Continuity and Coherence: Building A Strong Mathematics Curriculum
Wednesday, 15 September 2021
The importance of a carefully and effectively structured curriculum cannot be understated: in schools with one, every lesson counts and both pupils and teachers can navigate through the maths world guided by the North Star of their curriculum intent. Without one, learners are stumbling in the dark trying to make connections between seemingly disparate ideas.
For many schools, the emphasis on curriculum under the new Ofsted inspection framework will have come as a welcome change, moving the focus away from exam results and towards the complete education of a pupil from the beginning to the end of their time in school.
Maths is in many ways unique when it comes to curriculum planning — there are no novels, case studies, or contextual investigations to act as structured stepping stones from concept to concept, but rather a sprawling and vast universe where tiny granules of knowledge are paradoxically both connected and separate to one another. In this lies the subject’s beauty, but also its challenge: how does one begin to chart their way through it all? In what order should concepts be approached so that pupils’ knowledge backpacks are never so full as to be overflowing, but never so empty that they are left without a vital piece of information at the point when it’s suddenly needed?
Such was the challenge we faced at Complete Maths when we set out to write our own curriculum, using the combined wisdom of our maths team and their decades of classroom experience, and of course our own founder and North Star, Mark McCourt. In this blog, we shall attempt to share some of our findings and their implications for school leaders.
Principles of effective curriculum design
Before diving into the finer details of curriculum design, schools need to consider the guiding principles against which they will work. Let’s start with those that we believe to be core to creating effective curriculums:
Use the National Curriculum / Curriculum for Excellence
In the buzz and excitement of replanning a curriculum, and imagining all the possibilities that come with it, we can’t forget that for most schools in the UK the starting point must be either the National Curriculum or the Curriculum for Excellence. There is no need to add unnecessary complications by attempting to rewrite what is already there — let the relevant curriculum act as foundation, and focus on how it can best be implemented in your school and your context.
Play the long game
At minimum, a strong curriculum thinks five years ahead — those working with younger pupils might be thinking in even longer time frames than that. We’ve spoken before about the links between sport and education, and once again when it comes to curriculum design we find much common ground. A typical Olympic cycle — by which I mean the structured period of training before an athlete peaks at an Olympic games — is four to eight years long. This means World Championships, even other Olympic games, will pass by without themselves being the point at which an athlete’s performance peaks. This involves an extraordinary amount of foresight and planning, but a team of people come together to do it because the pay-off of a gold medal is worth it.
We want our pupils, too, to perform at their absolute best on the days when it really counts — but actually, we want our curriculum to deliver even more than that. We also want our pupils to leave formal education and spend the rest of their lives knowing they can be successful in maths, applying their problem-solving skills whenever they can, embracing challenges, and relishing a puzzle. With that outcome in mind, we then work backwards and ask ourselves what the steps leading up to it look like.
Continuity is key here: as much as possible, a strong curriculum teaches ideas and methods for the first time in such a way that they do not need to be retaught in a different way further down the line. If a tiny granule of knowledge needs to be understood in a particular way for an eighteen year old to be successful, then it must be taught in that way from the moment it is first introduced, even if that moment occurs when a pupil is only five or six years old. An ambitious aim indeed, but one that will pay off.
It also means providing numerous opportunities for pupils to be successful, so they know what success feels like throughout their education and are motivated to keep going. If pupils are asked to move on before they are ready, then their rate of success — and, in turn, their motivation — will rapidly decrease. The strongest curriculums know and reflect this, as we’ll explore in more detail later on when considering Ofsted’s recent maths subject report.
Embed challenge at all levels
‘Challenge’ can sometimes be seen as the exclusive domain of high attainers — a constant struggle to stretch the most able, while allowing the rest of the class to progress at the ‘expected’ pace. Conversely, for those SEND pupils whose learning needs might mean they need more time and scaffolding to grasp ideas, challenge can simply mean ‘keeping up with everyone else’, and that in itself is considered enough.
A strong curriculum offers every pupil the chance to work at the boundary of their current ability, wherever that ability might lie. When I talk about embedding challenge here, what I really mean is differentiation, a more detailed definition of which can be found in Volume 3 of AskMark. It is important to note that a pupil’s boundary can vary hugely depending on the topic, and to assume that top set pupils will always require more challenging material than those in middle or bottom sets is to overlook the non-linear path learning often follows.
Many schools we’ve spoken to since launching CLASSROOM and, more recently, TUTOR, tell us that they use our readiness assessments to personalise learning as much as possible by pre-identifying the ability boundary for each pupil, and for each specific learning objective. Some even use these same assessments to re-set pupils. This isn’t possible in every context, but it is possible for every curriculum to allow a degree of flexibility. A subject lead who claims to be able to tell you exactly what topic a particular class will be studying in three months’ time has missed the point — for all their planning ahead, the best curriculums also have the ability to adapt to the learners following them.
Plan time to revisit concepts
The science here is indisputable — the best long-term learning happens when pupils have time to revisit knowledge and skills regularly, and in small chunks. If geometry is introduced at the beginning of one academic year, and subsequently not touched again until the beginning of the following academic year, then pupils will need to spend a significant amount of time re-learning core knowledge.
On the contrary, a curriculum that routinely loops back on itself to ‘check in’ on previous topics, either through revision quizzes or more formal assessment, sets pupils up for success by helping them move knowledge from their short-term, working memory into their long-term memory. As an added bonus, doing so can also allow pupils to make links between topics and remind them of the importance of core knowledge, like times tables, in even advanced maths.
Drivers among us will remember the feeling as a learner when changing gears or accelerating smoothly required so much active concentration that maintaining full awareness of what other cars were doing seemed a distant dream — the reason, of course, why driving instructors have their own brake pedal! Eventually these things become second nature, and we become safer, more effective drivers who can devote our attention to our surroundings. By the same logic, regular practice of core, and frequently-used, skills like multiplication helps to shift them into procedural memory, the equivalent of an experienced driver changing gears without a second thought. The upshot of this is pupils’ limited working memory is saved for the tougher, newer skills.
A curriculum that moves too fast, or attempts to cover too much at once, denies pupils the chance to make this conversion, and in the absence of our own teacher-version of the emergency brake pedal, pupils risk a crash.
Be ready to adapt
Although we’ve touched on this already, it nonetheless bears repeating. Andrew Jeffrey’s session at our July MathsConfMini used the perfect metaphor to describe this with his session titled ‘Year 3 and Smart Motorways’. A smart motorway, Andrew explains, identifies a problem up ahead and then slows down traffic further back. In this way, bottlenecks and lengthy tailbacks can be avoided and, although traffic is slowed earlier than needed, this ultimately means it travels faster in the long run. Andrew’s session explored how he applied this principle in the classroom by designing a three-week programme to address the Covid-induced learning gaps he predicted would cause problems for year three pupils further down the line.
A school could hire the best maths teachers and leaders in the world to design the most comprehensive maths curriculum ever written, but if on day one the pupils turned out not to know a chunk of what we thought they knew, and teachers pressed on regardless, the curriculum’s value would plummet. All of the principles above eventually boil down to this one: teach individual pupils what they need to know, in the right order, so that they can understand the next thing, and the next, and the next.
What do Ofsted say about effective maths curricula?
Ofsted’s subject report for maths was released at the end of the last academic year, and while it is not intended to be read as a list of things schools should be doing, it is a valuable resource as a summary of what the most successful schools are doing, offering up a wealth of good practice. When it comes to curriculum planning, there are clear common threads running through the report which might offer schools practical guidance.
The report repeatedly emphasises the importance of looking at the curriculum as a whole, advising that “[...] teachers need to prioritise ‘forward-facing’ knowledge. This goes beyond important facts of number. It includes the mathematical methods that pupils will take with them on their journey.” To implement this involves an element of information-gathering across all stages: the backward-planning discussed earlier, coupled with a consistent approach to teaching methods.
The report also explores the structure of the curriculum, particularly in terms of the core concepts taught at primary level. “In countries where pupils do well,” it observes, “pupils are able to attempt more advanced aspects of multiplication and division in Year 4 if they have been given more time on basic arithmetic in Year 1. This may explain why successful curriculum approaches tend to emphasise core knowledge early on.” It also advises against trying to cover too much too soon, adding that “A focus on core knowledge in younger year groups can be achieved by focusing on depth over breadth.” Pupils will be more successful later if they have mastered, early on, basic core concepts such as number bonds, times tables, and place value.
This also influences the structure of what comes next, with the report advising, “Strategies for solving problem types are then best taught and learned once pupils can recall and deploy facts and methods with speed and accuracy.” As discussed earlier with the learner driver analogy, our aim is to free up as much thinking space as possible for challenging new concepts and ideas, which isn’t possible for those pupils for whom the basics aren’t secure.
This focus on the basics has another benefit, as highlighted by the report: success breeds motivation, so when pupils have the chance to be successful early on with straightforward, but important, recall or fact-based tasks, they are instilled with the belief that they can be mathematical. The report goes so far as to link maths anxiety to a failure to grasp core concepts early on, the impact of which is often carried by a pupil for the rest of their education. Once again, consistency and cohesion across the whole curriculum, underpinned by a strong focus on basics at the very beginning, sets pupils up for success later on.
For pupils who arrive without these core concepts in place, or for those identified as at risk of falling behind, the report advises giving them more time rather than a separate curriculum — the latter risks pupils missing out on content, and in turn being locked out of the more challenging problems that would give them access to the highest grades. This impacts the way in which schools plan their intervention as part of their curriculum; in some cases it might even inform the number of hours of maths a pupil studies each week. “Curriculum progression,” it concludes, “is by intelligent design rather than by choice or chance.”
Building a strong curriculum means taking into account a huge range of pedagogical factors; the report emphasises that “Successful curriculums illustrate the importance of detail, sequencing and alignment of content, instruction, rehearsal, assessment and mechanisms to continually upgrade”. But perhaps the clearest takeaway for maths teams can be summarised by the following quote, taken from the section on curriculum progression: “Successful curriculum progression is planned from the beginning of a pupil’s education through focusing on core content, to develop pupils’ motivation and to allow more breadth and depth later.”
Planning for success
How, then, does a school begin to plan a curriculum like those described above?
To see long-term, sustainable changes at one end of the school, the seeds must be sown at the other end. This requires patience and forward-thinking: if year 11 results require improvement, then the work starts in year 7. If it’s possible to go even further back, as might be the case for all-through schools or those with close relationships between feeder primaries and secondaries, then even better.
It’s also important to hear from teachers at every stage. In secondaries, where teachers are likely to work across multiple year groups, this is much easier, but primaries too should create opportunities for the EYFS and year 6 / primary 7 teachers to swap notes. What topics continually cause problems? Where do pupils get stuck? What does assessment data suggest they struggle with? How will the answers to these questions influence the curriculum?
Time is the biggest barrier to these conversations, but if time can be found early on then the payoff further down the line will surely be worth it.
Under the microscope: the Complete Maths curriculum
We know our maths curriculum isn’t the only one out there, but we call ourselves Complete Mathematics for a reason: ours is the most comprehensive. The prerequisite links add a unique level of extra detail which you won’t find anywhere else, and which we believe facilitates the kind of forward-planning that both Ofsted and our own combined experience in school tells us works.
Schools often approach us wanting to map their own curriculum against ours — and schools following the White Rose schemes of work, for instance, will find these already available on CLASSROOM and ready to use. Some schools create their own entirely bespoke schemes of work mapped against our curriculum points, either independently or with help from our support team. We know that in many cases, hours of work have already gone into planning a school’s own curriculum, and we want schools to be able to combine this with our platform in the way they see fit.
That said, whenever possible we encourage schools to follow our curriculum. Put simply, this is because all of the principles described earlier have gone into planning it, and we believe it works. Our curriculum spans early number sense all the way to further maths, so we’ve been able to make the links between and across ideas that are so rarely possible when teaching within one of either primary or secondary phases.
Our curriculum isn’t labelled according to year group, but rather according to stage; this is a deliberate decision which reflects our approach to mastery and our belief that all pupils should be working at the boundary of their own individual learning, and not at that prescribed by their age. Our readiness quizzes, and the detailed pedagogical notes, accompanying each objective, are there to make it easier for teachers to identify that boundary and to start classes at an appropriate point.
St Peter’s RC Primary School in Wales is a great example of a school switching to the Complete Maths curriculum to help embed a mastery approach, with pupils not moving onto a new topic until they have scored 80% on the current one. Following our curriculum also meant teachers were able to channel more time into other areas of their practice, and the school has subsequently been moved out of Special Measures — you can read more about how they have done this on the case study section of our website.
For us, the how is just as important as the what. Our instructional videos and teaching notes frequently refer to manipulatives, metaphor and analogies to help make challenging concepts more concrete for pupils to understand. Mark McCourt explores this in detail in one of his own blog posts, Models, Metaphors, Example and Instruction, and you’ll see the approaches he discusses echoed across our own resources, CPD and videos.
Planning for the Future