We now have a goal – it is highly specific to the pupil and, through dialogue with the teacher, the pupil can set about undertaking more practise with an awareness of that goal – they can be looking out for the common mistake they make and can try to reduce the number of times they falter to, say, just two times in every forty questions. Purposeful practise can be carried out independently at home because the pupil has a success metric to give them continual feedback and spur them on.

Purposeful practise keeps the pupil at the limit of their competence and, therefore, creates the cognitive conditions for learning to occur. So, the next step in differentiation is how the teacher reacts to the pupil’s need for purposeful practise – varying the amount of practice, the goals and the feedback to best realise the individual pupil’s limitless potential to learn. A pupil can significantly improve their mathematical skill through purposeful practise. But it does have its limitations, since purposeful practise leaves the pupil to determine how best to overcome their common mistakes.

The next stage in differentiation is, therefore, how the teacher responds to the pupil’s progress with their personal purposeful practise by deciding what type of deliberate practise to provide to the individual pupil. Deliberate practise is also goal driven, but draws upon what is already known in a domain to improve performance. With the pupil above, who has been forgetting to multiply the second term, the teacher can coach them in overcoming the problem by telling them about tried and tested ways for doing so. In other words, in the deliberate practise phase, the teacher trains the pupil in the approaches that experts in the domain have developed and used to overcome the very specific problem they are facing.

The final stage of practise is designed to help further assimilate the new learning with the pupil’s developing schema of knowledge. Now, practise problems are randomly mixed with problems of earlier learnt ideas – this removes recency and cue from the pupil’s practise exercise and forces them to retrieve previously learnt skills and to identify when to select certain mathematical tools.

The final stage in differentiation is, therefore, the teacher’s reaction to a pupil’s agility in selecting appropriate methods in mixed problems – all pupils will improve their method selection at different rates, so the teacher carefully judges the amount of practise required and supports the individual pupil as required.

This view of differentiation can be thought of as the oft quoted idea of learning being like building an enormous edifice. Constructing a mighty building requires very careful placement and gradual levels of scaffolding. Here, the teacher is the scaffold, providing all the necessary support and rigour needed for the pupil to fulfil their potential.

And just like the construction of an edifice, it is key that the scaffolding is removed at the right moment to let the building shine.

David Burns (@mrburnsmaths) asked:

My daughter started Primary 1 in August 2020. What advice would you give a new parent, like me, to help support and develop mathematical ideas/concepts?

Thank you for your question, David. I am sure it will come as no surprise to you if I offer this very short, initial response: Get a huge bag of Cuisenaire rods and let your daughter play with them.

And always be on the lookout to extend her natural play into opportunities to behave mathematically. For example:

  • How many orange rods does it take to surround your favourite toy (Perimeter)
  • Can you make a huge yellow and green snake? What other snakes can we make? (Sequences)
  • Or, a favourite; making sandwiches using two long rods and 'filling' them with rods that add up to the length of the 'bread'

And beyond Cuisenaire? Well, all children want to learn about the environment around them, whether it is learning to count objects around them or learning about the value of numbers. Take all the opportunities you can to convey your joy of all things mathematical – make mathematics an integral and natural part of everyday life and conversation. This could be as simple as allowing your daughter to learn about measures by cooking in the kitchen or talking about money and time.

Most importantly let her enquire and discover the mathematics all around her.

Got your own question for Mark? Simply tweet @LaSalleEd using the hashtag #AskMark and he’ll answer as many of your questions as he can.

Next week: Catering for mixed ability classes, including the challenges faced by KS1 teachers; how to cater for pupils who finish independent work ahead of others; and the impact of White Rose in Primary schools.

#AskMark Volume 2

Tuesday, 06 April 2021

In the second of our new weekly #AskMark series, we are putting another question to our founder, Mark McCourt. This week, we’re asking how to implement a mastery model in a mixed ability class.

Don't forget you can submit your own questions too - simply tweet @LaSalleEd using the hashtag #AskMark .

What advice would you give to a teacher with only mixed-ability classes, who wants to follow a mastery model but is worried about how to implement it?

Firstly, it is important to pause and think about all classes. Any class of pupils with more than one pupil in it is a mixed ability class. Ability is an index of learning rate – it is about how readily a pupil acquires understanding of a novel idea. It is not fixed and can change from topic to topic. In a mastery approach, we want pupils to progress through learning mathematics together. So, very large differences in learning rate can introduce difficulties in timing this progression. There is no way around this, there is no way of aligning pupils’ learning rates. But there are easy and effective strategies for addressing the issue. For example, different pupils could have different prep or consolidation activities.

It is also true that all classes containing more than one pupil are mixed attainment classes. Attainment is a measure of where a pupil has reached in their learning of a domain. There are no particularly accurate ways of measuring this, so, at best, it’s a pretty broad statement. Working in a mastery approach with pupils of mixed attainment is necessary because all classes are. The issue is when the difference in attainment becomes large.

Nobody would suggest, for example, putting a pupil who cannot yet count in a class with a pupil who is working on second-order differential equations is a good approach. So, there is general agreement that there is some point at which the differences in attainment becomes large enough to warrant pupils following different courses.

The question is, how large? And how large are schools being asked to cope with?

Well, actually, the differences can be really rather large and a mastery approach can still be highly effective. A key ingredient of a mastery approach is diagnosing and fixing any gaps in prerequisite knowledge before pupils begin to learn a new idea. Done well, this can ensure that pupils with quite different prior attainment can work on new ideas at the same time and at pace.

However, some schools are being asked to work with differences that are so large that the effectiveness of the approach is compromised too severely. What to do? Well, perhaps, as many schools have done, use in class groupings. This can work with a mastery approach, but it is extraordinarily complex and creates a huge work burden for teachers.

Fundamentally, a mastery approach is just not compatible with a large attainment gap. I would therefore advise any school wishing to use a mastery approach to avoid mixed attainment classes.

Got your own question for Mark? Simply tweet @LaSalleEd using the hashtag #AskMark and he’ll answer as many of your questions as he can.

Next week: Advice for parents of young children on supporting and developing mathematical thinking, and guidance for trainee teachers on effective differentiation.

#AskMark Volume 1

Tuesday, 30 March 2021

2021 promises to be an exciting year for all of us at La Salle with the expansion of Virtual Maths School planned for the spring, a return to face-to-face conferences, the growth of our Teacher CPD College, more new features added to our Complete Maths platform, and more dedicated staff joining our team behind the scenes. Plus, we have a big surprise to share with you in time for the summer holidays, so watch this space!

Our community of mathematicians is at the heart of what we do — because when teachers are given the time and opportunity to share their insights and learn from one another, everyone benefits. It is in this spirit that we are excited to launch ‘#AskMark’, a new series in which we invite you to put your questions to our founder, Mark McCourt. Over the course of his career Mark has accumulated decades of experience both in and out of the classroom, so this is your chance to benefit from it. Simply tweet @LaSalleEd using the hashtag #AskMark and he’ll answer as many of your questions as he can.

To kick off the series, we’re starting with two questions from our team - next week, it’s your turn!

Earlier in your career, you were yourself a teacher - in what ways have pupils changed since you were in the classroom?

They haven’t. Kids are kids are kids.

I know it’s tempting to bemoan each new generation of pupils for lacking some great disposition that your generation had, but I don’t buy it. At heart, they have the same ambitions, same fears, same potential, same hopes.

And I think we should have the same goal for them as I know my teachers had for me: to become educated. To become aware of the origins and growth of knowledge and knowledge systems; to be familiar with the intellectual and creative processes by which the best which has been thought and said has been produced; to learn how to participate in what Robert Maynard Hutchins once called ‘The Great Conversation’.

The tools and technologies they use might be different, but it is by enabling them to become learned that we future-proof our children.

A huge part of La Salle Education is its community of Maths teachers - why is it so important to you for that community not just to exist, but to keep growing?

There is so much knowledge within that community. If we all knew each other, if we all shared and debated our theories then, together, we can iterate towards shared standards of excellence, which would give us our best defence against mindless initiatives, fads and fashions.

And… just because I’ve never met a lovelier bunch of people to have a pint or two of beer with.

Got your own question for Mark? Simply tweet @LaSalleEd using the hashtag #AskMark and he’ll answer as many of your questions as he can.

In Conversation with Mark McCourt

Written by Hannah Gillott Friday, 26 March 2021

Mark McCourt founded La Salle Education with a view to bringing together teachers across the world and uniting them in a mission to improve mathematics education for all pupils. It’s an ambitious goal - but one he is en route to achieving. Undeterred by the challenges Covid has posed, more than 10,000 teachers have shared ideas and learned from one another at La Salle events in the past twelve months. I joined the team just in time for #MathsConf25, which proved the perfect introduction to the energy and potential within our community.

For Mark, La Salle Education is much more than an EdTech company — it is the culmination of decades of experience in the world of education at every level. Mark’s belief in the passion and skill of the existing teacher workforce infuses every element of La Salle, from the growth of the Teacher CPD College to the open platform offered by MathsConf, at which teachers at every stage of their career are invited to share their insights.

As the newest member of the team, I was keen to learn more about the educational philosophies and beliefs which underpin La Salle Education - so I put some questions to Mark. Our conversation is a compelling reminder of the importance of mathematics, and the role teachers can play in transforming pupils’ lives.

Hannah: Tell me about the story of La Salle Education - how and why did you first decide to create it?

Mark: I had been running large-scale education reform programmes for government here in the UK and overseas, and found it increasingly difficult to align my values with the typical strategic approach that education ministries take – basically, that the solution to underperforming education systems is to blame teachers and find ways of replacing them with newer, shinier teachers who would not be bogged down with all that had been before. Don’t get me wrong, without exception those reform programmes were staffed by great people who were all on the side of teachers and wanted the very best for pupils in our schools, but the strategy is doomed from the outset. Continual reactionary initiatives, designed to alienate and castigate existing (and particularly older) teachers in order to wipe out any previous government’s approach, serve only to heighten the teacher retention problem and create a never ending cycle of reinventing wheels.

I always felt (and vigorously argued for at all levels) that the answer lay within the existing workforce. Teachers are extraordinarily devoted to the core reasons why they entered the profession: helping pupils to learn, develop and go on to be able to lead purposeful and meaningful adult lives with autonomy and joy. Teachers will work unusually hard to make this happen – I say that from a point of view of having worked in many different industries and having run many different reform programmes for a whole host of different workforces. Teachers are not a normal cross-section of society – there is something about them; the vocation, I guess. They are more enthusiastic, more driven and more open to improvement than any other group I know of. But initiatives that treat teachers as the problem, or training that patronises, lead — understandably — to a reduction in enthusiasm and commitment.

I wanted to break away from the constraints of having to toe a party line. I wanted to create an environment in which teachers could collaborate and support each other over the long term – free from passing fads, free from short term policy and initiative. Always driven by one simple belief: teachers are intelligent professionals.

A natural place to begin was with my own subject area, mathematics.

Looking at just the UK, for example, around 350,000 people are involved in the teaching of mathematics in the primary, secondary and FE sectors. Every one of these teachers carries out thousands of micro-research experiments, day in, day out. That professional body knows a huge amount about teaching mathematics. Imagine if all of that knowledge could be untapped. Imagine if every one of those teachers knew each other well enough such that they felt no fear in discussing their own struggles in teaching mathematics and such that they could support their colleagues with their own expertise. That’s what I was interested in doing.

One of the projects I used to have responsibility for was the National Centre for Excellence in the Teaching of Mathematics (NCETM). My heart used to sink each year when we held our national conference – hosted on a school day, in central London and in the most extravagantly expensive locations (the Royal Opera House in Covent Garden was a particular insult to schools struggling to buy enough mathematical equipment). These events epitomised the disconnect between real teachers and those who occupied positions of apparent authority over them. In the typical delegate list, a tiny handful of actual teachers were there.

I repeatedly argued for a different way, once suggesting we hold it on a Saturday in Kettering. This was met with such a derisory and mocking response that, when La Salle started running MathsConf, I chose a Saturday in Kettering. Teachers came in their hundreds.

So, I mulled over these issues for a few years and realised the only way to bring about large-scale collaboration was to create a blend of online and face-to-face environments. That’s what led to Complete Mathematics – a club for mathematics teachers to work together, draw on the canon of knowledge that already exists, become friends, and form a long-term, sustainable, free-from-diktat, professional learning network.

H: La Salle Education is underpinned by a belief in the importance of ‘mathematical thinking’ — can you explain what you mean by that term, and why you see it as such a vital skill for students to acquire?

M: I will offer you this quote from my book, Teaching for Mastery:

“I take ‘mathematics’ to mean a way of existing in the universe. Mathematicians are curious in all aspects of their lives. Mathematicians, when faced with a problem, enjoy the state of not yet knowing the resolution (indeed, knowing there may not even be a resolution). Because they are curious, mathematicians, when faced with a problem, ask themselves questions of it. They can specialise, pattern-spot, conjecture, generalise, try to disprove, argue with themselves, monitor their own thinking, reflect and notice how these new encounters have changed them as a human being. That is to say, mathematics is an epistemological model: a way of considering the very nature of knowledge."

“Sadly, in many Western countries, children have been conditioned to believe that mathematics is about wading through questions, getting ‘right’ or ‘wrong’ answers. This is confusing to mathematicians, since it does not represent our domain at all. Mathematicians are not in the business of answering lists of questions. Rather, they meet scenarios and, driven by their curiosity, create their own questions and follow their own lines of enquiry. Many of these lines of enquiry result in unexpected results, but we do not consider these to be ‘wrong’, simply not what we thought would happen. Often, great discoveries in mathematics have resulted from lines of enquiry that lead to unexpected results. Mathematicians enjoy being stuck. They revel in the initial apparent impenetrability of a scenario and understand that by attacking it in a structured way, enlightenment can arise.”

H: How might all teachers promote mathematical thinking in their classroom?

M: I guess the simple answer to that is: be mathematical in front of pupils and give them space to be mathematical too. It takes a little bit of forced stepping out of oneself as a teacher – after all, we are already experts in the mathematics that we want our pupils to grasp, so we need to recognise our view of the mathematics we are talking to pupils about is not the same as their novice view. I like to stand in front of a class and narrate aloud my novice internal monologue. It is, of course, an invented and affected monologue – I am acting. But it is important that novices are shown effective ways of thinking about a mathematical problem. For instance, I might write a problem on the board and say out loud ‘Hmmm, I wonder what this is. I wonder how I might go about resolving this.’

We want pupils to realise that mathematics is not about seeing a problem and instantly knowing how to resolve it. That’s not what life is like for a mathematician – much of our time is spent wondering, struggling, playing around with ideas, testing, breaking, retrying and, sometimes, simply getting lost. We want pupils to realise that mathematics requires purposeful effort and that, with such effort, not only do we arrive at a resolution to the problems we are working on, but we also have a fascinating time getting there.

It is easy to spot the difference between a class in which pupils think of mathematics as being about ticks on a page and a classroom full of pupils who have been conditioned to be mathematical. In the first classroom, when the teacher writes on a question on the board and asks the pupils to work on it, lots of hands shoot up into the air and a chorus of ‘I can’t do it’ is heard. In the mathematical classroom, puzzled faces stare at the problem and pupils think, ‘I can’t do this, yet.’

That ‘yet’ is so important. Pupils realise that mathematicians enjoy being stuck because that is the opportunity to do something meaningful.

Of course, as teachers, it is part of our art that we keep the mathematics that pupils are working on just at the very limits of their current knowledge and understanding – so, although there will always be struggle, that struggle will result in success. And success breeds motivation. The cycle is virtuous.

H: You’ve spoken before about the huge number of non-specialists teaching Maths in schools today. Imagine I am one of them — my line manager has just dropped in and informed me they haven’t managed to recruit and I’ll be teaching Maths next term. What are the most time-efficient and impactful things I could be doing to turn myself into an effective Maths teacher?

M: Perhaps the single most useful (and perhaps most calming) point to note first is that, as a profession, we know a heck of a lot about teaching mathematics and all of that knowledge is there for you to share in. The profession is typified by a willingness to support colleagues. So, to begin, make sure you have really informed guidance to help you in all aspects of your lessons. This is why we built the Complete Mathematics platform – to create a central repository of information about every single lesson. Non-specialist teachers using the platform will find all manner of support materials and exemplification to help them prepare for lessons. Of course, you’ll incrementally become expert too – and, as you do, you can also add your expertise for others to learn from. That’s why the platform is ever-growing.

Get to know mathematics teachers. We are a tremendously friendly bunch, I promise you. Come to a MathsConf, have a drink. Knowing other mathematics teachers means there is always someone to drop an email to or give a call or join a text message group with. We all have tough lessons – even the most experienced, specialist teachers – so knowing that there are friendly mathematics teachers around to bounce ideas off or just have a chat with is a great way of getting comfortable with teaching a new subject.

Don’t reinvent wheels. Spend your time role playing in your mind the pedagogic decisions you will make throughout the lessons that you are teaching tomorrow rather than making resources or writing plans – these all already exist.

Finally, don’t expect to be an expert mathematics teacher from day one. Like all things, it takes time and deliberate practice. By drawing on all the support that exists around you – like Complete Mathematics, MathsConf, and the Teacher CPD College – you’ll be able to deliver effective mathematics lessons whilst continuing to grow and develop your expertise.

H: What are your thoughts on the growth of private tutors in the UK?

M: In the UK, approximately 25% of all pupils aged 8-15 have a regular private tutor for mathematics who provides supplementary education; working on misconceptions, strengthening understanding, consolidating classroom learning, supporting with homework and revision amongst much else.

Clearly, high value is placed on learning by the pupil’s family (who are often sacrificing other things in order to fund the tuition). And it is often assumed by policy makers that 75% of families who do not engage a tutor place a lower value on education. But this just is not true. Time and time again, surveys reveal that the majority of the 75% do indeed want their child to also have supplementary education, but they simply are not able to afford to purchase it.

How can that be right? In what view of the world is that possibly ok?

Here is a tool which clearly helps pupils to excel in mathematics and is clearly an ambition of all families, yet is reserved for just a small minority. Closing the gap would be easy; ban all tuition. But this is not the right thing to do. The right thing to do is to remove all barriers that limit pupils.

Supplementary education is expensive because of the human resource cost. But what if it was possible to bring about all the benefits of a human tutor using a different approach? That’s what we are doing. It does, of course, take an enormous amount of work to create a system which can plan for and devise responses for every single possible twist that can arise when a pupil is learning mathematics – but something being incredibly difficult is no reason for not doing it, in fact for me it is the very reason for doing it!

H: La Salle Education’s mission is to improve mathematics education for all students. What role does technology play in closing the gap between students from the most and least disadvantaged families?

M: Firstly, I’ll say that I do not think that closing the gap between pupils is a useful or right focus. The focus on closing the gap too often morphs into holding the most advanced pupils back – I don’t think this is a helpful thing for humanity. What I am interested in is helping all pupils to excel. All pupils have the potential to excel in mathematics. It cannot be acceptable that some are prevented from doing so. The answer is to remove all barriers that limit pupils. Secondly, I should also say that pretty much everything is a technology – a pencil, the school curriculum, the printed word, etc. Teachers do, and always have, deployed technologies in order to best support their pupils’ learning. And teachers recognise that technologies need to be used critically – technologies should have a purpose in mind.

o, in brief, the purpose of technologies in education is to remove barriers that limit any individual pupil from excelling. Creating such technologies is complex and requires a deep understanding of learning. We shouldn’t stop until all barriers have been broken down.

H: What’s next for La Salle Education?

M: At the most basic level, there are two things that keep me awake at night:

Firstly, the impact that a teacher has on the life of an individual pupil is profound. So, all teachers must be able to continue to grow and develop throughout their careers and have a platform for articulating and testing their own theories.

Secondly, every individual pupil has the potential to excel. So, all barriers that limit them from doing so must be removed.

Next for La Salle? Well, it is to continue to play our small part in helping teachers have a profound impact on pupils and helping pupils to excel.

The three strands of what we do are:

  • Complete Mathematics – a teaching and learning community consisting of an online environment and face-to-face events.
  • The Teacher CPD College – an online repository of self-study courses for teachers
  • The Virtual Mathematics School – a school staffed by virtual tutors to enable all pupils, from all economic backgrounds, to access supplementary education

We will continue to develop these strands – the process of improvement is unending – and continue to work with teachers to ensure we meet their needs. We started with mathematics because mathematics has a liberating impact on an individual’s life, but we are not stopping there. In the future, we’ll support teachers and pupils of other subjects in the same way.

I think it’s worth ending with our mission statement. This is why I go to work in the morning:

“Our mission is simple: we want to improve mathematics education for all pupils.

The reason we are on this mission is also simple: being mathematically literate transforms a life.

Mathematical competence is the foundation for being able to lead an autonomous and rewarding adult life. Being mathematical means being able to overcome challenges and navigate through life with purpose.

All children have the potential to become mathematical. All children have the potential to leave school intellectually equipped to be successful.

La Salle Education exists to help those potentials be realised.”


Look out for the launch of our new #AskMark series, your chance to put your questions to Mark McCourt. Make sure you follow @LaSalleEd on Twitter for updates.

What we learnt from our latest #MathsConf25

Written by Hannah Gillott Friday, 19 March 2021

A year on from our last in-person event, Saturday’s virtual #MathsConf25 made the most of the benefits of moving online. Bringing together workshop leaders and delegates from around the world, the day was an energetic and passionate celebration of mathematics not just as a subject to be taught, but as one to be enjoyed, debated and puzzled over for the pure joy of it. Although #MathsConf26 will return to the original in-person format, such was the success of the online version that we will be keeping it for future #MathsConfMini events.

As ever, #MathsConf25 kicked off with a Friday night social, with Rob Smith at the helm of a packed schedule of activities. Our community of mathematics educators forms the foundation of La Salle Education, and founder Mark McCourt passionately believes in creating a space where, at the end of a tough lesson or a challenging day, teachers know they will find someone ready to offer support. That so many of you logged on ready to crack codes, fold origami and debate the number of holes in a t-shirt is testament to the success of that vision, while the headline performance from tutor-by-day-musician-by-night Atul Rana was a brilliant reminder of the many talents of our community.

Fresh approaches to familiar concepts

#MathsConf25 was our biggest yet, with over 50 workshops taking place across six periods and nine rooms throughout Saturday. The generous sponsorship from AQA meant attendees could access every session for just £5, with all recordings made available after the event. As ever, our workshop leaders covered a huge range of topics from the role of storytelling in EYFS classrooms to Data Science for A Level Maths students. The #MathsConf25 hashtag on Twitter was abuzz throughout the day with teachers sharing their excitement over new strategies and insights to incorporate into their lessons — alongside photos of entries for the cake competition, fittingly themed around pi. Many of the workshops challenged delegates to unpick concepts they might take for granted, instead looking in detail at the methodology and didactics behind seemingly simple processes like counting or calculating percentages. Perhaps the most valuable component of #MathsConf is the opportunity to be a learner again, approaching familiar problems with fresh eyes and having the time to share your insights with equally curious colleagues.

Bridging the gap between Primary and Secondary

The “false dichotomy” between Primary and Secondary emerged as another theme throughout the day, with Secondary colleagues embracing the chance to learn more about how mathematical foundations are laid early on. There is clearly a strong desire for more collaboration between the two stages, but the demands of the typical school day often prevent this. Many delegates gladly seized the opportunity to learn from those teaching in different contexts. Larissa Chan’s early session on Discalculia offered a particular reminder of the challenges facing some students, alongside practical strategies that could be implemented in any classroom at any stage of learning.

Celebrating learning, not outcomes

Upon the conclusion of the main workshops, Atul Rana led the Post-MathsConf25 debrief on Twitter — a 90 minute livestreamed discussion with eight other workshop leaders reflecting on the day’s learning. Limes Wright’s workshop on maths anxiety particularly struck a note, with much discussion of the role teachers can play in relieving anxiety for students and transferring, instead, their passion for the subject. Teachers, the debrief concluded, must lead by example, creating an environment in which every member of the class — teacher included — is free to make mistakes. Wrong answers are a rich and often untapped source of learning — raising their status in the classroom opens up opportunities to correct misconceptions and celebrate the mistakes that eventually lead learners to the right solution. The benefits behind such mathematical thinking extend beyond the classroom — what can’t our students achieve when they can reframe failure as another step towards success?

A community for all teachers

#MathsConf25, like those before it, sought to provide a platform for teachers to share their expertise with colleagues. As Mark McCourt so aptly put it during the debrief, “Teachers are intellectuals. They have theories and those theories are worth sharing.” One of the most discussed workshops of the timetable was delivered by Nathan Day, who is nearing the completion of his ITT. The success of his session reinforces how much we have to gain when we open the floor to teachers from all backgrounds and all levels of experience, recognising that every one of us has something to learn but every one of us also has something to share. Workshop leaders will now be offered free attendance to future Complete Mathematics conferences for life, so we would urge anyone considering running one to get in touch with their proposal.

With #MathsConf26 pencilled in for Saturday 10th July in Kettering, we hope to see as many of you as possible in person to continue learning from one another and building our community of maths educators and enthusiasts.


#MathsConf25 ticket holders can relive the conference on our website, with all workshop videos now available to watch back. If you missed out, sign up to our Teacher CPD College where workshop videos are available with your subscription.

As ever, we would like to thank all of our sponsors for making #MathsConf25 possible: AQA, Pearson, Whiterose Maths, Maplesoft, OCR, Collins, The OR Society, WJEC Eduqas, Tarquin Books, Arc Education and SAGE Publications Ltd.

The New Lesson Page on Complete Mathematics

Written by Josh George Monday, 22 February 2021

Following the recent release of our Enhanced Curriculum page, the natural next step was to roll this update out across the areas of the platform that draw from the Curriculum—none more so than Lesson Planning. Today we are excited to release the brand new Lesson page on the Complete Mathematics platform.

The last year has seen record activity on the platform, with teachers across the UK and beyond using Complete Mathematics to plan, deliver, and review their mathematics lessons. We’ve enjoyed discussing implementation and use with the Complete Mathematics community, and as part of this update have incorporated loads of your ideas and feature requests to make Lessons on the platform more useful, impactful, and user-friendly than ever before.

Here's how it works:

Below we highlight some of the key elements of this release and where this will take us next, grouped by the following themes:

If you have any questions or are a prospective user who wants to learn more, get in touch with our support team — This email address is being protected from spambots. You need JavaScript enabled to view it..


Pedagogy & Accessibility

We've now fully embedded the enhanced curriculum into lesson planning. This brings a whole package of benefits to the Lesson page.

Updated Objective Search Tool

Find the exact objective you want to involve in your lesson with greater ease now that objectives and their descriptions are shown as your search results. View where that objective sits in the context of the curriculum, or your scheme, at the click of a button.

Refined Prerequisite Mapping

Discover the required understanding for each objective with even more confidence, and explore the threads through the Curriculum Universe in depth.

Models and Didactics

Objectives in your lessons will now include the recently added support material sections, Models and Didactics. We explore these sections and their implementation further in this blog.

Adding the next recommended Objective from a scheme to your lesson, and reviewing the materials.

Teaching Progress Review

Use the Class Scheme page to explore how a Class is progressing through their assigned scheme as before, plus, now review this same teaching progress against the full Curriculum, Mathematical Groups, and Mathematical Topics.

Device Optimisation & Fonts

More effectively access, amend and review your lesson plans on-the-go with improvements to the mobile lesson experience. Plus, edit the font used in the Curriculum for further accessibility, or just personal preference (yes, Comic Sans).


Flexibility & Class Communication

We’ve worked a number of user-requested features into this release, in particular, to further assist the Complete Mathematics community with their remote teaching, and their blended learning practice more generally.

Planning Status Control

Freely set a lesson’s status to Planned or Taught without having to add an Objective, enabling you to mark revision, assessment, or other off-timetable lessons as green on your timetable.

Lesson Notes

Add bespoke notes to your lesson, instead of your Objectives, for jotting down your lesson layout, or recording an off-timetable lesson.

Notes for Pupils

Create and share notes with your pupils, for class-wide announcements, reminders, instructions, or otherwise. Control the pupil visibility of your notes at the click of a button.

Creating lesson notes for yourself, your colleagues, and now for your pupils to access too.

Assignment Creation and Review

Easily assign work from one or multiple Objectives with dedicated sections for Classwork and Homework and a brand new creation tool. Plus, review assignments and monitor pupil activity fully within your lesson.

Voice Note Assignment Feedback

Record yourself giving feedback on each pupil’s classwork or homework submissions, for your pupils to log in and listen to, mirroring the classroom experience.


Mastery & Beyond

Along with the new functionality and benefits we've previewed so far, this update also lays the foundations for further, major additions to the platform—some included in this release, and some to follow in the near future.

Prerequisite Quizzes

Assess your classes understanding of the required knowledge for your upcoming Objectives—a key aspect of teaching for mastery—with brand new readiness quizzes. Select to include first, second, or third level prerequisites, implementing the refined mapping directly in assessment.

New Quiz Builder

Build date range and prerequisite lesson quizzes with improved control and visibility of the objectives covered. Review and edit the granules to be included, with a summary of the quantity and approximate level of content covered available too.

Generating a date range quiz, reviewing the recently taught objectives to be drawn from.

Lesson Readiness Insights

Inspect relevant assessment results data for your class as soon as you add an Objective to your lesson, to help you plan your next teaching steps. Review results for the added objective, its linked prerequisites, as well as the appropriate mathematical groups and strands.

What Next?

Following the completed release of the new Lesson page there are a number of exciting platform projects we will be working on, both short-term and long-term, including: importing scheme progression from other classes and previous academic years; saving, re-using, and sharing lesson plans; lesson schedule builder; pupil knowledge security insights...and lots more!

Complete Mathematics users can log in now to use the updated Lesson page.

For prospective users, you can book a webinar with one of our team to learn more, or get in touch at This email address is being protected from spambots. You need JavaScript enabled to view it..

Satisfying Government Requirements for Remote Education Provision with Complete Mathematics

Written by Josh George Sunday, 27 September 2020

Updated 11th January 2021

With the continued disruption to schools around the world, governments are setting out the type of provision institutions must put in place to prepare for pupil absences, as well as local or national lockdowns. As an example of requirements being outlined, you can find the full guidance for schools in England here.

Using this guidance as context, we wanted to explore how the Complete Mathematics platform satisfies remote education requirements.

If you are already a Complete Mathematics user, you can be confident in the suitability of your existing provision. If you are not with Complete Mathematics and are looking for a new platform to fulfil these, or similar, requirements for your institution then you can book in a conversation and demonstration with one of our School Support team.

We are always happy to talk through how Complete Mathematics might support your department and pupils, both in the current disruption and beyond, using existing or catch up funding to make a real impact on your school’s blended learning environment.

The Government Conditions:

Teach a planned and well-sequenced curriculum so that knowledge and skills are built incrementally, with a good level of clarity about what is intended to be taught and practised in each subject so that pupils can progress through the school’s curriculum.

The Complete Mathematics curriculum is a single, continuous, coherent and fully supported learning journey through school mathematics. Assign your bespoke scheme of work to each class to follow and track progression throughout each year. Plan lessons to be accessed by pupils both ahead of time, on the day, and retrospectively as desired.

A classes scheme of work progress with next recommended objective for planning
Select a digital platform for remote education provision that will be used consistently across the school in order to allow interaction, assessment and feedback and make sure staff are trained and confident in its use.

One single subscription includes unlimited teacher and pupil accounts, for full and unrestricted access across your institution. The platform is used for planning, teaching, assessing, feedback and reporting, and we offer a free department-wide training session for all subscriber institutions, ontop of the ongoing support available with our knowledge base and live chat.

Complete access for all teachers, classes, and pupils in your institution.
Overcome barriers to digital access for pupils by providing printed resources, such as textbooks and workbooks, to structure learning, supplemented with other forms of communication to keep pupils on track or answer questions about work.

All worksheets, quizzes, and tests can be downloaded and printed. Completed offline work can be added to the platform to retain full tracking of submissions, and inclusion of results into the platform analytics.

Downloading a quiz for printing
Have systems for checking, daily, whether pupils are engaging with their work, and work with families to rapidly identify effective solutions where engagement is a concern.

Along with live tracking of viewing and completion of assigned work in each lesson, the platform includes pupil lesson objective progression judgement for each piece of learning, based on the pupil's regular checking of I Can statements.

Use your pupils activity on assigned work to inform your own judgements on progression.
Identify a named senior leader with overarching responsibility for the quality and delivery of remote education, including that provision meets expectations for remote education.

Each account has a lead user, along with manager and teacher permissions to give a schools senior leadership team full oversight.

Across the platform there are reporting tools to enable senior leadership to assess impact and progression across classes, cohorts, and pupil groups.

When teaching pupils remotely, we expect schools to set meaningful and ambitious work each day in an appropriate range of subjects.

Build your lesson plans and assign work for your pupils to access online. Monitor, track, and feedback on submissions remotely. Set specific work for particular pupil groups within classes including high-attaining pupils or intervention groups. Plus, generate bespoke quizzes to provide personalised, automated remediation activities based on each pupil’s results.

Assigned and trackable classwork and homework for whole classes or pupil groups.
Plan a programme that is of equivalent length to the core teaching pupils would receive in school.

Use the same continuous timetable, seamlessly transitioning between in-school or at-home learning, preparing lessons and quizzes in the same space for pupils to access in a familiar way, at a familiar time. Use the same continuous learning journey, no matter the disruption in environment.

One consistent timetable, to follow each scheduled maths lesson at-home or in-school.
Providing frequent, clear explanations of new content, delivered by a teacher or through high-quality curriculum resources or videos.

Each granular objective in Complete Mathematics is supported by: an overview and context, common misconceptions, example questions with worked solutions, key learning points, models, didactics, resources, & linked tutorial videos. These materials are specific to each granule, and there are 1800+ objectives across the whole journey. Choose which objective(s) you want to teach in a lesson, and pick the materials you want to include in your teaching, fully accessible to pupils on their side of the platform.

Each granular objective is full of supporting materials and teaching resources.
Providing opportunities for interactivity, including questioning, eliciting and reflective discussion.

In an assignment activity summary teachers can see whether pupils have viewed, submitted, reflected upon, or got in touch about each piece of assigned work.

Both pupils and teachers can start a discussion around an assignment, to request help, offer feedback or otherwise.

Providing scaffolded practice and opportunities to apply new knowledge.

Regular quizzes collating recently taught ideas for recall and application, followed by further, unlimited, dynamic pupil quizzes based on any of the objectives covered for additional practice.

Personalised pupil remediation guidance following each quiz.
Enabling pupils to receive timely and frequent feedback on how to progress, using digitally-facilitated or whole-class feedback where appropriate.

The platform facilitates teacher feedback on assignments as well as discussions between teacher and pupil.

Additionally, following quizzes and tests the platform displays results feedback for pupils, summarising the assessed topics of mathematics and which might be the priority for improvement.

Using assessment to ensure teaching is responsive to pupils’ needs and addresses any critical gaps in pupils’ knowledge.

Generate weekly, low-stakes, formative quizzes based on recently taught content, automatically marked and showing you in real time how successfully each idea has been grasped. Analysis of all previous results data highlights the key recently taught objectives that are not yet secure and require further teaching before moving forward, so you can add these objectives to your next maths lessons. Pupils can access immediate remediation guidance on each objective they have struggled with within each quiz taken, with opportunity to learn and practice beyond their lessons.

Inform your next teaching with recent quiz results, with insecure objectives highlighted for you.
Avoiding an over-reliance on long-term projects or internet research activities.

Complete Mathematics breaks mathematics down into granular objectives and supports the teaching and learning of each granule with specific support materials, examples questions, and questions.

The Complete Mathematics Curriculum, complete with resources and tutorial videos.
We expect schools to consider these expectations in relation to the pupils’ age, stage of development or special educational needs, for example where this would place significant demands on parents’ help or support.

Complete Mathematics covers all of school mathematics from Primary to Further Education and is fluid to each pupils stage of mathematical maturation. Each pupils MathsAge progression is tracked, with bespoke guidance available for each pupil at every point.

The platform can be implemented through teacher direction, independent pupil use, as well as assisted by parents or tutors.

Recognise that younger pupils and some pupils with SEND may not be able to access remote education without adult support and so schools should work with families to deliver a broad and ambitious curriculum.

All users can login from home and have access to the entire journey of the school mathematics, with digital manipulatives embedded to explore mathematics at an accessible level, on any device.

Accessible, digital Manipulatives on the platform.

If you would like to learn more about the Complete Mathematics platform and subscription then don't hesitate to get in touch. You can book a demo here, or email This email address is being protected from spambots. You need JavaScript enabled to view it. and we'll be happy to answer any queries you have.

Money Back Guarantee on Complete Maths Subscription

Thursday, 09 July 2020

At La Salle Education, we are so confident that the Complete Mathematics platform will benefit your department that we are offering a simple Money Back Guarantee on subscription.

If you are not 100% satisfied with Complete Mathematics at the end of the first year of your subscription, we will fully reimburse the cost.

We can offer this Money Back Guarantee because we have so many school subscribers with years of experience of using the platform and seeing standards rise. Of course, just as with any product, the efficacy comes through using it as designed.

So we are happy to give you the absolute assurance of quality, and impact on standards of teaching and learning, by guaranteeing your purchase if your school commits to using Complete Mathematics as it is designed to be used. At your end, this means:

  • Ensuring all members of staff receive the free getting started training session, delivered by one of the La Salle school support team
  • Ensuring all members of staff know how to access ongoing support
  • Following the Complete Mathematics Curriculum Journey
  • Planning all lessons through the easy to use, time saving platform planner
  • Running at least one formative quiz per week for each class
  • Ensuring all staff monitor and consider the detailed quiz results and analytics
  • Setting at least one homework assignment per week for each pupil
  • Ensuring that all pupils complete at least one hour of activity on Complete Mathematics at home each week (this may include quizzes, assignments, times tables, tutorials, etc)

And that’s it!

We know that when Complete Mathematics is implemented as set out, the impact on teaching and learning is significant, both in terms of pupil attainment and in terms of enhancing teacher practice.

Subscribe to Complete Maths today and hit the ground running in September.

Mathematical Maturation

Written by Mark McCourt Thursday, 18 June 2020

As a Complete Mathematics subscriber, you will be familiar with seeing several different metrics for communicating pupil attainment within the platform, whether it be our own MathsAge, GCSE grades or National 5.

But what lies behind these metrics and how are they assigned?

At La Salle, we are interested in the journey that a pupil takes in learning mathematics from counting through to calculus. As pupils learn more and their schema of knowledge develops, they become more and more ‘mathematically mature’.

Taking a view of the curriculum in terms of mathematical maturation is incredibly important if we are to provide pupils with a truly meaningful, interconnected view of mathematics.

As pupils mature mathematically, they move through phases, or levels, of typical dispositions, behaviours, knowledge and understanding.

Pupils are growing mathematically. And hopefully heading to becoming young mathematicians themselves, who will be inspired to continue to study, use and love mathematics well beyond leaving school.

Behind every element of Complete Mathematics is a sense of this ‘mathematical maturation’, which we communicate by considering the cognitive Demand Criteria Level (DCL).

We thought you might like to know a little bit more about DCL and to see the descriptions of the levels that drive our metrics.

The Complete Mathematics DCL range from Level 0 to Level 22.

Here is every DCL in detail, followed by a discussion of how these levels map to attainment grades in the platform.

Complete Mathematics – Demand Criteria

The demand criteria are broad descriptions of mathematical maturation from the point of no mathematical education through to becoming a mathematician. It is not age related. It is not intended to be treated as a strict ladder of progression. Rather, the levels within the demand criteria aim to give a general sense of the capabilities and dispositions of a person learning mathematics as they reach stages of mathematical maturity.

Level 0

A pupil operating at DCL0 is assumed to have biologically primary mathematical senses, including cardinal and ordinal numerosity up to three. Pupils can distinguish between simple objects arranged in order of size and colour.

Level 1

A pupil operating at DCL1 is beginning to learn about mathematics beyond the intuitive and biologically primary mathematics that they have encountered and have a sense of from early childhood. The mathematics at DCL1 requires explicit teaching, particularly to move beyond natural numerosity. Pupils will develop number sense beyond first, second, third and one, two, three. They can order, compare and perform arithmetic within 20, though may require the support of concrete materials to do so. Pupils can count forwards and backwards within 100. They have an emerging sense of fractionness through a simplistic understanding of ‘half’ and ‘quarter’. Pupils can make simple statements about the relative position of an object and can name simple 2D shapes. They have an emerging sense of length, height, weight and capacity and can order objects based on the properties where the values are simple. They are beginning to tell the time and handle money with simple denominations.

Level 2

A pupil operating at DCL2 is developing key foundational knowledge in mathematics. They can order and compare within 100 using appropriate symbolism and count forwards and backwards in steps of 1, 2, 3, 5 and 10. Pupils are beginning to realise that not all calculations should give exact responses, rather that it is sometimes more appropriate to provide and estimate. They can work with arithmetic within 100 and are particularly confident when the calculations are in the context of money. Pupils are beginning to appreciate an inverse relationship between addition and subtraction / multiplication and division, as well as some simple constraints that apply to the operators. Pupils are developing a stronger sense of fractionness through imagery and objects. When describing simple shapes, pupils can make statements about symmetry and other basic properties. Pupils can gather information about length, weight, capacity, mass and temperature by using appropriate apparatus with simple scales. Pupils can represent information in a small number of very simple formats, including tables, tally charts, bar charts and pictograms, where both the values and scales are straightforward and discrete.

Level 3

A pupil operating at DCL3 is formalising key foundational techniques. They can work with addition, subtraction, multiplication and division with 3-digit numerals in a variety of problems, checking their answers where necessary and using quick mental recall of multiplication facts related to 1, 2, 3, 5, 4, 8 and 10 times tables. They understand the place value of digits within numerals to 1000. Pupils can identify, represent and solve simple problems with non-unit fractions where the denominator is small. They appreciate the meaning of the denominator as indicating how many equal parts a quantity or object is being split into. Pupils use appropriate units when working with money, length, weight, capacity and mass. They can describe properties of simple shapes, even when the orientation is changed, including perimeter and basic geometrical properties relating to angles, including identifying perpendicular and parallel sides. They accurately use the language of acute, obtuse and right angled. Pupils can calculate using simple time intervals.

Level 4

A pupil operating at DCL4 is becoming fluent in key foundational knowledge and techniques. Pupils are comfortable in communicating with numbers to at least 1000000. They quickly recall multiplication facts up to 12x12 and use their knowledge of factor pairs in working confidently with arithmetic within 10000, for which they use formal written algorithms and can solve problems in a variety of contexts. Pupils can confidently order and compare numbers and have a good understanding of place value. When counting backwards, they can bridge across zero into negative numbers. They can add and subtract with fractions and recognise and understand the decimal equivalence of simple fractions. Pupils can round numbers, including answers, to the nearest 10, 100 or 1000, or, when working with decimals, to the nearest whole number or to one decimal place. Pupils can convert between time formats between different units of measure, such as metres to kilometres. They can calculate perimeter and area of rectilinear shapes, though they may still rely on the use of counting squares to find area. Pupils communicate information more clearly, adding bar charts and time graphs to their repertoire. A DCL4 pupil is beginning to be able to hold mathematical conversations when solving problems, using correct terminology and appropriate formal algorithms.

Level 5

A pupil operating at DCL5 is becoming increasingly confident in using foundational knowledge and techniques to work in a range of situations. They have extended their mathematical literacy to include working with numerals to 3 decimal places, recognising and working with common multiples or factors, and using both square and cube numbers. Pupils understand what it means for a number to be primar and can establish whether a number up to 100 is prime. Their mathematical communication is largely through efficient formal methods. Pupils now use rounding as a method for checking calculations and can convert confidently between measures, recognising some common equivalences. Pupils use of fractions in problems is increasingly confident, including situations where they must work with improper fractions or identify equivalent fractions. Using their foundational knowledge, pupils are expanding their geometrical repertoire. They use standard units when work with measures and can find perimeters of composite shapes. They are working with polygons, volumes, nets and 2D isometric representations of 3D shapes. Pupils can describe the effect that straightforward reflection or translation has on simple shapes. They can state and describe simple angle facts.

Level 6

A pupil operating at DCL6 is confident with arithmetic, including with decimals, has sound mental calculations skills, can round numbers to a required degree of accuracy, and works comfortable with problems involving a mix of fractions, decimals and percentages, including when it is necessary to convert between those forms. They can do this because their foundational knowledge is embedded and well-rehearsed. Pupils are building on their foundational number knowledge to understand the general case. They are beginning to appreciate pattern and can work with linear number sequences as their sense of algebra starts to emerge. Pupils understand how points can be expressed on a plane using a coordinate system. When working with shapes, including triangles and parallelograms, pupils use standard units, can state simple angle facts and use formulas for finding area. In 3D, they use standard units for volume and formulas for finding volumes. A DCL6 pupil is on the cusp of becoming mathematically functional.

Level 7

A pupil operating at DCL7 can be considered mathematically functional. They can access and understand key mathematical information commonly encountered in day-to-day life. In addition to their already established foundational knowledge, pupils use negative numbers in context, can solve percentage problems and are able to use estimation as a method for attacking problems. Pupils can list combinations of two variables and can express missing number problems algebraically. Pupils can work with simple formulae. Their geometry repertoire has been extended to include translating shapes on a coordinate grid and working with simple scale factors. They are familiar with key parts of circles and how to name and label them. When working with data, pupils use pie charts and line graphs. They can find the mean average of a set of data and understand its meaning.

Level 8

A pupil operating at DCL8 is beginning to move beyond a simple functional use of mathematics, seeing more relationships between areas of mathematics and understanding more about its applications. Pupils can express one quantity as a fraction of another, work with roots in addition to powers, can round numbers to a required number of significant figures and understand the rules relating to the order of operations. Pupils are increasingly sophisticated in their appropriate use of calculators and other technologies to enhance their mathematical work. They routinely convert between standard units in a range of problems. Pupils use of ratio is becoming more useful, particularly now that they understand the purpose of reducing a ratio to its simplest form. When examining generality, pupils understand the meaning of simple expressions and know when they are used in equations. Their use of algebraic notation is consistent and appropriate to the simple problems that they work on, including problems involving substitution, generating sequences, simplifying expressions, collecting like terms and multiplying a single term over a bracket. The linear equations they solve are confined to those in one variable. Pupils use formulae when solving volume problems. They understand properties of parallel and perpendicular lines and a range of other properties of 2D shapes. Their use of coordinates is accurate in all four quadrants and they use this knowledge to plot linear graphs, understanding the meaning of gradient. Pupils can identify congruency in triangles and can interpret scale drawings. When working with data, pupils consistently make appropriate choices for best representations including frequency tables, bar charts, pie charts, and pictograms. Pupils understand the probability scale.

Level 9

A pupil operating at DCL9 has well established formal written methods for working with arithmetic, which they do with confidence and accuracy. Building on their appreciation of proportion, they can solve a range of problems involving ratio. Their use of percentages in solving a wide variety of problems continues to expand. Pupils can express numbers as multiples of primes by decomposition. Given a straightforward linear sequence, pupils can determine an expression for the nth term in the sequence. Pupils solve linear equations, including those that first require rearrangement or factorisation. Pupils use their established understanding of gradients and their use of graphs of linear functions to work confidently with conversion graphs. Pupils understand and can identify alternate and corresponding angles. They can construct bisectors of lines and angles and can interpret and construct loci. Pupils work with scale drawings in a range of problems and can construct enlargements. Pupils understand simple sets and unions and can express these in a range of ways including Carroll and Venn diagrams. Pupils established use of area is extended further to include composite shapes with circular parts and finding surface area of prisms or cylinders. When working with data, pupils can carry out a statistical project, using a range of representations including scatter graphs and the three averages.

Level 10

A pupil operating at DCL10 can be considered mathematically literate. Their mathematical knowledge and skills are at a level suitable for the majority of generalist jobs and can transfer across a range of workplace requirements. They have strong arithmetical skills, including arithmetic with mixed numbers and negatives, they can work with percentage change problems and understand direct proportion. Pupils can work with, and solve problems involving, a range of compound measures. Pupils can expand binomials, change the subject of a formula and solve equations with an unknown on both sides, including those with fractional expressions. Pupils knowledge of equations is built further by their new understanding of simultaneous equations, which they can solve in cases where both equations are linear. Pupils can plot and interpret graphs of quadratic and cubic functions. They have a good understanding of angle sum properties and fully appreciate congruence. At DCL10, pupils first embark on the formal study of trigonometry, starting with an understanding of Pythagoras’ Theorem. When working with data, pupils can use sample spaces, stem and leaf diagrams and frequency polygons.

Level 11

A pupil operating at DCL11 is moving beyond the mathematical knowledge required for a general level of mathematical literacy. Their representation of number now includes standard index and they can work with inverse proportion. Pupils can plot and interpret graphs of exponential and reciprocal functions. They are able to generate and find general terms in geometric sequences. Pupils can model real life situations using formulae and graphs. Their knowledge of simultaneous equations now includes situations where one equation is non-linear. Pupils formal study of trigonometry continues with an appreciation of the sine, cosine and tangent functions. Pupils can better describe data and its limitations through the use of interquartile range and box and whisker diagrams.

Level 12

A pupil operating at DCL12 can draw on a range of sophisticated mathematical techniques. Their understanding of the connections and equivalences across fractions, decimals, percentages and ratio is matured. They can use equivalent ratios, find reverse percentage change and compound interest or other repeated percentage change measures. Pupils representations of number are enhanced by effective use of index laws and they can perform calculations where numbers are in standard index form. Pupils further appreciate the impracticality of finding precise solutions and have added trial and improvement as a numerical method to their repertoire. Pupils work confidently with sequences involving triangular, square and cube numbers. They can solve equations involving direct and inverse proportion. Pupils can find the equation of straight lines and they use function notation appropriately. Working with non-linear expressions, pupils can solve quadratic equations through a variety of methods including the use of factorising and completing the square. Pupils have more formal approaches to defining turn and position, including the accurate use of bearings. They know the sine and cosine rules and can find missing lengths and angles using Pythagoras’ Theorem and trigonometry. They have committed to memory special trigonometric angles in simple cases. Pupils can use trigonometry to find the area of triangles. They are confident in using scale factors. When working with data and carrying out statistical projects, pupils adopt systematic listing strategies. They understand the probability of mutually exclusive events. A DCL12 pupil should be encouraged to study mathematics at a higher level in adult life.

Level 13

A pupil operating at DCL13 works fluently with number. They can perform calculations with surds, can find upper and lower bounds and use recursive formulae. Their confidence in working algebraically is enhanced by their new ability to work with algebraic fractions and transform functions. Pupils understand similarity and can identify a range of properties of similar shapes, using these to perform calculations. Presented with information about shapes in given formulae, pupils can undertake a process of dimensional analysis enabling them to state the type of property being expressed. Pupils are familiar with and can sketch graphs of sine, cosine and tangent functions. Pupils know a range of circle theorems and can use these in formal geometrical proofs. Pupils understand conditional probability and can accurately produce a tree diagram to describe and experiment and calculate probabilities of theoretical events.

Level 14

A pupil operating at DCL14 is an emerging mathematician. They are able to work with mathematics eloquently, using sophisticated mathematical terminology and conventions. Their use of number is matured, enabling them to choose appropriate levels of accuracy with which to communicate solutions; choosing to express values in significant figures, work with expressions involving quantities in ranges of lower and upper bounds, and reliably work in surd form. At DCL14, pupils can express generality correctly in a wide range of scenarios, including solving simultaneous equations graphically and algebraically. They can find roots of equations by transforming expressions and can work with simple algebraic fractions. Pupils’ understanding of geometry is well developed. They can work accurately with trigonometric functions to solve problems in two-dimensions. Pupils can find volumes and surface areas of frustums and spheres and are able to define and interpret simple vectors. At DCL14, pupils work with difficult probability questions, including ‘without replacement’ problems. A DCL14 pupil would be expected to pursue further study of mathematics to an advanced and higher level.

Level 15

A pupil operating at DCL15 is a young mathematician who will be able to specialise in a mathematical field at higher education and in their career. Their well developed number work allows them to tackle growth and decay problems. They can find the general term in quadratic sequences and solve both linear and quadratic inequalities. They are increasingly interested in and capable of producing mathematical proofs. Their understanding of geometry is expanded by the addition of new skills in using negative scale factors, describing combinations of transformations, working reliably with plans and elevations, and performing calculations involving arcs and sectors. When using data, pupils understand the meaning of and can produce cumulative frequency diagrams. They are confident in working with grouped data and can produce and interpret histograms, including those involving unequal grouping.

Level 16

A pupil operating at DCL16 is becoming increasingly specialised and sophisticated in their use of mathematics. In particular, they are studying pre-calculus in a meaningful way and are making more articulate and accurate use of formal mathematical argument in proofs. In pre-calculus, pupils appreciate characteristics of rates of change and can describe the meaning of gradients at points on curves and, in simple cases, calculate these gradients. Pupils understanding and use of vectors is further developed to a point where they can use combinations of vectors in geometrical proofs. Pupils can describe and plot coordinates in 3D. When undertaking statistical projects, pupils understand implications for sampling populations. In their analysis, they are able to discard outlying data by considering central tendency and measures of spread. Pupils understand correlation and can add a line of best fit to data, using it to produce commentary about the trends and make predictions.

Level 17

A pupil operating at DCL17 can use mathematics reliably in a wide variety of situations, particularly in describing the real world. They understand the laws of indices for all rational exponents and are able to rationalise denominators. Pupils understand and can use force, weight, displacement, speed velocity and acceleration. In solving non-linear problems, pupils can use the factor theorem and their knowledge of the discriminant of a quadratic function. They can represent linear and quadratic inequalities graphically and can describe asymptotes. Pupils use intersection points of graphs to solve equations. They can transform graphs and know the gradient conditions for two straight lines to be parallel or perpendicular. Pupils use the equation of a circle in solving problems. When working with data, pupils select sampling techniques based on their knowledge of the population. Pupils interpret regressions lines for bivariate data.

Level 18

A pupil operating at DCL18 uses well-reasoned mathematical argument that is accurate, appropriate and concise. They use proof by deduction, proof by exhaustion, disproof by counter example and proof by contradiction. Pupils understand and can use exponential and logarithmic functions and their graphs. They know the laws of logarithms and can use them to solve equations. Pupils can perform vector addition and multiplication by scalars. At DCL18, pupils are beginning to specialise in areas of mathematics that will enable them to continue to study the discipline at a high level. If choosing to pursue a mechanics / dynamics path, pupils understand and use Newton's first law and second law for motion in a straight line under gravity. If choosing to pursue a statistics path, pupils understand informal interpretations of correlation and can describe why correlation does not imply causation. Their analysis of data is further refined by sophisticated use of standard deviation.

Level 19

A pupil operating at DCL19 is beginning to understand the implications of The Calculus. They understand and can use differentiation from first principles for small positive integer powers of x. They can find the second derivative and know that this represents a rate of change of gradient. They apply differentiation to find gradients, tangents and normals, maxima and minima and stationary points. This allows them to identify a function as increasing or decreasing. Pupils know the Fundamental Theorem of Calculus. If pursuing a mechanics / dynamics path, pupils understand and can use Newton's third law. They can calculate the distance between two points represented by position vectors. If pursuing a statistics pathway, pupils can express solutions through correct use of 'and' and 'or', or through set notation. They know the binomial expansion of (a + bx)n and reliably use the notations n! and nCr.

Level 20

A pupil operating at DCL20 is a young mathematician who is continuing to understand implications of The Calculus. They understand integration as being anti-differentiation and can integrate expressions. They can find a definite integral and know this to represent the area under a curve. If pursuing a mechanics / dynamics path, pupils will expertly use vectors to solve problems in in context, including forces. If pursuing at statistics path, pupils can clean data and undertake statistical hypothesis testing. They select an appropriate probability distribution for a context, including when not to use the binomial or Normal model.

Level 21

A pupil operating at DCL21 is expanding their mathematical knowledge base to enable later study of more complex situations to be addressed with calculus. They work with increasing decreasing and periodic sequences, understand and use sigma notation, understand finite geometric series and the sum to infinity of a convergent geometric series. Pupils reliably use modulus notation. Pupils accurately perform algebraic division in simple cases and understand the modulus of a linear function. They can use composite functions; inverse functions and their graphs and are able to determine combinations of transformations. Their geometrical communication is enhanced by their use of radian measure and they know the exact values of sine, cosine and tangent of special radian measures. Pupils understand secant, cosecant and cotangent, arcsin, arccos and arctan. If pursuing a mechanics / dynamics path, pupils use vectors in three dimensions to solve problems in kinematics. They use trigonometric functions context, including problems involving vectors, kinematics and forces. Pupils use formulae for constant acceleration for motion in 2 dimensions using vectors and use calculus in kinematics for motion in 2 dimensions using vectors. If pursuing a statistics path, pupils extend the binomial theorem to any rational n and know its uses in approximation.

Level 22

A pupil operating at DCL22 is a mathematician ready to advance to higher education study in mathematics or a closely related field. Their competence in working with a significant range of mathematical problems is secure. They are articulate and able to communicate mathematically with precision and elegance. Their study of The Calculus means they can reliably differentiate and integrate expressions containing a wide variety of functions including trigonometric, exponential and logarithmic terms. They know the connection between the second derivative and convex and concave sections of curves and points of inflection. When differentiating, they use the product rule, the quotient rule and the chain rule. When integrating, they use integration by substitution, integration by parts and integration using partial fractions. They know integration as the limit of a sum. They understand limitations of methods and are able to deploy appropriate numerical methods such as Newton-Raphson. Pupils understand parametric equations of curves and can find the first derivative of simple functions defined implicitly or parametrically. Pupils produce clear and elegant geometrical proofs using, for example, double angle formulae and other trigonometric identities. Pupils can construct simple differential equations in context. If pursuing a mechanics / dynamics path, pupils can model motion under gravity in a vertical plane, resolve forces in 2 dimensions and understand equilibrium of a particle under coplanar forces. Pupils use addition of forces. They understand the impact of and can model motion of a body on a rough surface. They understand and use moments in simple static contexts. If pursuing a statistics path, pupils understand the conditional probability formula and use this when modelling with probability. They use Normal distribution as a model and can interpret a given correlation coefficient using a given p-value or critical value. A pupil operating at DCL22 has studied mathematics with determination and flair throughout their time at school and should be encouraged and supported by the mathematics teacher to follow a career as a mathematician.

So now you know more about the DCL framework and the broad phases that pupils pass through as they become more and more mathematically mature.

How then do these cognitive Demand Level Criteria inform the attainment metrics in Complete Mathematics?

Our MathsAge has been carefully mapped against the DCL and against other useful metrics, including GCSE, National 5, Core Maths, A Level, Higher and more.

Let us take a look at how the DCL line up with some of our most commonly used metrics.

DCL CM Stage Maths Age GCSE (Eng) GCSE (Wal) Nat 5 Core Maths A Level Higher NC Level
1 1 6 1 Low G Fail N/A N/A N/A 1
2 2 7 1 High G Fail N/A N/A N/A 2
3 3 8 1 Low F Fail N/A N/A N/A 3
4 4 9 2 High F Fail N/A N/A N/A 3
5 5 10 2 Low E Fail N/A N/A N/A 4
6 6 11 3 High E Fail N/A N/A N/A 5
7 6 11 3 Low D D N/A N/A N/A 5
8 7 12 4 High D D N/A N/A N/A 5
9 8 13 4 Low C C N/A N/A N/A 6
10 9 14 5 High C C E N/A N/A 7
11 9 14 5 Low B B E N/A N/A 8
12 10 15 6 High B B D N/A N/A 8
13 10 15 7 Low A A D N/A N/A 8
14 11 16 8 High A A C N/A N/A Above NC Levels
15 11 16 8 Low A* Upper A C N/A N/A Above NC Levels
16 11 16 9 High A* Upper A C N/A N/A Above NC Levels
17 12 16 Above GCSE Above GCSE Above Nat 5 B E N/A Above NC Levels
18 12 16 Above GCSE Above GCSE Above Nat 5 B D D Above NC Levels
19 12 17 Above GCSE Above GCSE Above Nat 5 A C C Above NC Levels
20 12 17 Above GCSE Above GCSE Above Nat 5 N/A B B Above NC Levels
21 13 18 Above GCSE Above GCSE Above Nat 5 N/A A A Above NC Levels
22 13 18 Above GCSE Above GCSE Above Nat 5 N/A A* Upper A Above NC Levels

As you will be able to see, the DCL often span multiple grades in a metric system. It is not the intention to convey the sense that a DCL or a grade can be pinned down accurately to a certain question of task – many tasks span multiple DCL and grades are a reflection of the performance of the population. Rather, what we are interested in is the pupil’s own development as a mathematician, the knowledge and skillset they acquire along the way and how these are articulated through the way in which a pupil behaves mathematically.

So, perhaps the next time you are looking at the MarkBook inside the Complete Mathematics platform, you can notice the attributes and dispositions those individual pupils exhibit in the classroom and see them as maturing gradually and know that, no matter what their current stage, they can continue to grow to become a successful young mathematician.

Using the Complete Mathematics Platform

Written by Abbie Brownlie Thursday, 26 March 2020

During this ongoing period of disruption in education, pupils and teachers have been working incredibly hard to keep on track.

Teachers and schools have done an incredible job with the support and provision they put in place in an exceptionally short time under remarkably challenging circumstances.

Our school support team have published a series of short training videos to show exactly how the Complete Mathematics platform can be used by both teachers and students to maximise the impact of blended learning.

This includes episodes on:

  1. Adding Pupils
  2. Lesson Planning
  3. Assigning Work
  4. Times Table Practice
  5. Remote Quizzes
  6. The Pupil App
  7. Accessing Support

Find out more about Complete Mathematics subscription with a demo from one of our team!

You can watch the whole ‘Using Complete Mathematics for Remote Learning’ playlist here!


Adding Pupils

A video covering how to easily add pupils to the platform, the generation of their login details for school and home access, as well as how your pupils log in.
Remember, you get unlimited, free pupil accounts with a Complete Mathematics subscription!


Lesson Planning

This video will show you how to create daily maths lessons from school or home, with appropriate support materials, resources and tutorial videos (including how to add your own). Plus, how your pupils can access these planned lessons on the platform, be it in the lesson or remotely.


Assigning Work

This talkthrough video will cover assigning content for your classes as both classwork and homework and your ability to do that from school or at home. While also taking a look at how pupils can access and interact with this assigned work, including: submission, adding their reflections, and asking you questions!


Times Table Practice

In this video you will see how pupils can take daily times tables quizzes with multiple representation of times table facts. Then find out how teachers can track progress and see their pupil's progression.


Remote Quizzes

This video will show you how to create informed, automatically marked, low-stakes quizzes based on your planning. See how pupils find and complete their assigned quizzes online and the opportunities for independent work via their results analysis view, either at home or in class.


The Pupil App

This video provides your pupils with a walkthrough of how exactly they can use the Complete Maths platform. This video is made to be shared directly with your pupils, which will show them where to find upcoming lessons, their assignments and quizzes that are due. Using the timetable to explore past and future lessons, and the quiz results page.


Accessing Support

We will discuss where to find our knowledge base, using the platform support hub, and other available sources of support.

Models & Didactics on the Complete Mathematics Platform

Written by Mark McCourt Friday, 29 May 2020

Pupils make sense of new mathematical ideas from a range of models, metaphors, examples and instruction. This means teachers need to be equipped with a set of approaches that they can call upon in the classroom.

Well established models, tried and tested over many years, such as the use of Cuisenaire rods, double number lines, algebra tiles, Dienes blocks, geoboards and many more are central to the subject specific pedagogical knowledge for everyone teaching mathematics.

We are adding these models to Complete Mathematics, tagged against the appropriate mathematical ideas so that teachers have immediate reminders of powerful pedagogies for the lessons they are planning.

Examples of models being added to Complete Mathematics

The specific technical details behind a mathematical idea – the building blocks of the idea itself – are critically important to understand and communicate precisely if pupils are able to make connections across the universe of mathematics as they grow and learn more. These technical details are known as the didactics of the mathematics.

Didactics act as a bridge between the teaching process and the learning process. An understanding of didactics enables teachers to translate the mathematical competencies they themselves have – the mathematics the teacher understands and can work with easily and without having to think – in to a communicated form of the mathematics such that pupils who have never before encountered it can make meaning and grip the idea at hand.

As Jim Fey writes in the seminal book ‘Didactics of Mathematics as a Scientific Discipline’ (1994) preparing mathematics for teaching can be conceived of as elementarization, that is, “the translation of mathematical concepts, principles, techniques, and reasoning methods from the forms in which they are discovered and then verified by formal reasoning to forms that can be learned readily by a broad audience of students.”

We are adding the didactics of mathematics to Complete Mathematics so that all teachers can readily access the technical detail of the ideas they are communicating and see how these mature over time.

Examples of didactics being added to Complete Mathematics

These and further updates to the curriculum will be live on the Complete Maths platform in the next few weeks. Stayed tuned for further investigations into this release soon.

La Salle Education raises £1M to make high quality maths tuition accessible to all students

Tuesday, 20 April 2021

La Salle’s platform includes everything needed to teach, learn and assess mathematics from primary to A-Level, both in-school and remotely.

London, United Kingdom: La Salle has successfully completed a new round of funding of just under £1M. La Salle’s existing shareholders supported the round and were joined by new investors who see a substantial opportunity to address a new, direct-to-consumer market by utilising La Salle’s digital technologies.

La Salle’s existing product, Complete Mathematics, is already well used by schools in the UK and overseas, supporting teachers to plan and deliver highly impactful lessons on a platform underpinned by the world’s most comprehensive maths curriculum. Building on its existing technology, La Salle has now developed a new, complementary offering in response to the post-COVID demand from parents and families seeking high-quality, comprehensive maths tuition online. This new ‘digital tutor’ will bring about all the benefits of a real-life tutor but at a fraction of the cost to schools and families.

“I am delighted that current and new investors are supporting the business to launch our new ‘digital tutor’ product” says Mark McCourt, CEO at La Salle. “We know that around 25% of pupils in the UK already benefit from private tuition, but those benefits are not accessible to the rest of the population due to the high cost of a real life tutor. This new funding will support the rollout of our ‘digital tutor’, thereby levelling the playing field and bringing maths tuition to all families regardless of income level. It’s a hugely exciting initiative that will really help pupils from all backgrounds and from anywhere in the world be successful in learning mathematics.”

This new round of funding will be used to grow the customer base in the UK, where catch-up funding has been made available for schools to invest in high quality tuition aids and internationally, bringing universal access to high quality mathematics education and support.

In 2021, La Salle will:

  • Launch the world’s first fully comprehensive ‘digital tutor’ for mathematics, covering all areas of the curriculum for all age and ability levels.
  • Provide ‘summer school’ access for pupils free of charge, with the intent of ensuring low income families are equally well-equipped for a successful academic year.
  • Meet international demand for individually tailored maths tuition that removes the need for a conventional tutor - with hyper-personalised digital tuition delivered at a truly affordable price.

About La Salle:

La Salle Education was founded in 2013 to support teachers of mathematics. The community of teachers using its services has grown rapidly, with thousands of teachers from around the world regularly attending La Salle events and receiving professional development from the company. La Salle supports teachers effectively and efficiently through its online teaching, learning, assessment and monitoring platform, Complete Mathematics. In 2020, La Salle launched trials of its ‘digital tutor’ product, attracting an immediate user base. The company is now taking the next step in helping pupils by using its intelligent technologies and comprehensive content to offer online tuition at a genuinely affordable price.

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