Thursday, 05 May 2022
Our TUTOR platform provides an affordable alternative to 1-1 tuition, offering pupils access to 1800 individual maths lessons from counting to calculus. At present, TUTOR is only available through schools; you should contact your child’s school directly if you are interested in accessing the platform straightaway. Alternatively, click here to be added to the mailing list and notified when TUTOR for Families is released, allowing you to purchase access to TUTOR directly.
Anxiety in the run up to a big exam is completely normal, and a feeling all of us will be familiar with. But for some children, exam anxiety can start to impact their day-to-day lives and ultimately prevent them performing at their best on the day.
In the final part of our series on supporting your child through exams, we’ll look at ways of coping with anxiety not just in the run up to exams, but any time they are confronted with a stressful situation.
Understanding where exam anxiety comes from
Understanding why we react with anxiety to exams, and other similar events, is an important first step to managing that reaction. The part of our brain that controls our fear response, the amygdala, hasn’t changed much since our days roaming the plains avoiding sabre-toothed cats. The reasoning parts of the brain know that, logically, an exam doesn’t pose a threat to life — instead, the threat is failure, or at least not meeting your potential.
However, the nuance of this message is lost by the time it reaches the amygdala; instead, it hears ‘DANGER’ and reacts accordingly, without knowing the severity of the threat. The amygdala then does everything it can to save your life: speeds up your heartbeat so your limbs are primed and ready to run; makes your breathing shallow so a potential predator can’t see you hiding; prepares you to fight, flooding you with adrenaline, or to freeze until the threat has passed.
These responses would be incredibly helpful if you really were facing a physical threat to life, but when actually all you need to be able to do is calmly focus on one task, your brain is in some ways working against you. There are two challenges we face when trying to deal with anxiety: the first is how to manage the fear response once it’s kicked in, and the second is how to prevent it kicking in to begin with. Let’s look at how.
Getting rid of the fear response
Firstly, don’t be angry at your brain for reacting the way it has — it’s trying to keep you safe. The next steps are all about convincing it that everything is okay.
The common advice to take deep breaths is common for a reason: it’s the first step to telling your body that it can relax. In doing so, you are telling your brain ‘I don’t need to hide and be still anymore, the danger has passed.’
If your child is upset or anxious, have them take a deep breath in through their nose, counting slowly to five, hold it for a second, and then let the breath slowly out through their mouth, counting slowly to five. After several breaths in and out, start to focus on relaxing different parts of the body: let the shoulders drop, the forehead relax, and the feet rest gently on the floor.
There are many guided meditations and breathing exercises available for free on YouTube, or through the paid app Headspace — encourage your child to put one on and spend a few minutes every day thinking only about their breathing and relaxing their whole body so that, when it is needed, they are practised in controlling their fear response.
Preventing the fear response
If at all possible, ask your child’s teacher if they can spend some time in the room where the exam will take place practising their breathing techniques. This will help them build positive, calming associations with that space. They might even sit with a teacher, talking through the exam routine and what will happen when they first enter the room. This may not always be possible — but the good news is visualisation strategies can be just as useful. An NLP [Neuro Linguistic Programming] technique known as the Circle of Excellence can be particularly effective for exams, but also works well when preparing for big presentations, competitions, or any other situation where you need to perform at your best under pressure. You can read more about the technique here, but the basic steps are outlined below.
Start by standing up, and imagining a circle on the floor in front of you. The circle could be made of light, or could be made of fire, depending on the state of mind you want to get into.
As you look at the circle, think again about the state of mind you want to get into: you might wish to feel calm and in control, or powerful and strong. Imagine a time when you felt that way — if you are doing this with your child, have them describe to you a situation or time when they felt that way. Alternatively, they might think of someone they associate with that state of mind: a friend, a fictional character, or a famous person they admire.
Try to recreate in your mind, or have your child describe to you, what it’s like to be in that state of mind: how do you stand? What emotions are you feeling? What are you thinking? What are you doing with your arms and legs? Then step forward into the circle on the ground, taking that state of mind with you.
Stand in the circle for as long as you can maintain the feeling of strength, calmness, or power that you created in step 3. When you feel it start to fade, step back out of the circle.
Now imagine that the circle is shrinking, until it is small enough to pick up. Imagine you want to carry it with you — maybe you’ll put it into your pocket, or clip it round your neck or wrist like a watch or a piece of jewellery. Carry it with you until you need it again, and then go back to step 1.
Avoiding fear and anxiety from taking over in the first place is about changing your brain’s response to an event — when it comes to exams, where there is little opportunity to practice ahead of the real thing, visualisation is key.
Above all, keep reminding your child that their brain is on their side and is trying to keep them safe; understanding why a fear-based response happens is the first step to challenging how their brain reacts. Breathing and visualisation techniques allow us to use the logical, thinking part of our brains to alter the response of the instinctive, animal reaction, and are tried-and-tested techniques used by high performers the world over. And as with anything, the more they practise, the better they will get at maintaining calm in the face of stress.
Monday, 25 April 2022
We put all of the latest learning theory into our TUTOR platform, designed to offer an affordable alternative to 1-1 tuition. This three-part series for parents seeks to share some of these ideas behind learning and revision, alongside practical strategies parents and carers can use at home to help ensure their child enters the exam hall feeling as calm and prepared as possible.
Whilst the strategies explored in the other blogs in this series mostly focus on ‘neurotypical’ revision strategies, if your child is struggling with ‘typical’ revision methods, why not give these a go, whether or not they have been diagnosed with a ‘neurodivergent’ condition such as ASC, ADHD, Dyslexia, OCD, Tourette’s Syndrome, Anxiety and Depression etc.
The variety of ways our brains work is called ‘neurodiversity’. Whilst many children struggle with time management, focusing on topics that aren’t of special interest, and anxiety when out of routine, some will have (diagnosed or undiagnosed) conditions which can be described as ‘neurodivergent’, as opposed to ‘neurotypical’. For more information, this is a helpful article.
Disclaimer: this article does not present an exhaustive list for each condition that might be classed as neurodivergent, and is not an alternative to diagnosis and medical advice. It is intended to support young people, diagnosed or otherwise, with some alternative revision strategies. Since many ND individuals present with more than one condition and lots of strategies overlap, these are some strategies to try alongside the methods suggested in the first part of this series. Specific conditions have only been specified where this adds clarity.
Neurodivergent individuals are known for being creative thinkers, a trait which is starting to be recognised by employers as enriching to their organisations. This article explains this in relation to dyslexia and STEM careers.
Visuals: Drawing pictures, creating mind maps and making models are visual ways of helping to remember information. Using visual representations and physical objects to ‘manipulate’ in teaching maths is becoming much more common and supports neurodivergent and neurotypical learners. TUTOR tuition videos use this approach. For a free Times Tables resource with this approach, click here.
Recording voice/video: Recording voice notes or making videos of themselves could be a fun and effective revision technique for children, as long as they are actually using this to revise! ‘Teaching’ others helps them to review and retrieve the information or skill they will need in the exam, and making a video means they can then watch it back again. People with dyslexia and dysgraphia are particularly likely to benefit from this approach, due to difficulties with writing.
Use their Imagination: Your child might invent stories that help to remember a sequence of events or use mnemonics (eg remembering the order of rainbow colours using Richard Of York…). They might turn the revision itself into a game - answering 10 questions in 10 minutes and they then get a small ‘treat’...
Rewards: The previous blog in this series discussed rewarding your child for their effort and this is absolutely relevant here. Particularly if your child has difficulty with attention regulation, reward based on what is achievable for them, rather than what you think they should be doing. They may well be really trying to focus on their work even if it doesn’t look like they’ve done anything - asking them to rate their effort can help with understanding how your child is feeling and help them with self-awareness and being kind to themselves on difficult days.
Focussing on Tasks
People with ND conditions often get labelled as ‘lazy’, even though they might be really trying to focus. There are some things which may help to regulate attention: urgency, interest, challenge and novelty. Keeping these in mind, hopefully you’ll have found some strategies for ‘interest’ and ‘challenge’ above, and changing up revision strategies or subjects can bring ‘novelty’ in too. The next two suggestions tackle ‘urgency’:
Timers: Creating a sense of urgency with an achievable goal/deadline can really help ND-brains to focus on a task. One example of this is the pomodoro technique based on working for 25 minutes and then having a 5 minute break. However, if this is too long for your child, alter the timings — for instance, a 10 minute session with a 3 minute break.
Alternatively, try another trick similar to one suggested in our previous blog, adapted for ND brains that struggle to start tasks — set a timer for 5 minutes, and your child needs to only work this time. You may well find that when the timer is up, they have just got stuck in and want to carry on… let them (within reason)!
Body-doubling: Do you feel like your child only focuses when you are in the room? They might benefit from a technique called ‘body-doubling’. The accountability of having someone else in the room focused on a task (it doesn’t need to be the same task — you could be reading a book or cooking dinner) can help to create urgency and regulate attention. If you are not able to be physically present, this can also work virtually - depending on your child’s age and their friends, they could ‘body-double’ over video call, or with a grandparent, even if they live too far to visit. If none of these are viable options, youtube has many ‘study with me’ videos that follow the pomodoro technique and feel like you are on a call with someone.
Music: Whilst many people find listening to music relaxing, people with autism and/or anxiety may find additional benefit from music in regulating or expressing their emotions. Moreover, people with ADHD, who we described at the start of this section as needing novelty, interest, challenge and urgency, might benefit from the ‘interest’ music promotes whilst they work on a task they would not otherwise be interested in. As we will see in the next section, having a dance break may help too…
Movement helps with focus and reducing anxiety in many ND-conditions.
‘Stimming’ - repetitive movements such as flicking fingers, jiggling feet, twirling/jumping, pulling hair and stroking or rearranging objects, where not harmful to the individual, can provide relief from stressful or overstimulating environments.
Breaks: Taking frequent breaks to move can help ND brains with reducing stress and releasing dopamine which is needed for regulating attention and, in individuals with ADHD, is lacking. Why not combine this with the pomodoro technique above!
If your child is undiagnosed/not entitled to breaks or a separate room for exams, you might look for ways your child can move that supports them to focus on their exam.
Stimulation: Bouncing their leg up and down and fiddling with blu tack or a rubber band are ways your child can move whilst in an exam hall without distracting others.
Counting numbers of e.g. ceiling tiles or seats in a row can also help to reduce anxiety by distracting from the stressful environment.
For sensory stimulation, stroking a small section of fabric (look for offcuts at your local fabric shop) or a small sensory toy that fits discreetly in a pocket can reduce anxiety, particularly in people with autism.
If your child has a diagnosed condition, it is not unreasonable to ask for accommodations in line with their normal way of working; in fact ‘reasonable adjustments’ is a requirement by law (Equality Act, 2010). However, it is important to apply for this early as schools need time to process this - if your child is in year 5, 10 or 12 and will be sitting exams next year, you should ask about this in your next SEND support meeting with your child’s school.
Reasonable adjustments for exams can range from being allowed a fidget/sensory toy, to assistive software or a reader/scribe/prompt, to having extra time and a smaller room. Familiarising themselves with what the exams will be like can help too - knowing which room the exam will take place in, how long each exam is, whether your child will be allowed movement breaks during the exam etc. Your child may well benefit from talking and/or walking through what they can expect in the exam, including changes such as using a transparent pencil case rather than their usual one for GCSE/A-level exams.
Many ND people have difficulty with time management and organisation, and changes to their usual routines. Keeping as much as possible the same within this revision period will help - this is not a good time to try out new foods if your child finds this difficult, for example. You might also find that even activities they enjoy might trigger meltdowns because of the additional exam stress.
Being flexible and allowing your child to choose not to do certain activities at this time might be just what they need to cope with a period which may be overwhelming for them, and help to develop their independence. At first, your child may need more support deciding what they should do in their revision session, although presenting them with a couple of choices still gives them agency over their revision. As they learn what helps them, they should find it easier to know what they need to do and which techniques help them to focus and achieve their goals.
Many parents of children with and without ND-conditions will be feeling frustrated by exam season - but it's important to remember that it takes time to develop these skills. You are not alone.
At present, TUTOR is only available through schools; you should contact your child’s school directly if you are interested in accessing the platform straightaway. Alternatively, click here to be added to the mailing list and notified when TUTOR for Families is released, allowing you to purchase access to TUTOR directly.
Thursday, 21 April 2022
We put all of the latest learning theory into our TUTOR platform, designed to offer an affordable alternative to 1-1 tuition. This three-part series for parents seeks to share some of these ideas behind learning and revision, alongside practical strategies parents and carers can use at home to help ensure their child enters the exam hall feeling as calm and prepared as possible.
Revision for exams will dominate most lessons at this time of the year, and teachers up and down the country will be telling their pupils to make sure they are revising at home too. But what does successful revision look like, and how can parents and carers help their children in revising effectively? Read on for our eight top tips for supporting your child in developing effective revision strategies.
Make a revision timetable
It can be tempting to spend two hours designing a beautiful, colourful revision timetable that is then never looked at again — but a truly useful revision timetable is less about appearance and more about the timing of topics.
It goes without saying that revision time should be spread across subjects, with more time spent on those subjects your child finds the hardest. But research also tells us that the time between topics is just as important as the time spent on them. If, after learning something new, your child looks at the content again the next day, and again two days later, and again a week later, they will remember significantly more than if they waited a week to revisit it. A well thought-out revision timetable should include short time slots for revisiting, in brief, previous learning to make sure it sticks.
Practise recalling information without notes
Another revision technique often favoured by children is reading over notes or textbooks, highlighting chunks of material and then copying out notes — but while it can feel like they are working hard, this is not an effective way of remembering information.
Every revision session should include a test on the material without access to notes — this is what teachers call ‘retrieval practice’, and it’s all about firing up your memory by forcing you to find, or ‘retrieve’, old material.
Use two colours of pen to support retrieval practice
Have your child turn their notes over and then, in one colour of pen, write down or mindmap everything they can remember about a topic within a time limit (say, five minutes). After the time is up, have them turn their notes over and use a second colour of pen to add in everything they’ve missed. Finally, have them repeat the first part of the task, recalling everything they can without access to notes. They should see that, on the second time around, they are able to remember much more.
As a follow-up, have your child look over all the material they wrote down using the second pen colour — this represents all the knowledge they are not yet confident in. This is the material they should add to their next revision timetable slot for that subject, using the timings identified in tip 1.
Go phone-free for part of the day
Every time you ask your brain to switch between tasks, there is a short drop in efficiency - a bit like a relay team, whose 4 x 100m time will always be slower than the sum of their individual 100m times, because of the time lost in passing the baton. This is known as ‘context switching’, and the many distractions of the internet have made it a huge problem for anyone trying to focus on one thing.
Every time your child is distracted by a new message, or a video, their brain is being asked to context switch, and this will slow them down. Phones and other internet-enabled devices can be incredibly valuable learning tools, but as soon as your child needs to focus on offline tasks like answering practice questions or completing the retrieval task in tip 3, encourage them to go phone-free. This only needs to be for short periods of time, but it will be a huge help in their ability to focus.
Make sure your child is getting plenty of sleep
While we are sleeping, our brains decide what information needs storing in the long-term memory and what can be discarded. Studies have even shown that a good night’s sleep can improve memory retention by as much as 40% — you can read more about this in Matthew Walker’s excellent book ‘Why We Sleep’. For older children in particular, the temptation to stay up late revising is huge — but this will actually be preventing your child from learning as effectively as they could. As much as possible, children should be switching off devices at least an hour before bedtime and prioritising a full night of uninterrupted sleep.
Mix questions from different topics
When first starting to revise, keeping topics separate is helpful for improving recall and confidence. However, exams will demand children to move swiftly between topics, choosing the appropriate method and approach each time. It is important that children practise this ahead of time — this could mean working through sample exam papers, but it could be as simple as writing a range of questions and answers onto flashcards, and having a friend or family member test them on different topics at random.
Break revision down into small chunks
Revision doesn’t have to mean hours sitting at a desk surrounded by books — it can be as simple as pulling a set of flashcards out of your pocket while waiting for a bus, or using the ad break of a TV show to answer a practice question, before checking it in the next ad break. For many children, simply sitting down to start feels like a huge task, so a little-and-often approach can go a long way. Encourage your child to set a timer if they are struggling to get going — and stop when the timer runs out, even if they feel like they can keep going. The more revision sessions end with the feeling that it wasn’t so bad and they could have carried on, the less put off they’ll be at the thought of starting again the next day.
Remember that success breeds motivation, and not the other way around
We like to think that the most successful people are also the most motivated, but this isn’t really the case: motivation comes from feeling that you can be successful, and the most successful people are those who can keep going even when they aren’t motivated.
If your child has plenty of opportunities to feel successful, even if that success is spending five minutes on a single task without looking at their phone, then they are more likely to keep going. Conversely, long and late-running revision sessions where they are too tired to retain information and frustrated from what they perceive to be a lack of progress is only going to add to their anxiety and stress.
Let your child teach you things that they know and you don’t; ask them questions you know they know the answer to; challenge them to work for slightly less time than you believe they can manage; celebrate every completed task or extra mark gained. The exam is of course important, but it’s more important for your child to believe in their own potential — this will take them further than any grade.
We’ve designed TUTOR with all of the tips above taken into account:
- Short, 3-5 minute videos designed to chunk learning
- Quizzes focused on single topics, followed by quizzes mixing a range of topics
- Regular opportunities for retrieval practice in every Goal
- XP points earned for every interaction in the platform
- An Awards panel celebrating your child’s success
- Parent Accounts so you can support your child’s progress
At present, TUTOR is only available through schools; you should contact your child’s school directly if you are interested in accessing the platform straightaway. Alternatively, click here to be added to the mailing list and notified when TUTOR for Families is released, allowing you to purchase access to TUTOR directly.
Wednesday, 15 September 2021
As a relative newcomer to the Complete Mathematics team, I’d watched the build-up to MathsConf27 from afar (well, the other side of the office) for several weeks and experienced the virtual MathsConfMini in September. I was therefore very aware of the fact that for Complete Mathematics, MathsConfs are an organisational Everest. From my interaction with MathsConfMini, I also knew just how much the team invested in making sure that the workshops were extremely high quality and ran as seamlessly as possible. With a 10-year background in teaching myself, I was hugely impressed by their dedication to creating a collegiate, accessible and massively comprehensive CPD event for maths teachers across the country.
However, whilst I had all of the theory, what would it be like in practice?
Well — in practice, it started with me behind the wheel at 6.45am on a Saturday morning. As I peered blearily ahead, I thought of all the places I knew other MathsConf27 attendees were travelling from and felt immediately abashed – if they could take on the journey from as far afield as Brussels and Birmingham, then I could manage the Blackwall Tunnel.
Not long after arriving, the atmosphere began to take on a real buzz. Everybody, it seemed, felt excited to be doing this again in-person. Especially given that in-person, among other things, meant proceeding with what I like to refer to as the ultimate ‘comforting conference trio’: a real, paper copy of the day’s events in one hand, a croissant in the other, and a Complete Mathematics tote bag over your shoulder.You don’t get this on Zoom, I thought to myself, as I surreptitiously sipped my second coffee in ten minutes (well, it was an early start).
Your adrenaline levels as a first-timer are very much dictated by whether or not you are also presenting a workshop, but either way, the Complete Mathematics team are there to greet and guide you through the event from the very beginning. Your welcome pack includes a comprehensive programme of events for the day, with a map of the venue, to ensure you can navigate your way around as smoothly as possible. Be sure to visit the refreshments before embarking on the day, in order to refuel from your journey and chat to like-minded attendees. It’s also very worth taking a moment to speak to the team from AQA, and take advantage of the opportunity to ask any of those exam board related questions you might have. Also, don’t forget to visit the wide range of exhibitors in order to see what they have on offer, as well as collect your clues for the Treasure Hunt! If you are presenting, then someone from the team will take you to your room, and make sure it is set up as you would like, so you feel as comfortable as possible in advance of your workshop. You will also get a chance to do all of the above during the lunch break – they’re there for you all day!
In his opening presentation, which provided invaluable guidance on the current exams landscape, Andrew Taylor from AQA acknowledged the collective feeling by kindly noting that he felt slightly ‘nervous’ due to the amount of ‘real people in a real room’. From the morning’s interactions, I knew that this feeling was also shared by some of the workshop leaders, whether this was their first time at MathsConf or not. I felt it myself too: simply not knowing exactly what to expect, and not having the facility to turn the camera off on a Zoom call whilst I collected myself, was enough to get the adrenaline going.
Subsequently, Mark McCourt’s rousing presentation further added to the sense of unity in the room, as he reminded everyone about one of the key facets of the day: the ability to talk and network with each other, and the power that could come about as a result of the ongoing communication and support between the 350,000 maths teachers nationwide. Mark also introduced the newest offering from Complete Mathematics — TUTOR — to rapturous applause, sparking the catalyst for a huge amount of interest throughout the rest of the day in a platform — and movement — that we are hugely excited about.
As a member of the Complete Mathematics team, I’d been privy to the growing menu of workshops over the preceding weeks and had continued to be impressed by the sheer variety of choices on offer. There were offerings for maths teachers of every level of experience, from niche interests to wide-reaching pedagogical concepts. I’d been assigned to a particular room for the morning’s sessions, and was excited that the schedule would include a combination of ideas with which I was both familiar and unfamiliar: a great mix.
Mathematical Minds Beyond the Curriculum
First up was Jennifer Obaditch from AQA with her workshop on the EPQ. Having been an EPQ coordinator during my time as a teacher, and found it exciting but challenging at times, I was looking forward to Jennifer’s insight. She reminded us of the intrinsic value of the EPQ to a KS5 pupil – one who is currently studying in a climate of very content-rich A Levels, assessed at the end of two-year period – who won’t have necessarily amassed the wider study/interview/research skills they will need to flourish at university.
Furthermore, and even more notably, was the way in which Jennifer shone a light on a reality of the EPQ that I had certainly experienced myself (and almost definitely not questioned as much as I should have done) regarding the strong Humanities/English drive and frame of reference that often either delivers the skills, or surrounds the pupil projects. It was fascinating to be shown a whole host of EPQ project titles that had come from a mathematical perspective, and understand the ways in which the non-essay, ‘artefact’ option could be the perfect outcome for a pupil wanting to apply their mathematical knowledge to a more practical project. Having invested in lessons fairly recently, I for one would love to know exactly ‘how dimples affect the aerodynamic drag of a golf ball’— they just don’t teach you that at the driving range! Those in the room were certainly inspired by the prospect that EPQ projects could be opportunities for pupils with a strong interest in maths to explore it in an exciting, self-led, ‘real world’ way, beyond the confines of the classroom.
Teaching with Technology
Depending on when you trained, and your own personal philosophies on such controversial matters as: ‘Real Books vs Kindles’ and ‘Filofaxes vs The Calendar App on your Smartphone’, your feelings about incorporating technology into your teaching may be hugely positive, negative, or somewhere in between. No doubt over the past few years you have had an experience with technology that may well have been, shall we say, less than favourable.
However, if the past couple of years have taught us anything, it’s that now is the time to embrace what technology can do for teaching and learning like never before. For me, this workshop was the one where I felt on the backfoot from the start — not quite a filofax aficionado, but certainly someone who prefers a real book over a kindle. No fear though, as it was being run by the inimitable duo: Douglas Butler and Mark Hatsell (one of our very own developers at Complete Mathematics). Having met by chance at their local sailing club, they guided me through the rough seas of my own ignorance expertly, as they demonstrated the possibilities of their quite frankly genius creation: Autograph.
With their first Windows version released in 2001, they were celebrating the 20th anniversary of what has become the most popular and widely used dynamic maths application in UK schools. Having grown from 2D graphing and coordinate geometry, Douglas and Mark have worked on adding more features, and Autograph now includes Statistics, 3D graphing/coordinate geometry and Complex Numbers. Their session walked us through how to use the variety of features to create not only visual, but seemingly ‘alive’ representations of the above concepts. The workshop was an invaluable opportunity to play around with the features and understand more about how this (free!) resource could be so easily slotted into your teaching practice, to the huge benefit of your pupils’ understanding. Using Autograph is such a quick technological win — you can quite literally mesmerise your pupils with maths!
Learning from the Literature
At this point, given the new subheading and my heavy reliance on alliteration throughout this blog, it seems right to reveal that whilst I’ve been a teacher and head of department for 10 years, my subject specialism is English — not maths. Imposter alert! I know what you’re thinking — no wonder she was clueless in the Autograph workshop.
Therefore, when it came to Rhiannon Rainbow and Peter Mattock’s workshop on running a pedagogical book club, I felt I was sailing on slightly calmer waters. Rhiannon explained the (extremely appealing) format of the events: open to all, with manageable extracts from a wide variety of the latest educational research, many of which come with an audiobook for ultimate ease of access. With a member each time responsible for writing up the ‘takeaway’ from each session (or illustrating it beautifully!) Rhiannon explained how the book club took a truly inclusive and accessible approach — helpful for teachers for whom confidence with educational literary texts might be on the lower side.
In recent times, the book club had gained such ground that the authors of the texts themselves had also attended — adding an extra depth and dimension to the conversation. Indeed, Peter Mattock’s initiation of a discussion about his recent book, Visible Maths, created a discernible buzz in the room, and his past comments on the reciprocity of the feedback process summed up the mutual benefits of the book club, as he described how it was ‘priceless as an author to hear real-time feedback on the content.’
What I didn’t expect however, was for Rhiannon to drop in the name of a writer I was very familiar with, from my own teacher training some 11 years ago — David Didau. David is a key figure in the world of English teaching, and I had spent many a seminar and meeting discussing his ideas about pedagogy and teaching approaches in English over the last decade. This is where this workshop became really transformative. As Rhiannon extolled and evidenced the benefits of branching out beyond just maths research, a small seedling that had taken root in my mind since starting at Complete Mathematics began to sprout leaves: the untapped value of two core subjects sharing pedagogical approaches and insights was potentially huge.
A Connected and Cohesive Curriculum
Having been given the permission to be my authentic English teacher-self by Rhiannon, and by a passing comment from an innocent bystander who claimed that I ‘seemed like an English teacher’ (what does that even mean?!) I was intrigued as to what the afternoon’s workshops would bring, and whether they would enhance the sparks of synthesis between two supposedly binary subjects that had begun to form in my mind.
Jo Morgan’s session after lunch, ‘Curriculum Sequencing’, constituted the perfect continuation, as she explicitly compared the English and Maths curriculums in terms of the sense of choice (or lack thereof) when it came to choosing the content to actually teach. That in itself was a thought that had never occurred to me, and one which when broken down by Jo, I found supremely useful — to the extent that a sort of Venn diagram started to form in my mind, where I realised that despite blatantly obvious differences, there were also some strong opportunities where the subjects overlapped.
With less choice on content in maths, Jo focused on what maths teachers could control: sequencing, pedagogy, resources, methods and manipulatives. For me, the focus on sequencing and ordering the curriculum really resonated; the importance of creating a perfectly balanced, interconnected jigsaw, which focused on prerequisite skills and seamless building of knowledge over time. From my short time at Complete Mathematics, I knew this was a fundamental foundation of their products, and had been built directly into both CLASSROOM and the recently launched TUTOR. However, I hadn’t really considered it properly in relation to my own subject area, and made such an explicit connection between the higher order principles of curriculum design that would benefit pupils hugely in both English and maths — two core subjects that we know can greatly impact a child’s options and prospects later on in life.
There’s Narrative in the Numbers
With my mind awhirl, I was looking forward to Mark McCourt’s workshop, ‘Limitless’, where he would build on what he’d touched upon earlier in the day: the premise that all pupils could progress and succeed in mathematics, given equal access to teaching practices which focus on mastery and consistency, and not low expectations or the ‘conveyor belt’ approach.
Having seen these values in practice, in the fabric of the company and design of the products, I found this portion of Mark’s presentation truly — sorry in advance Mark — inspirational. What came next however, set off fireworks in my brain.
Mark was explaining a teaching point, and discussing scenarios around taking a mastery approach to quizzing pupils on their learning, when I heard two words that were part of my everyday professional vernacular: metaphor and narrative. In role as a teacher who hadn’t managed to communicate the maths learning point to its fullest extent, Mark said: ‘I haven’t found the story for them to receive this meaning.’This is why he calls it ‘mathematical literacy’, I thought to myself incredulously, as I heard English teachers all around the country stand and applaud. Was this the epitome of the overlap between the subjects? Was this why in essence, good maths pedagogical theory could help influence good English practice, and vice versa? For many years I’ve been able to see the maths in English —the structure and sequences of sentences, paragraphs and whole stories are in themselves, a beautiful kind of equation. But English in maths, on an overarching, pedagogically-principled level, was really new to me.
As the day drew to a close — I reflected on what I had learnt, like any good pupil. Given my subject specialism, I’d had my eye on Jenny Hill-Parker’s Literacy in Mathematics workshop for some weeks (sorry to have missed you, Jenny!) but in the end was assigned to supervise workshops. What I realised was that it was no fortuitous coincidence that I, as a non-maths specialist, had not only improved my knowledge of maths pedagogical content, but that actually, so much of what I had experienced during the day bore close relation to, and could certainly enhance, my experience of being a practitioner in what is generally considered to be a subject which is totally at odds with maths. In school, we often barely have time to deep dive with our own departments, let alone others. We all know the benefits of cross-curricular collaboration — but what about cross-core collaboration?
Whether you’re a newbie like me, or a seasoned pro, join us for our next MathsConf, and take your pick from an amazing array of top quality maths CPD! Dates for your diaries (or filofaxes) are: Saturday 12th March 2022 for #MathsConf28, our next in-person conference! See you there!
Buy your tickets here: completemaths.com/mathsconf/28
• Mark McCourt @EmathsUK
• Rhiannon Rainbow @Noni_Rainbow
• Jenny Hill-Parker @JennyHillParker
Tuesday, 26 October 2021
For all the thousands of hours children spend in school or college each year, what actually happens when they’re there can seem a mystery to those waiting at home. Parents and teachers have few opportunities within the year to speak, and when they do, it’s on the clock. Parents are often left to rely on their children for a full explanation of what they’re learning at school or college and how it’s going, but this is of course only part of the story.
Maths is one of the few subjects in which the core content of what’s being taught hardly varies — every generation will recall learning times tables, or converting fractions to percentages. But how it’s taught varies hugely, and if you’re a parent without a maths teaching background wanting to know how to support your child, it’s hard to know where to start. That’s why we asked our maths team, all of whom are experienced qualified maths teachers across a range of ages and contexts, for their advice.
Read on for the ten things they’d like parents to know about learning maths:
It’s okay to count on fingers!
We often associate using fingers as a negative sign that a child isn’t secure in counting, but in fact, it is thought to reduce what’s known as ‘cognitive load’. Cognitive load is comparable to tabs on a computer — the more tabs are open at any one time, the slower the computer is likely to run until eventually it crashes. Each new task or challenge is like opening a new tab on the computer. When introducing a new skill or a tougher problem, teachers will always aim to reduce pupils’ cognitive load as much as possible so that they can focus on just one thing. Counting on fingers means pupils’ brains have one less thing to think about, therefore freeing up precious space for the challenge in front of them.
Finger counting can also make it easier to perceive quantities without having to count them. This is known as subitising, and it’s something most of us do without thinking — recall the last time, for instance, you rolled a dice. If you knew the number you’d rolled without having to count each individual spot, you were subitising.
Of course, finger counting has its limitations and as children develop their understanding of mathematics, we want them to move to more efficient methods.
You can read more about the role of finger counting, and how to move children to other methods, here.
The methods teachers use might be different to the ones you learned in school, but that’s okay!
It might sometimes seem like the methods taught in maths class "nowadays" bear little or no resemblance to what you remember seeing as a child. Whilst it is true that mathematics teachers may decide to teach a variety of methods, often this is done to help with understanding or to provide a method that will work across a range of problems. It is also quite often the case that the methods will be superficially different but are really the same procedure.
If you are helping your child with a problem, ask them to explain the method their teacher has shown them. Having them explain the method to you is a great way to boost their own understanding. If they are unable to explain to you then their teacher needs to know that they haven't understood what is expected of them. Try to resist the urge to have your child do the question using "your" method. Most teachers would prefer a little note next to the homework saying "Jonny didn't understand what to do here" so they can address it next lesson.
Tricks and shortcuts sometimes do more harm than good
As tempting as it can be to teach your child a trick that will get them 'the right answer', this can be damaging to their mathematical development, ignoring the need for a greater depth of understanding in favour of the correct answer. There is a false economy in teaching and using these tricks, as further down the line the prerequisite idea isn't understood fully and building on this knowledge isn't possible.
Nix the Tricks is an American book which looks at avoiding these shortcuts that limit mathematical development — it’s free to download, and offers alternative explanations to some of the tricks you might yourself have learned while in education.
Find the mathematics around you
As students become more mathematically able, the opportunities for finding the maths around you become more limited. Many of the ideas students work with as they approach end-of-school examinations don't scream 'real life', but it is important that children discuss ideas that they've been working with.
Very young children find great pleasure in counting the number of steps they're climbing, counting down from five before they 'blast off' and love to play 'shops', setting up a toy shop, taking money from you and giving you your change before bagging up the toy and wishing you a good day, As they become more mathematically literate, their focus might shift to the idea of time and knowing how long it is until dinner time, or how long they have left on their scooter, and they can be engaged here.
As they continue to grow, and their mathematical powers heighten, the opportunities might seem to dry up, but estimating the food bill whilst doing the weekly shop is an excellent way to engage a child with maths and you can sweeten the deal with incentives ("If you're within £2 of the total, I'll get you a sausage roll!"). The opportunities may not be there when students are mastering trickier concepts at GCSE and A-level, but just asking "What have you been looking at in maths today?" and "Tell me a little about it..." will go a long way to provide opportunities for retrieval of these ideas in a very casual manner.
Parents of young children might enjoy seeing below some of the maths Jonny Hall, one of our maths team, uses at home with his young son.
Anyone can have success with school mathematics
Whilst some children are able to understand school or college mathematics more easily and quickly than others, that doesn't mean that all children can't achieve the same level of understanding given sufficient time and support.
Many schools and colleges will set pupils based on prior attainment (how they have done in tests). This is often taken as a measure of a pupil's potential, but this should not be the case. Sets are a way of placing pupils in a class where they will get a level of support (and challenge) best suited to their current level of mathematical development. Being in a lower set should not mean that expectations are lower and that pupils can not still go on to achieve at the highest levels of school maths.
In most cases, success with school maths comes from hard work and from the belief that hard work will see pupils improve. Intelligence is not fixed. We are all capable of accruing knowledge and understanding complex ideas if we are determined and we work hard at it.
Being quick at maths isn’t the same as being good at it
It is commonly assumed that being "good" at mathematics means being fast, solving problems before anyone else or finishing first on tests. This is not the case, and praising speed can have lasting negative effects.
Early in their mathematical development, many pupils experience short, single-step problems. Often, these problems can be answered quickly but we should try not to praise speed and instead praise effort, accuracy or clarity of explanation. As problems get more complicated, pupils who have received praise in the past for being fast will inevitably rush and make mistakes — often these are the pupils who show little or no working out in their answers. On the other hand, pupils who have been praised for accuracy, understanding and communication will often have more success.
Whilst it is true that quick recall of addition and multiplication facts can be helpful, it is far more beneficial in the long run for pupils to take an extra few seconds to think a little more deeply about strategy and communication.
Numbers come before numerals
‘Number’ refers to the quantity of something, whereas a ‘numeral’ is simply the symbol we use to write that number. Young children should always start with numbers — for instance, recognising who has rolled the highest number on a dice, matching numbers on dominoes, or (for a more seasonal example) spotting who has the most conkers.
In teaching terms, we talk about this as developing ‘number sense’, a key skill early on which involves pupils understanding numbers in a more concrete sense. A child might be able to count from one to ten, for instance, but without a strong number sense they are just repeating sounds by rote. Counting piles of conkers, or the spots on dice, or dividing socks into pairs of two can all help to relate an abstract sound — ‘three’ or ‘seven’ — to a concrete example.
Once children have an understanding of what a number means, they are then ready to learn the numeral by which it is represented on paper.
Introduce larger numbers through tens and units at first
Subitising, the process referenced earlier through which a person perceives a quantity without counting it, isn’t possible once a child begins working with larger numbers. Imagine the difficulties in trying to subitise the spots after each roll of a 20-sided dice!
Instead, children can be introduced to larger numbers by breaking them down into tens and units. ‘Twenty three’ is an abstract concept — but ‘two tens and a three’ will make more sense to a younger child, and can be easily modelled using groups of physical objects. Cuisenaire rods are a great resource to help with this — a standard set consists of a range of rods in different colours, where each colour of rod can be used to represent a different quantity. The rods are all in proportion to one another, so children of different ages can benefit from using them: younger children might explore how ten ‘ones’ are the same as one ‘ten’, while slightly older children can use them to explore fractions and decimals, for instance.
Very, very large numbers can be even harder to visualise, but here the internet has stepped up. Last year, a viral video on TikTok showed the wealth of Jeff Bezos represented by grains of rice — a perfectly-modelled example of the difference between a million and a billion, and one which can be replicated at home!
Mathematical thinking is about more than just ‘doing maths’
We talk a lot at Complete Maths about helping children to be ‘more mathematical’. By this, we don’t mean being ‘better at maths’ in the sense of achieving higher test scores or exam grades (though of course these are both by-products of a child becoming ‘more mathematical’). Instead, we mean to help children think more like mathematicians, a skill that transcends the classroom and will help them in every area of their lives.
Thinking mathematically is ultimately about adopting a problem-solving approach to life. When you set out on a long journey and are given the choice of two routes, one over a shorter distance but navigating slower country roads, the other a longer distance but using only motorways, you are thinking mathematically when you make your decision. You are considering the relationship between speed, distance and time, as well as using probability to calculate the likelihood of heavy traffic at different times of the day. If you have a desired arrival time in mind, you are combining your knowledge of time with your knowledge of estimation and subtraction as you work backwards to determine when to leave.
Every time you go to the supermarket and must compare the cost of a branded pack of 10 to a Value pack of 15, you are thinking mathematically.
Every time you dismiss a takeaway restaurant with a single five-star review in favour of one with three hundred reviews and a four-point-five star average, you are thinking mathematically.
Every time you calculate how many points your team needs to take from its next three matches to win the league, or to qualify for a tournament, or to avoid relegation, you are thinking mathematically.
Your children, too, will be using their mathematical skills on a daily basis, but they may not have made the connection between this and what they are doing in their lessons at school. If children can learn to recognise when they are being mathematical, even at home, it can go someway to overcoming a belief that they are ‘bad at maths’ just because they find parts of it difficult. This very much feeds into our final point:
How you talk about maths matters
There is a social stigma attached to illiteracy, and often a sense of shame around an adult admitting they can’t read. Almost the opposite is true of maths — just think how many times a day you hear casual throwaway remarks like “I’m terrible at maths”, “I can’t do maths without a calculator” or “I was rubbish at maths too, you must get it from me”. What message does this send? Firstly, that being ‘bad at maths’ isn’t just common, it’s expected. Secondly, that being ‘bad at maths’ is both a fixed and an inherited trait. Neither of these mentalities helps children.
Just think: the greatest maths brains throughout history spent most of their time being stuck on a mathematical problem. When framed like this, it becomes easier to see being stuck as what is meant to happen as you improve. At its core, maths is about using what you know to try and find a solution to something you don’t. Think how dull a murder mystery would be if the very first episode told us who the killer was and how they did it, and then apply that same approach to learning maths. This is exactly the attitude we want to foster in children: the idea that if they can solve something immediately it’s too easy; the belief that getting stuck on or having to think hard about a problem isn’t just a positive, it’s the whole point; and above all the importance of the word ‘yet’.
“It doesn’t make sense — yet.”
“I don’t understand it — yet.”
“I can’t do it — yet.”
If children can adopt a ‘yet’ mindset, they won’t just be successful in maths, they’ll be successful in everything they put their minds to. As adults, we can model this mentality by never speaking about a subject in absolutes, and always looking beyond the immediate difficulty towards the future success.
Dyscalculia and Maths
As a parent, your child's school might have spoken to you about Dyscalculia. Dyscalculia is understood by many to be a specific learning difficulty related to maths in which children will struggle with basic skills like number bonds or counting backwards. It is far less widely known and researched compared to dyslexia, which refers to specific trouble with reading and letters, but pupils diagnosed with dyscalculia may need similar levels of intervention and support to be successful in maths. You can read more about some of the signs of dyscalculia here and, if you think they apply to your child, should make an appointment to speak to the SENCO at your child’s school as soon as possible.
All of the tips above will be just as helpful for a child with dyscalculia, and it should not be seen as a barrier to their success.
Keeping In Touch
To access more advice for parents from our maths team, and to be kept updated on parent access for TUTOR, you can join our parent mailing list by clicking here
You can learn more about how to use Cuisenaire rods as a maths learning aid, and purchase your own set, by visiting their website: http://www.cuisenaire.co.uk
This Maths Fluency Set contains 100 Maths Fluency cubes in 10 colours, 15 double-sided activity cards, and a multilingual activity pack.
This article contains suggestions for maths games that can be played at home, and do not require any specific maths resources.
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
Thursday, 19 August 2021
In May we posted a blog exploring the importance of secure foundations when preparing pupils to move from primary to secondary school. In this blog, we aim to explore in more detail how teachers can use, and assess, prerequisite knowledge to identify weaknesses in pupils’ foundational knowledge, and to demonstrate how each of our platforms is designed to support this process.
Few teachers would dispute the notion that pupils will be more successful in studying a topic if they have understood the underpinning principles and knowledge behind it. On the flip side, few teachers would argue that large class sizes, mixed-ability sets or age-not-stage curriculums are optimal when it comes to ensuring every pupil has understood the previous step before moving onto the next one. That is not, however, to say it is impossible: with the right tools, and the right training, teachers can take a mastery approach whatever the context in which they are teaching.
In AskMark Volume 2, our founder Mark McCourt wrote that “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.” Several factors need to be in place for this to happen:
- Teachers need to know the underpinning prerequisite knowledge for the topic they are about to teach
- Teachers need both the resources and the opportunity to test this prerequisite knowledge before beginning a new topic
- Teachers need time to explore the results of any prerequisite test
- Teachers need the freedom and opportunity to adapt what they planned to teach when prerequisite test results tell them it is required
- Teachers need the confidence and the expertise to know how to deviate from the lesson plan or scheme of work
- Teachers need the time and resources to differentiate their lesson, and any follow-up interventions, to suit the individual needs of each pupil within their lesson (for a more detailed exploration of what we believe effective differentiation in a maths classroom looks like, see Volume 3 of AskMark)
For non-specialists, NQTs or RQTs, and any teacher whose workload frequently impinges on their planning time (in short: for any teacher!) the list above is bound to pose at least one challenge. Luckily for us, Mark McCourt quite literally wrote the book on mastery and it is under his guidance and leadership that we’ve developed each of our platforms with a view to making it easier for teachers to follow this approach. This blog offers an exploration of how Complete Maths CPD, Complete Maths CLASSROOM and Complete Maths TUTOR can support teachers at every stage in delivering effective, mastery-based lessons.
Understanding Effective Mastery Teaching with
Our online CPD platform currently contains over 200 courses, covering pedagogy relevant to teachers of every age group and at every stage of their career. If you are looking to kickstart your mastery journey, or perhaps to refresh your knowledge and revisit the basics, then we would recommend starting with our hugely popular Mastery Learning course.
For teachers who are confident in the core principles of mastery, and now wish to explore the role of assessment and prerequisite knowledge in more detail, we recommend the following courses:
In Responsive Teaching, Gary Lamb explores different strategies through which teachers can know whether true understanding has taken place, as well as research and models for effective corrective teaching.
Doing the Right Stuff, delivered by Dave Taylor, covers practical exercises classrooms teachers can use to identify the right level of maths to teach, and how to get pupils doing the right stuff from the moment they enter in year 7 through improved curriculum planning.
Getting Teacher Assessment Right, led by Mark McCourt, explores how and why we assess. Originally recorded in light of Centre Assessed Grades, it remains a thought-provoking discussion of the role of assessment within the classroom.
Numerous schools and organisations are now launching what we have nicknamed a ‘CPD Bookclub’ using our online bank of CPD courses. This involves a group of teachers agreeing a focus, selecting relevant courses from Complete Maths CPD, and then either watching and discussing the course together in small chunks, or watching the course at home then meeting to discuss it at a later date. If you’d like to read more about how Sir James Smith’s School in Cornwall implemented a CPD Bookclub, check out their case study on our website.
Embedding Prerequisites into Lessons with
Earlier this year we launched our new lesson page, which provides teachers with the prerequisite knowledge for every objective added to a lesson. The video below demonstrates how CLASSROOM can be used to create readiness quizzes for pupils to complete either at home or on devices in lesson before beginning a new topic, with the level of depth specified by the teacher in advance:
Even if you don’t have access to devices in school, the prerequisite strands can be a valuable tool in lesson planning. Fiona Wilmot from Bewdley School talks about this in depth in her school’s case study, also available on our website. In the case study, Fiona describes how “[One of our staff] was using it as a very detailed diagnostic: ‘This is what they need to know beforehand. Do they?’ Testing that they were okay before moving on. And it made his teaching far more effective and far more secure.” Each objective is also supported by detailed teaching notes including common misconceptions, which non-Maths specialists in particular report finding incredibly useful when following a mastery approach.
It is also worth noting that these readiness quizzes can be exported as a pdf and printed off – schools have then used them as revision resources, sources for quiz questions, or as the backbone to readiness work in lessons.
New and existing CLASSROOM subscribers have access to free platform training as part of their subscription; if this applies to you we would encourage you to take advantage of the offer if you haven’t already and see how the new lesson page, and especially the readiness quizzes, could work in your school.
Testing and Fixing Prerequisite Knowledge with
Private tuition is considered the gold standard simply because a tutor’s modus operandi is finding out what a pupil doesn’t know and then addressing this, something that becomes far more challenging in a group. TUTOR was built to ensure every single pupil could have access to bespoke maths teaching, delivered by an expert, and thus providing the benefits of tuition without the sometimes prohibitive cost.
There are two ways in which pupils can approach learning on TUTOR: the first is to allow the platform to assess them and pinpoint a starting point in the maths universe. From here, pupils will have a bespoke course mapped out in front of them, which responds to and re-charts their journey following every completed quiz. This iteration of TUTOR is in its final stages of development at the time of writing, and will be available to pupils very soon.
The second option is for pupils to enrol themselves on, or for teachers to enrol pupils on, a ready-made course. Each course is made up of ideas, and each idea is broken into a number of goals; each goal in turn begins with pupils being shown the prerequisite knowledge required, and prompted to take a short quiz testing this knowledge. At the end of the readiness quiz, if pupils score less than 80%, they will be prompted to ‘fix’ any gaps and directed to Learn and Do videos exploring the specific topic in more depth.
In both cases, the testing of and response to prerequisite knowledge is key. In an earlier blog post, we explored the ways in which TUTOR could be used to support teaching and learning in and out of the classroom. When it comes to prerequisites, our focus today, TUTOR is most valuable as a resource not just for assessing the readiness of a pupil to move on to a new topic, but also as a form of intervention after the lesson if a teacher believes gaps remain.
Many schools have told us they plan to use TUTOR this year in place of normal lunchtime or after-school intervention — rather than have one maths teacher deliver the same lesson to a slightly smaller group, a member of staff from any team or department will instead supervise pupils working on TUTOR. Pupils working from home can similarly benefit, meaning those not included in school-based interventions won’t miss out. In this way, even teams made up of non-specialists, or with lower levels of experience, can ensure every pupil’s knowledge of prerequisites is continually checked and, when needed, addressed.
Putting Prerequisites at the Heart of the Curriculum
Prerequisite knowledge is core to what we do at Complete Maths: we want to map it, test it, and help teachers react to it at every possible opportunity. If your team feels the same, or if you want our help moving your team in this direction, then we want to hear from you.
- Could you collaborate with us on a retrospective case study, sharing with other schools how you used our platforms to embed the testing and teaching of prerequisites in your school?
- Would you like to collaborate with us on a forward-facing case study, in which we work together to help your team use our platforms to develop your use of prerequisites?
- Could you deliver a workshop at our next MathsConf sharing how you use prerequisites in your classroom?
Monday, 16 August 2021
Since Summer School launched in July, its pupils have watched over 81 days worth of videos and completed tens of thousands of quizzes, generating reams of data which schools are no doubt eager to use as they reopen their doors for a new academic year. From September, Summer School’s evolution into TUTOR will bring with it a host of new features and advanced functionality which will support teachers even more, both in their interventions, and in their day-to-day teaching and learning.
Whether your pupils have been active on Summer School, or you are just getting started in September, we want you to get the most out of TUTOR as part of your teaching and learning programme this academic year. Read on for six ways to use TUTOR in the classroom:
Let TUTOR create a bespoke pathway for your pupils
TUTOR’s most impressive feature is, arguably, its ability to assess a pupil, set a customised starting point for their learning, and then guide them through our entire maths curriculum. If you’re looking for a powerful, personalised intervention tool, this is it. You might decide to use TUTOR in place of group intervention — simply provide pupils with a laptop and a supervising member of staff, and let them work at their own pace. Alternatively, have pupils log on at home and use TUTOR as the backbone of their home learning. From the teacher side, you’ll be able to view their progress through the curriculum as well as their quiz scores, making it easy to track both usage and assessment outcomes.
Use the Stage courses to help pupils revise content from the previous academic year
If your pupils enjoyed the discrete courses available through Summer School, then we have good news: these are staying put on TUTOR. Our Stage courses are mapped against the Complete Maths curriculum, with each stage roughly corresponding to the equivalent academic year — so in Stage 6 you will find content you might expect to see in confident classrooms in Year 6 (England and Wales), Year 7 (Northern Ireland), or Primary 7 (Scotland). For pupils who have begun the year below the expected standard for their age, these courses can offer a great catch-up resource for them to work through at home.
Build your own bespoke course
Perhaps you have an assessment approaching and need pupils to revise a specific set of topics. You might have a class or a group of pupils who’ve missed a particular unit and you need them to catch up so the rest of the cohort can move on. These are just two of the reasons why we’ve created the option to build your own bespoke course using the content already available on TUTOR. Pick and choose the appropriate Ideas for your pupils, and we’ll supply the Readiness assessment, videos, and Goal quizzes.
Challenge your most able using the Topic courses
When a pupil discovers a topic in maths that ignites a spark in their imagination, we want to nurture that spark into a flame. Topic courses include all the content we have available on a particular slice of mathematics, from Transformations to Trigonometry. If one of your pupils shows particular aptitude or interest in a topic, then these courses are perfect for stretching them beyond what they might cover in the classroom and letting them explore in more depth. For older pupils struggling to grasp a complex idea, returning to the beginning of the relevant maths could help tackle misconceptions or cement key knowledge that may otherwise be missed.
Set Goal targets as a tangible method of revision
For pupils who struggle to plan revision, or who lack motivation, setting a target number of Goals to complete could be the answer. Each goal follows the same structure and offers a bite-size, straightforward way to revise, especially during the week. It’s easy for both teachers and parents or carers to track progress through a Goal, too, making a Goal target — such as completing one Goal per night during the week and two each weekend — a manageable and measurable way for those at home to support their children. The Awards on TUTOR add an extra layer of motivation, with pupils rewarded not just for high scores on quizzes, but also for the number of log-ins and the number of Goals completed.
Set Readiness quizzes as homework before teaching a topic
If you’re following a Mastery curriculum then you already know the vital importance of assessing pupils’ existing knowledge before attempting to teach a new topic. In reality, when teaching a class of 30 pupils, each of whom comes to the lesson with their own strengths and weaknesses, it can be hard to assess quickly who is ready to go and who needs extra support. The Readiness step on TUTOR can do this for you, by sharing with pupils the prior knowledge they will need to have in place, assessing them against it, and then recommending which topics need ‘Fixing’ along with supporting resources. Asking pupils to complete Readiness quizzes at home, before the lesson, means you’ll begin teaching armed with the knowledge of which pupils need further support and which are ready to stretch.
STEP Academy Trust partners with Complete Maths to make high quality maths training available to all their teachers
Friday, 25 June 2021
STEP will use Complete Mathematics products across their schools — including everything needed to improve the teaching of Mathematics across all year groups.
London, United Kingdom: Complete Mathematics and STEP Academy Trust inaugurate a partnership to mark their shared commitment of providing the best training to Maths Teachers at the most accessible and inclusive price. Dubbed the “perfect partnership” by both sides thanks to the natural alignment in their values, STEP Academy Trust and Complete Maths believe what they are able to accomplish together is as close to true mastery as one can get in the current system. STEP Academy Trust has been singularly focused on embedding mastery within all its schools, and believe they have found the perfect product to align with their mission, and their approach. The impact of this shared vision now culminates in their alliance with Complete Maths, and with it a joint commitment to providing the very best outcomes for pupils.
The Complete Maths product suite is designed to support teachers at every level: Complete Maths CLASSROOM contains everything teachers need to teach, learn and assess Maths in a way that is both highly flexible and inherently diagnostic: the core tenets of mastery teaching. Then, through Complete Maths TUTOR, teachers can enrol pupils on a bespoke, 1-1 tutoring programme covering Year 1 all the way to A Level for as little as £1 per pupil per week. Meanwhile Complete Maths CPD, nicknamed ‘The Netflix of Professional Development in Maths’, offers access to over 170 online courses targeting every element of mathematical pedagogy, with new courses added every week.
This partnership between Complete Maths and STEP Ahead Trust was developed in three phases:
- Phase 1: STEP signs half of their schools to Complete Maths CLASSROOM, the online teaching, learning, assessment and monitoring platform for teachers to use with pupils in their classroom.
- Phase 2: After trialling, STEP signs all of all their 18 schools to Complete Maths CLASSROOM.
- Phase 3: Facing increasing demand for online training solutions developed for educators, by experts, STEP looks for training solutions that can be used consistently across entire departments as opposed to individually or as a one-off solution. STEP signs all of their schools to Complete Maths CPD & inaugurates partnership with Complete Maths.
As part of this agreement, STEP Academy Trust also brokered a deal for its teaching school hub, STEP Ahead: they will offer a 30% discount for all schools across East Sussex, and Brighton and Hove to sign up to the training platform — less than £5 per teacher per month — making it truly accessible, and the most high value solution for teacher training today.
Celebrating the new partnership, Director of Education Dominic Bristow says, “The Complete Maths team has been dedicated to the improvement of Maths teaching since Mark [McCourt] first began conducting in-school CPD with colleagues as La Salle Education. Now, with a growing group of advocates and a well-evidenced set of products behind us, we look forward to continuing our mission to improve the life chances of children through this new partnership. STEP have provided us with the ideal opportunity to showcase the best of classroom practice and teacher development, so we are delighted to help them to reach all the schools in their training hub.”
Deputy Head of the STEP Teaching School, Matt Swain says: "At STEP, we know that secure subject knowledge and pedagogy is essential for responsive and lean teaching of mathematics. So having Complete Maths CPD at our teachers’ fingertips allows leaders and teachers at any time to tap into the deep subject knowledge of maths experts at every developmental stage.
“Working with Complete Mathematics has further supported us in embedding consistently good practice across our schools. Using the data provided in CLASSROOM, particularly the assessment features, has empowered our policy of same-day intervention for our pupils, so that no one is ever left behind. We now have a responsive curriculum, bespoke to every individual pupil — and we are excited to supplement this with the upcoming TUTOR programme, which absolutely aligns with our mission.
“Because as a Trust, we are committed to improving the life chances of all children: where we have the capacity to make a difference, we are morally bound to do so. Our partnership with Complete Maths will help us accomplish this.
“We were delighted when Mark visited one of our primary schools in June and described what we do as ‘mind-blowing’ after observing our lessons and intervention in progress. We are so proud of our teachers and pupils, and thrilled that visitors, too, can see the impact of our work with Complete Maths.”
About Complete Maths:
The Complete Maths Group (previously known as 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 Complete Maths events and receiving professional development from the company. Complete Maths supports teachers effectively and efficiently through its online teaching, learning, assessment and monitoring platform, Complete Maths Classroom. In 2020, Complete Maths launched trials of its ‘digital tutor’ product, ‘TUTOR’ 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.
STEP Ahead is a Teaching School Hub based at Angel Oak Academy in Southwark and is part of STEP Academy Trust. After becoming a designated teaching school in 2018, STEP Ahead has shared their teaching approach with a growing alliance of member schools, working in both South London and East Sussex. Earlier this year, they were awarded hub status and given a specific remit to support schools in East Sussex and Brighton and Hove from September 2021. The expert team of teachers and school leaders passionately believe in developing teachers as the only sustainable way to drive up standards in schools and ultimately, improve the outcomes of their pupils. STEP Ahead is now expanding their reach by working with a number of strategic partners and members who share their passion for teacher development.
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.
Friday, 23 April 2021
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.
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.
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.
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:
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:
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:
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.
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:
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:
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.
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.
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.