Michaela’s Blog
Michaela Community School, Wembley

Michaela’s Blog

First term at Michaela

In September, I started teaching at Michaela Community School. This last term has been a brilliant introduction to a warm and happy school that has learning at its very core.

I started at Michaela after completing the Teach First LDP, where I spent two years teaching Maths in east London. Lessons were frequently disrupted by poor behaviour. Teachers were bogged down in time-consuming marking policies and the data collection of anything and everything in the hope that this would please Ofsted. We were not giving pupils the academic education that would enrich their lives and enable them to access universities like Oxford and Cambridge.

I wanted to be part of a school that was genuinely enabling upward social mobility, and that fulfilled the ultimate test – would I send my own child there? I had been following Michaela for a while and loved everything I heard. Joining the school felt like a natural step.

The on-boarding process of becoming a Michaela teacher begins long before September. I was given plenty of reading material immediately after accepting the position, and throughout the year. I visited and observed lessons, and received advice on how to embed the routines.
At Michaela, we talk about staff “singing from the same hymn sheet” and “rowing together”. The reason the school works so well is because there is such a high degree of consistency in what everyone does. Everyone’s vision is aligned with that of the school. The on-boarding process ensures that this is the case.

The teaching at Michaela is excellent. I am regularly amazed by the history our pupils know, the complexity of the mathematical problems they can solve, and the French they speak with aplomb and perfect accents. Part of the reason every teacher is so good is because our observations and feedback system are so unusual. In my first two years of teaching, I remember spending days poring over powerpoints, hours writing incredibly elaborate lesson plans and creating resources, and countless more hours marking books…and then marking pupils’ responses to my marking. The whole process was stressful and not always helpful.

At Michaela, I am observed at least three times a week. We have an open-door culture where anyone can walk into any lesson and give feedback instantly via email. When I am observed, the feedback is always swift, practical, and something I can act upon instantly. It isn’t stressful because it is entirely low-stakes. There are no grades awarded. If observations in schools were truly about improving people’s practice, they would always be entirely low-stakes and there would be no need for grades.However, at too many schools, observations have become a performance management tool, rather than an opportunity to actually improve teaching. At Michaela, we are never told to do something because Ofsted (supposedly) want to see it. All feedback is given because it will genuinely improve pupil learning.
Two weeks into the term, I remember thinking I’d already learnt more than I had in the last two years.
After seven weeks, my teaching already looks so different to last year – and my pupils are better off for it. I’m now keen to be observed as much as possible.

Since starting, I’ve realised it is possible to enjoy CPD! We have CPD every Monday, and new staff have extra CPD with Katharine every Wednesday. This is an opportunity to discuss a thought provoking topic, such as grammar schools or candour. The atmosphere during CPD reminds me of my university days. It has a similar scholarly feel because it is an informed discussion with people bouncing off each other’s points and ideas, in the same way we would do during Cambridge supervisions. The pursuit of knowledge is not limited to pupils at Michaela – there is clearly a culture of learning among the staff too. Everyone loves to read and engage with educational ideas.

Starting at Michaela has been made easier because so many of our systems are centralised. Gone are the days when I would have to give up time to run my own detentions. As resourcing is also centralised, I no longer spend evenings trawling through TES in the hope of finding a good resource (or spend time creating it myself). Now, my lesson planning involves looking through the relevant booklet, thinking about the exercises I want pupils to complete, as well as potential AFL questions. I have not created or even tweaked a single powerpoint this half term. The booklets that departments have created are of extremely high quality. The Maths booklets are infinitely better than any textbook I have seen. There is so much consideration given to the sequencing of topics, the writing of questions, or designing good AFL questions.

This term, one element of my practice I’ve focused on is economy of language. I had developed a bad habit of repeating myself because I was accustomed to pupils not always listening. I would repeat myself to ensure they had heard it, but I was reinforcing the notion that they did not have to listen to me the first time around. This is a habit I’ve been trying to break. Additionally, I grew accustomed to using lots of unnecessary words when a few would suffice. This makes explanations more cluttered and the key points less clear for pupils. The aim is for everything we say to be as efficient as possible, for the purposes of clarity and saving time. For example, when giving instructions, we might say “Books out. 20 seconds. Go.” or “Rulers on page 15. Line 3. Go”. Whilst I have made improvements with this over the last few weeks, I’ve still got a long way to go.

Finally, another aspect of Michaela I have quickly grown used to, and think is fantastic, is the culture of gratitude. As well as our lunchtime appreciations, pupils occasionally have the opportunity to write postcards to members of staff. Encouraging pupils to share their gratitude in this way strengthens the warm and positive relationships that exist in the school. Below are some examples. Receiving a stack of postcards on the last day was very touching, and a lovely way to end a brilliant first half-term.

If you’re interested in visiting the school and having lunch with the pupils, get in touch here. We’d love to have you see our school for yourself.

Interested in joining? We’re hiring a French teacher! https://mcsbrent.co.uk/teacher-vacancies/

Practicals in Science

Now that I’ve been at Michaela for over a year, I have come to see that one of the most enjoyable aspects of teaching science are our practical lessons. Whilst in the past this might have filled me with dread, the systems and logistics we have in place mean they run smoothly every time.


Before the practical

  • Knowledge – Firstly, we make sure pupils understand what the practical is about. For example, if year 8 pupils are completing a pondweed photosynthesis practical, we make sure pupils know the factors that limit photosynthesis. If year 9 pupils are completing a displacement reaction, we make sure they know the GCSE reactivity series off by heart. How we get the pupils to remember this knowledge can be found in a previous post here.
  • Practical skills lesson – before each practical lesson, we teach a practical skills lesson to discuss scientific skills, the equipment and terminology needed for pupils to make maximum gains during the practical lesson. This can be specific to the practical: for example, knowing the parts of microscope before looking at onion cells. On other occasions, we might look at broader aspects of practicals such as identifying hazards or risks, or possible variables. Pupils use self-quizzing to memorise these principles by heart.

Practical lesson

  • Expectations – In a practical lesson, the same high standards of behaviour are expected. Pupils are told that they must remain professional at all times. This includes everything from the use of lab coats and safety spectacles, to packing away equipment. Perhaps one element of practical lessons at Michaela that makes them markedly different from other schools that I have visited is the extremely low volume that our pupils maintain. At most, pupils may whisper when necessary in groups. We believe this is essential to ensure a calm and focused environment. We always model and practise ‘whisper voices’ before we begin, to clarify what this looks like.

Here are a couple of short videos of pupils completing practicals in class:

Miss Dyer : Heart Dissection 

Miss Cheng: Displacement reactions


  • Lesson outline – each practical lesson roughly has this outline.
    1. Recap (quick questions from the skills lesson);
    2. Reading instructions: equipment, risk assessment, method
    3. Practical
    4. Comprehension questions

After the pupils have done this a few times, these steps become fairly automatic.

  • Modelling – teachers always trial the practical before we teach it, taking pictures to model what the practical looks like. This leaves no ambiguity with things like setting up, and so on.


  • Visualiser – every Michaela teacher has one of these pieces of kit. It’s a great tool to have for all sorts of reasons, but in science, it can be brilliant for demonstrating practicals. For example, my year 10 class were recently dissecting hearts. I modelled this first on the visualiser so the pupils could identify all the parts and see exactly where to make incisions before they went ahead and did it themselves.


  • Routines– it is absolutely essential that strong routines are in place during practical lessons. This provides clarity and ensures safety at all times. For example, we assign different tasks to various pupils, such as sorting equipment into trays, or even carrying out different parts of a dissection. During the heart dissection practical, for example, pupil 1 made the first incision, pupil 2 made the second incision, and pupil 3 identified the blood vessels.


After practical

  • To make sure the practical was useful, and to consolidate the pupils’ learning, we give them a 25 mark exam paper on the practical. This ensures that the skills and knowledge they gained from the practical are linked closely to their theoretical understanding.

My HoD, Olivia Dyer has previously written about the principles behind practicals at Michaela here.

If you would like to visit Michaela and see us in action, email visits@mcsbrent.co.uk.

JLMFL – Teach French à Michaela

19 Sep 2017, Posted by admin in Michaela's Blog

Teach French à Michaela

We’re looking for an amazing French teacher.

Here’s some of what we do:

  1. The highest expectations of what pupils can – and do – achieve.
  2. The highest expectations of ourselves as teachers.
  3. Beautiful accents, exceptional pronunciation, fantastic range of knowledge.
  4. Lots of reading out loud, lots of questions, lots of translation.
  5. Unashamed teacher expertise and pride in our knowledge.
  6. Outstanding behaviour, allowing us to have great fun in lessons.
  7. Centralised resourcing, allowing you to focus 100% on your teaching.
  8. Great attention paid to minimising workload and maximising efficiency.
  9. Support in spades on an individual, department and whole-school level.
  10. Enormous amounts of love – of the kids, our subject and our school.

Here’s some of what we don’t do:

  1. Games, gimmicks, fads, learning styles, miming…
  2. Marking – we do whole class feedback, 15 minutes per class per week.
  3. Pictures – everything we do links to the written and spoken word.
  4. Individual or department detentions – it’s all centralised.
  5. Graded lesson observations, performance-related pay or targets.
  6. Anything that costs more in effort than we get in learning return.
  7. Bureaucracy and admin (at least, it is kept to the barest minimum!)
  8. Regular working at evenings and weekends.
  9. And…
  10. That’s all I can think of for now!

If you want to teach in a no-nonsense, equally challenging and supportive, joyful environment; with kids who know a lot and want to learn and arrive at every lesson ready to learn; with staff who work hard to be the best but have time to have a life outside of school, even during term time…


Is this the best we can do? Part 7: the spacing effect

This is part of a series offering my views on some problems with and solutions for UK maths education. The first part looked at the state of affairs with regards to GCSE and PISA results, the second part looked at my attempt at a diagnosis, the third part looked at pre-existing maths education success stories, the fourth part looked at how textbooks offer the largest potential to improve pedagogy across the nation, the fifth part examined the role of hard work and homework, and the sixth part diagnosed the problem of forgetting in maths education. This part examines one solution to improving maths memory: employing the spacing effect.

I remember it well. I’d reached the end of my first ever half term of teaching. But standing between me and the holidays was a large pile of assessments to mark. At first, the prospect didn’t seem too bad: I thought it would be relatively mindless work. But that belief was before I actually started the marking itself. My dreams of a mindless few hours rapidly transmogrified into a nightmare: the nightmare of realising how poorly my classes had done. Question after question hit me with the same realisation: ‘we’ve covered this in class – we even looked at this yesterday, in our revision lesson – how on earth did you get this question this wrong?’

test results

(scores are real, names have been changed)

The above table shows a representative sample of results from one class. Some pupils had turned in pretty good papers. But as a whole, across all my classes, the class average was pretty close to 50%. As a pupil myself, I could never have even comprehended getting 50% in a maths exam. As a teacher, I was even more baffled. I had taught 100% of the topics on the test – what on earth had happened to the other 50%?

Seeing my class results as a whole then pricked me with a deep fear. This was first half term as a teacher. What if I had been making some terrible mistake? What if I was a far worse teacher than I, or my tutors and mentors, had realised? With anxiety I found a fellow maths teacher in the staffroom and and told her of my predicament:

“So, my classes have done really poorly – I don’t understand what’s happened.”

“Oh no! What do you mean? How poorly did they do?”

“So, I think the class average is about… 50%? Most of them seem to be getting around 29 out of 55.”

“50%! Oh! DON’T worry about that, that’s fine! My classes get around that too.”

“Oh…. but how come they don’t get more? I taught it to them, after all.”

“They’ll pick it up later on. It’s fine!”

Thus ended my first half term as a teacher, and with it, my naive hopes of pupils remembering everything that I taught them.

Yet at the same time, it ignited within me a passion for maths pedagogy that eventually led to this blog. What if there is a better way than simply expecting so little from our teaching? What if there were some way of getting pupils to score close to 100% in tests? What if there were some way of getting pupils to remember nearly all of what you teach them?

This year, at Michaela, I’ve had the privilege of seeing 100% test scores happen before my very eyes – in a most unlikely place, too. I’ve taught a year 7 intervention group which contains some of our weakest pupils. They take a quiz every 3 weeks, and the scores are remarkable. What’s been going on? The secret is what we’ve been using:


Siegfried Engelmann’s Connecting Maths Concepts

The Connecting Math Concepts (CMC) curriculum forms the backbone of our intervention strategies at Michaela. It’s a brilliant curriculum: all the questions are provided, all the teacher explanations and examples are provided (in a book of scripts), all the sequencing is decided in minute detail… but most novel of all, one single typical 60 minute lesson will cover at least 8 different parts. Each part might tackle something like column addition, multiplication facts, describing shaded parts using fractions, subtraction with and without borrowing, word problems, two-way tables, finding the area and perimeter of shapes….

For example, here’s 30 minutes’ work from one lesson:

lesson 32 textbook

Then here’s another 15 minutes from that same lesson:

CMC lesson 32

Look at the diversity of topics covered! Every lesson is like this. Compare it to the typical way we teach topics: 10 minute starter followed by 45 minutes on one single topic; next lesson, move on to the next topic in the scheme of work.

In contrast, in CMC lessons: whenever a new topic is introduced, it’s done in a 5-to-10-minute chunk, where one tiny concept is introduced, then tested. How are the concepts broken down? For example, look at ‘part 4’ in the picture directly above. The question for the pupils to answer here would be something like: ‘write the multiplication fact shown here’ = ‘8 × 4 = 32’ and then ‘write the division fact shown here’ = ’32 / 8 = 4′. The question is simply testing whether pupils remember the locations of the quotient, divisor and dividend – separate from any actual division. And that’s it! Those two questions will be all that the pupils will be asked to do – for this lesson. Compare that with the average lesson on bus-stop division – I don’t think I’ve ever explicitly taught all the meanings that are implied by a filled-out bus stop. No wonder it’s so hard to get bus-stop division to stick for our weakest pupils.

But that’s just one lesson. The real secret is what happens over time. As we’ve seen, each CMC lesson contains a smorgasbord of mathematical activity, but there’s a big range of guidance in them. If you look closer at the two example lessons above, you’ll see that some of the lesson falls under ‘independent work’, and others don’t. Over several weeks, some topics are introduced for the first time, in tiny incremental parts; the topic complexity is gradually increased, and teacher instruction/reminders/guidance is gradually pared back, until the questions become part of the independent work for the pupils to do without any help.

As a whole, this is how CMC approaches each topic. What CMC never does is what we normally do in UK maths lessons: assign a topic to a single lesson or two, get pupils to practice it in the lesson, then recap briefly in the following lesson and move on for good.

The reason we do things this way is because this method works well for a good proportion of a class (the ~60% who achieved C+), but it’s clear that some of the children are left behind by it, floundering in misconceptions – the pace might be too quick, or the concepts (like remainders) too confusing. So, the average teacher addresses this by differentiating the work, but this has numerous disadvantages – to take just a few examples, some pupils never get as far as short division (so when will they learn it?) because they are still practising finding remainders; teachers find it difficult because they are trying to teach 3 lessons in one. (I have written on some of the pitfalls of differentiation before.)

Instead, this is the CMC method of breaking down a topic and designing a curriculum:

  1. break down a topic into parts which are so simple that they are nearly-impossible to get wrong.
  2. Then, practice that broken-down topic for a few minutes. Then – that’s that topic DONE for the day.
  3. Instead, to progress pupils further in that topic, re-introduce it in future lessons, gradually scaling up complexity, and scaling back guidance. This may take place over 5 lessons, before the topic is attempted independently.
  4. Repeat steps 1-3 for other topics, and then put them together in the same lesson.

Does this work? The results, to me, speak for themselves: the pupils (who are among the weakest in the year) enjoy it because they experience a lot of success; they work hard in lessons; and finally, in the included assessments, they get 90%+ each time. They are making clear progress on numerous topics and accurately doing difficult maths that, 11 months ago, seemed totally beyond them. In other words, CMC is another example of a ‘success story’ in maths education that we must learn from.

How is it so successful? One of the key designers of CMC is Siegfried Engelmann, noted for his application of pedagogical research to real-world curriculum, and this is no different. Let’s take a deeper look at the theory behind CMC’s’s unorthodox distribution of practice in smaller chunks, spread over extended periods of time:

The spacing effect

Distributing practice over time, instead of blocking it all in one lesson, has a name: the spacing effect. What is the spacing effect? Here’s Daniel Willingham, cognitive scientist extraordinaire:

Suppose a student is going to spend one hour learning a group of multiplication facts. How should that hour be allocated? Should the teacher schedule a single, one-hour session? Ten minutes each day for six days? Ten minutes each week for six weeks? The straightforward answer that we can draw from research evidence is that distributing study time over several sessions generally leads to better memory of the information than conducting a single study session. This phenomenon is called the spacing effect.


The graph above shows the results of Keppel’s experiment: an experiment that compared two ways of allocating teaching time. The two bars on the left show test results (higher is better) when practice is massed – i.e. allocated in bulk within a single lesson. As you can see, there is a huge dropoff in results in a test done one week later, compared to a test done the following day. In contrast, the bars on the right show test results when practice is distributed – i.e. split up into much smaller chunks to be studied over multiple days. Notice how there is barely any dropoff at all in the two bars: results one week later are almost as good as the results the day after the instruction finished. In other words, if you were to spend a 1-hour lesson teaching a topic, and followed it up the next day with 10 successfully-answered recap questions, then don’t expect pupils to retain what they’ve learnt for even a week. But if you employ the approach of CMC, and instead split up that same instruction and practice into four 15-minute chunks practised over four days, then watch and see your pupils retain it, for at least a week.

So, in short, the spacing effect tells us that one of the best ways to teach for memory is to teach topics in small chunks over extended periods of time. It’s an effect that has been proven time and time again in research. But strangely we just don’t teach like this! I suspect, as hinted at before, it’s because teaching in blocks works fine for the majority of pupils. But at its core, the spacing effect doesn’t have to be as complicated or detailed as the CMC curriculum; instead, it offers a very simple prescription: if you want pupils to remember something, you must continually return to it, again and again and again. Every day for a week. Every other day for a month. And then once every week for the rest of the year.

Because the spacing effect really is that simple, we do see it employed in other aspects of maths education – especially when it matters the most. If your schools runs a Year 11 intervention, one aspect of it probably looks like this: take pupils out of form time for 10 minutes, give them a standard GCSE C-grade question type, let them try it, re-teach it if they don’t know how. Repeat daily with similar topics and different topics in that crucial run-up to Easter, and watch them magically ‘improve a grade’. Similarly, more intensive hour-long intervention sessions rarely look like the usual 60-minute mono-topic lesson; instead, they’ll be filled with multiple topics, with the questions changing every 10 minutes, to be repeated over weeks and months. Can you see what’s happening here? It’s none other than the spacing effect: little chunks, repeated often, leading to long-term memory, fluency, and confidence. Heads of maths departments know this works well (for the previous GCSE at least) for year 11 GCSE. But if this is a successful method, then why don’t we make steps to implement this all the way down the school? In KS3? Year 7, even, for our struggling pupils? Why on earth wait until year 11 – when, for many pupils, it’s too late – to use one of the most powerful effects for effective memorisation?

A great revision resource from mathedup.co.uk… but what if every lesson looked a bit more like this?

That’s partly a rhetorical question. I’m aware that one big reason why not every lesson can look like CMC or a Y11 intervention session is because they are a intensive to resource and prepare. But that doesn’t mean we can’t find ways of making greater use of the spacing effect without overhauling our entire curriculum. Here are some suggestions for September.

Practical applications of the spacing effect

Immediate applications across all topics

The easiest application that all teachers must do: don’t take the default approach to ‘recap starters’, where you only spend 10 minutes recapping what you did the previous lesson. Keppel’s experiment shows that while pupils will be pretty successful in retrieving whatever you learnt yesterday, this is no guarantee that they’ll be able to retrieve much of it in a week’s time. Especially if you’ve moved onto another topic, make it your aim to recap the previous topic’s work on a daily basis for at least a week. Aside from the usual ‘5 minute recap of the previous lesson’, devote an extra 10 minutes of each lesson to wider recap: recap the previous topic, the topic before that, the topic before that too. Finished sequences and moved onto volume? Spend the next week, or even better, fortnight, devoting 10 (then decreasing to 5) minutes each lesson on sequences questions that they’ve answered before. Keep it minimal and sustainable – for example, just re-project the exact same practice questions of the sequences powerpoint you used before, or re-use questions from the previous week’s worksheets.

A good heuristic to see if you’re recapping frequently enough is if pupils can do it with as little guidance and prompting as possible. If, during recap, you find you’re effectively having to re-teach a topic from scratch, that’s a clear sign that you’ve left the topic untouched for far too long.

I wish I’d known this and made a better effort at doing this right from the start of my teaching career. The spacing effect research tells us it’s worth it.

Create starter booklets/infinite worksheets for high-leverage topics

High-leverage topics are those upon which many other topics hang: for example, written operations, fractions, negatives, simplifying algebra, equations, and so on.

We want our pupils to be 100% confident in such high-leverage topics. We want these topics to be ingrained in their minds. So, we ought to make use of the spacing effect. Since the spacing effect requires practice over long periods time, we want an easy way of having enough practice questions of increasing complexity to use on a daily basis. An easy way to get in enough practice without having to plan too far is to create starter booklets in advance: 10 minutes, every day, focused on practising key high-leverage skills.

William Emeny’s Numeracy Ninjas booklets are a brilliant free resource – we use them as our starter booklets for year 8. Similarly, setting Times Table Rock Stars as a daily homework over several terms is also a key foundation for future maths success. At Michaela, we’ve also sensed the need for something in between the two – TTRS focuses purely on multiplication & division facts, whereas Numeracy Ninjas aims to hit around 25 different key skills with just 1 question per topic per day. So we also occasionally make our own starter booklets for certain topics too.

I’ve uploaded one example of such a booklet, used with a bottom set Y7 class, on TEShere. Of course, it doesn’t have to be printed out as a booklet – projecting it also works. Either way, I’d recommend making them using random number generators in Excel, so you have an infinite variety of suitable questions, ready to project at a moment’s notice. And once you’ve gotten the hang of using Excel, and the pupils have gotten the hang of these questions, it’s really easy to increase or decrease the complexity of your question set. Other topics we’ve made booklets for include lots of practice with operations with negatives, decimal multiplication, and fraction operations. Or really, whenever you stumble upon a topic that your pupils keep on forgetting, just make a starter booklet! Give them 5 minutes of practice every lesson for a term, and complain no more.

Using the spacing effect to teach grammatical topics

The spacing effect also comes in very useful for those topics that are difficult to learn, and so easily confused and forgotten – in a previous blog I have called themgrammatical topics, As I wrote before:

The common issue with grammatical topics: just like the usage of full stops, they are easy to explain & understand poorly and very difficult to explain and understand comprehensively. There’s too much underlying logic that pulls in different directions. So, I’m increasingly convinced that with grammatical topics in maths, just as with language, understanding must follow grammatical fluency and familiarityOur understanding of them requires a mass of examples, together with a mass of practice, cohering together, over a stretch of time.

The final words ‘over a stretch of time’ are another way of saying: employ the spacing effect!

The same strategies I mentioned in that post come in handy here, too, so give it a read if you haven’t – in fact, I was really implicitly recommending the spacing effect there. The practical tips mentioned were:

  1. Reflect on how ‘grammatical’ the topic you want to teach is
  2. Start slow, build up
  3. Practice, practice, practice – over time
  4. Prepare to fly

Before I dive into more detail on these points, the first step is to be aware of your teaching group. Top sets generally do fine with traditional blocked practice plus recap and the odd revision lesson. Indeed, top sets generally find grammatical topics (like negatives, simplifying algebra, number work) pretty easy and intuitive. But middle sets would benefit more, and bottom sets would benefit most extensively, from a teaching approach that employs the spacing effect. That may determine how much you want or need to overhaul your resources.

So what does this look like? Here’s an example of how I taught bus-stop division, using lots of Excel random-number question generators, to my bottom set year 7 group:

  1. Ensure 100% confidence with division-facts: 10 minutes for two lessons.Division drill 1
  2. Ensure 100% in finding remainders. 10 mins over three lessons, overlapping with the previous lessons.Division drill 2
  3. Ensure 100% confidence in distinguishing the different answers when the divisor is bigger or smaller than the dividend. This was 10 mins over several later lessons. This was done with a mixture of these two layouts, to introduce to bus stop notation:
  4. Now, several lessons in, start to introduce the bus stop method, starting with questions where the answer is an integer. 15-30 mins over three lessons, again with some overlap with the previous lessons.Division drill 5
  5. Questions where division where the dividend is a decimal, to introduce pupils to having to add decimal points to their answer. 10 mins over three lessons, again with some overlap with the previous lessons.Division drill 6
  6. Questions where the answer is a decimal of one decimal place. (E.g. 47 ÷ 5 and 34 ÷ 4. No pictures from here, as I think you get the picture!)
  7. Questions where the answer is a decimal of multiple decimal places. (E.g. 255 ÷8)
  8. And finally, after several teaching chunks of understanding and using recurring notation, questions where the answer is a recurring decimal. (E.g. 124 ÷ 3, 235 ÷ 6)

Notice what’s different about the time allocation: instead of trying to fit all 8 of these question types into two or three hours’ worth of lessons, I spread them out in smaller chunks over 2 weeks of lessons, and it probably didn’t take that much longer in terms of actual time. If I had tried to ram all those teaching points into three consecutive lessons, some of my pupils would’ve gotten it all, but many more would’ve gotten stuck early on (and in my earlier days, I would have ‘differentiated’ to “solve” the problem: ‘okay, if you’re finding these new questions on recurring decimals hard, do more practice on these other types that you were finding hard before!’). That’s one big thing CMC taught me, and I’m very grateful for it – in our end of year assessment, the division question was well-answered by most of my class.

Design a spaced curriculum

I’ve left this one til last because it’s by far the most work, but it’s potentially the most rewarding. The three suggestions I’ve made above are somewhat ad hoc. But why not space practice throughout an entire curriculum? This is what Engelmann and co. did for CMC; my one flaw with it is that it’s designed for the US curriculum. I’d love to see an equivalent version that’s geared towards UK’s GCSE maths specification, or even KS2 SATs. An easier version of this may also be the topic of my next post, so stay tuned.


The spacing effect is one of the strongest effects in memory research. Yet we don’t use it nearly enough. Our failure to make more of it is at the heart of much of theperformance-learning gap: pupils can follow what we tell them to do in a lesson, but because we fail to revisit and re-practice over time, pupils’ lesson performance doesn’t turn into learning that endures in the long or even medium term. As I realised at the end of my first half-term as a teacher, this is disastrous for our pupils’ maths attainment, and thus their mathematical confidence and competency.

To improve maths results and fight off forgetting, we need to implement spaced repetition of practice throughout our teaching. I’ve suggested several methods we can do this:

  1. Increase the proportion of lesson time allocated to recap
  2. Make use of and create starter booklets to ensure spaced practice of high-leverage topics
  3. Use spaced practice to teach difficult grammatical topics over longer periods of time
  4. Create a spaced curriculum

In summary, real learning happens with focused practice, repeated constantly over time. It’s an intuitive insight – if you’ve ever learnt an instrument, or some skill, you’ll know this. The same is true of maths. Yet the way we teach maths is a bit like expecting a beginner musician to gain mastery of their instrument by learning a new piece every lesson. We must stop! And hopefully, in doing so, pupils will re-discover the music in the mathematics: the joy of playing a certain chord, again and again, the happiness found in repeating a favourite riff for a few minutes, the delight in focusing on nailing one particularly difficult phrase. I’ve seen it with my own pupils – the way they light up when they get to practise a drill; their celebrations when they get every question in the drill correct; the joy they get in knowing beyond a shadow of a doubt that they have mastered a very particular skill.

Is this the best we can do? Part 6: the problem of forgetting

JULY 26, 2017

“If you wish to forget anything on the spot, make a note that this thing is to be remembered.”

– Edgar Allen Poe

“The palest ink is better than the best memory.”

Chinese proverb

This is part of a series offering my views on some problems with UK maths education. The first part looked at the state of affairs with regards to GCSE and PISA results, the second part looked at my attempt at a diagnosis, the third part looked at pre-existing maths education success stories, the fourth part looked at how textbooks offer the largest potential to improve pedagogy across the nation, and the fifth part examined the role of hard work and homework. This part looks at the problem of memory in maths education.


The Persistence of Memory – Salvador Dali

In my career as a maths teacher, I flip regularly between deep job satisfaction and mild  despair.

Don’t get me wrong – I absolutely love my job. Most maths lessons I’m teaching pupils some sort of mathematical process. Most of the time pupils then perform that process with aplomb. Whiteboards with correct answers go up, I’m happy, the pupils are happy, I go around and mop up the remaining few pupils who weren’t quite 100%, put up some answers, and by the end of the lesson I and my pupils are feeling pretty pleased with ourselves.

But then comes the test. Even if it’s literally a few days later, and even if pupils face literally identical questions, their answers end up containing every confusion & misconception under the sun.

Whilst marking tests, I’ve often felt like Sisyphus – the mythical Greek figure condemned to roll a boulder up a mountain, watch it roll back down, and repeat forever. We strain and push our pupils up a mountain of mathematical understanding on a particular topic. After hours preparing pedagogy & teaching materials, then 60 minutes of communal graft in the lesson itself, it feels like we might have gotten somewhere with that topic. Then… fast-forward to tomorrow / the next week / the assessment, and those same pupils will literally deny having even heard of that topic before. ‘Gradient? What’s gradient?’


Yet I can’t write this solely as a complaint. After all, throughout my career, pupils have usually followed what I have told them. Lessons have gone well. I’ve explained things; pupils (seem to!) “get it”  and show they can do what I just modelled for them; we move on. So, if pupils are doing what I teach them, their forgetfulness must partly stem from a defect in my teaching. (Of course, I’m written previously on how hard work and homework plays a huge role in maths learning, so I can’t take sole responsibility for this failing: I constantly talk about the importance of independent revision, and many of my most forgetful pupils are those who give in shoddy, low-effort homework and never revise.) Nonetheless, there must be more that I can do.

The problem in a nutshell? I can’t think of a better way to put it than in Bodil’s phrase: ‘a lesson is the wrong unit of time‘. Much of our UK educational culture is focused on lessons: lesson objectives, lesson grading, showing progress within a lesson. As a result, we plan topics in lesson-sized chunks (‘addition this lesson, subtraction next lesson’), and we judge our success in lesson-sized chunks (‘the exit tickets show me they all learnt how to answer the key question this lesson’). But if we’re solely focused on lessons, then of course very little gets retained beyond that. Of course pupils forget. After all, as teachers, all we usually aim and plan for is for the pupils to ‘get it’ in the hour itself. To put the problem in reverse: we don’t teach with an eye for long-term memory; therefore pupils naturally don’t remember over the long term.

How big is this problem? And what can we do about it?

I know that my experience of mild despair isn’t unique. I’m sure that the majority of (non-selective school) maths teachers will know exactly what I mean. But in case you haven’t experienced it, choose one of your middle/bottom year 8 or year 9 classes, and set them this quick quiz:

  1. add 2 fractions
  2. multiply 2 fractions
  3. divide 2 fractions
  4. a) and b) find the area and perimeter of a triangle.
  5. a) b) and c) find the mean, median and mode of a set of 4 unordered numbers
  6. simplify 3c + 5c – 4 + 8
  7. simplify 6w × 3w.
  8. Round 3.449 to 1 significant figure.
  9. -3 + 1

The majority of these topics will have been taught again and again since Key Stage 2. These aren’t complicated questions. But if you’ve any experience teaching maths in a non-selective school, I’m sure you can smell all the inevitable misconceptions coming a mile off. My heart sinks simply in compiling this list. Despite the efforts of teachers over nearly a decade of maths education, pupils just don’t solidly remember what they’ve learnt. Therefore, it’s no surprise that UK maths numeracy is low: a 2013 OECD report found that ‘that a quarter of adults in England have the maths skills of a 10-year-old.’ Closer to the classroom, teachers experience the reality of forgetting every time a complex topic comes up. Teaching algebraic fractions to? “Well, let’s do a quick starter on mixed fraction operations… – oh dear. It looks like we’ll have to recap this…”

And this illustrates the first problem of forgetting: because we don’t teach with long-term memory as the goal, there’s a limit to pupils’ confidence in learning and using maths. For example, if pupils aren’t rock solid on the algebra in questions 6 and 7, then they won’t be comfortable in how to use algebra as a tool for modelling scenarios. If pupils aren’t rock solid on the averages in question 5, they won’t ever get how to use and understand statistics – and how not to be hoodwinked by misleading ones. Pupils will instead experience maths as a murky world of half-remembered instructions to follow – not a strong basis for becoming numerate adults.

Cognitive scientist Daniel Willingham puts the point in a more general way, explaining (as summarised by Joe Kirby):

The crucial cognitive structures of the mind are working memory, a bottleneck that is fixed, limited and easily overloaded, and long-term memory, a storehouse that is almost unlimited. So the aim of all instruction should be to improve long-term memory; if nothing has changed in long-term memory, nothing has been learned. Effective instruction is a simple equation: it minimises the overload of students’ working memories whilst maximizing the retention in their long-term memories.

Yet despite this research, and despite the consequences, the vast importance of memory is neglected. Year after year, maths teachers in staffrooms across the country will grumble that their pupils can’t remember the same fundamental skills that they’re meant to have learnt before. More worryingly, it seems that we often take forgetting for granted. I’ve seen it in comments and attitudes – my own, as well as in others around me: ‘ah, they lost loads of marks on surface area in the test; never mind, we’ll come back to it next year.

But this attitude always unnerved me: if pupils would simply come back to the topic afresh next year, why did we even bother spending the curriculum time to teach it this year? After all, the kids who understood it this year would’ve understood it if we’d left it to next year (or even the year after that). And as for the kids who didn’t understand it this year, what (besides getting a better teacher?) makes us think they’ll get lucky and understand it next year? We surely could have planned our schemes of work more effectively. This raises the second problem of forgetting: because we don’t teach with long-term memory as the goal, curriculum time is not used efficiently. This is especially concerning in light of the demands of the new GCSE, the extra content that must be taught, and the increasing complexity of questions. How are we meant to teach complex questions drawing on multiple topics, if pupils barely remember topics by themselves?

Of course, I don’t want to overstate it. Many pupils leave year 11 knowing a lot more maths than they did in year 7. In fairness to the staff room grumble, it is sometimes true that many pupils, who didn’t understand a topic one year, will find themselves grasping that topic in later years. But in my experience, this mainly applies to the pupils who’ve had a good experience of maths in primary school, who have supportive family backgrounds, and have the most highest intrinsic motivation. What about the rest?

As I’ve explored throughout this series, under the old GCSE specification there was a persistent ~40% of 16-year olds each year who couldn’t handle the minimal skills required for a C (the “good ol’ days” where grade C required nothing else but basic procedural recall). Clearly, a sizeable chunk of pupils each year have experienced futility themselves: namely, a secondary maths education where they have remembered very little mathematics. This is little surprise: the weakest pupils are those who have the most limited fluid intelligence and working memory capacity, and if they are to have any chance of succeeding in maths, they must rely heavily on their long-term memories of secure foundational mathematical knowledge to compensate for slower processing speed.

Image result for working memory long term memory

But if we don’t focus on long-term memory? Well, this is the third problem of forgetting: because we don’t teach with long-term memory as the goal, the weakest pupils are adversely affected: they forget (or even totally fail to learn) nearly everything we try to teach them.

So, why don’t we as a profession do more about helping pupils remember? Several reasons come to mind. Firstly, the vast majority of time in my experience of ITT and CPD was allocated to improving one’s day-to-day classroom practice – the craft of behaviour management, explanation, task design – together with broader subject knowledge. Notice what’s missing: in my whole training and CPD, I had a grand total of a 1 hour seminar on memory – and that was an optional seminar that I happened to choose. Aside from that, I had no training on the science of memory, or how to get pupils to remember. We don’t even seem to recognise it as a big problem.

This is reflected in the quality of advice I received early on in my career, when I told tutors and colleagues about my persistently-experienced problem of forgetting: just get pupils doing recap questions as starters, and revision lessons before tests. Yet this didn’t seem enough. Recap starters face these two problems:

  1. A bias towards recent topics. It’s easy to revisit last lesson’s topic, since it was so fresh in my memory. But topics from the previous week/fortnight/half-term? They were lucky if they came back out ever. And revisiting something just once isn’t a good recipe for remembering it in the long run.
  2. A lack of systematicity: even when I planned to revise something from a previous half-term, the widening of the time-frame made it very hard to know what to pick to revise. I would pick something that struck me as important, and we would revisit it briefly. But what of the other topics I passed over? Why choose one topic and not another?

Revision lessons before assessments also felt ineffectual: I usually had to re-teach the entire topic from scratch again as if we’d never studied it – so what was the point of studying it in the first place? – and even when I had explicitly ‘revised’ a topic with a weak class the day before their assessment, their seemed to be no consistent effect of that revision on their ability to get that question right on the following day’s assessment. This leads to the fourth problem of forgetting: because we don’t teach with long-term memory as the goal, we’re left with patchwork and ineffectual methods of getting pupils to remember.


To summarise: current maths pedagogy hardly ever focuses on teaching for long-term memory. Yet if this is not the goal of all our teaching, these consequences upon our pupils follow:

  1. It damages their mathematical foundations, having never secured them in the first place, and thus places limits on their confidence in maths and how high their mathematical understanding can go.
  2. Teachers are forced to reteach (essentially from scratch) the same topics, year after year, which is a highly inefficient use of curriculum time.
  3. The weakest pupils, as those who find remembering particularly difficult, are adversely affected and forget most of what they are ever taught in lessons – thus giving them very little hope of securing a decent mathematical education.
  4. We don’t use or even know proper methods of getting pupils to remember what they have been taught.

These are big problems. As I’ve asked repeatedly in this series, is this the best we can do? Thankfully not. As mentioned at the start, these problems are but the inevitable consequences of an education system focused on individual lessons – lesson observations, lesson grades, lesson plans. Recent years have seen some encouraging changes in this regard, particularly in the fact that Ofsted no longer grade observations. Yet unless we take advantage of this shift and implement a deeper focus on teaching for long term memory, pupil outcomes will not improve. Is this the best we can do? In terms of teaching for memory, I firmly believe no. I long to see a shift of focus towards memory in maths pedagogy; a refusal to accept ‘ah they’ll come back to it next year’; higher expectations of what we can do for all of our pupils.

How? My next post, on the spacing effect, looks at one way of improving memory.

Tabula Rasa – Yes He Can!

11 Jul 2017, Posted by admin in Michaela's Blog

Yes he can!

In my career as SENCO so far, I’ve often found myself at loggerheads with people who tell me that pupils with Special Needs can’t do things. Whether we’re talking about learning to read, writing for extended periods, or retaining focus in class, I always end up strongly disagreeing with anyone who tries to tell me that it is simply not possible for a child to do it.

This happens often, and every single time it does, I am appalled.

Sometimes, people blame the circumstances in the child’s life: “He’s got a lot going on at home, so he can’t do it.”

On other occasions, they blame the pressure the kid is under: “He’s really stressed, so he can’t do it.”

But most of the time, people use labels to explain the issue: “He has a special need, so he can’t do it.”

The people who spout these sorts of phrases seem to suggest (and in fact, sometimes explicitly assert) two maxims. Firstly, that some children don’t choose their behaviour. And secondly, that because they don’t choose their behaviour, we must never try to change it because that is unkind and uncaring.

This angers me because it imposes an obvious limit on the child’s capacity to improve. By telling schools, parents and pupils that struggling to learn or behave is a consequence of an irreparable issue within the child, we condemn the child to never being able to change.

This is precisely why, at Michaela, we believe that labels damage children. It doesn’t mean that I don’t recognise the challenges some children face, but rather I worry about the consequences of focusing purely on the ‘need’ instead of getting the behaviour we want in the long term. It is important that we at least aim for all children, regardless of need, to be able to learn and behave appropriately. If that’s not our aim, then what is?

I have met countless people who try to impose these astonishingly low expectations on the pupils at our school, and every time I do, I have to fight to keep expectations high. It seems mad to me that I have to defend the right to educate children effectively, but that is at the very heart of what it means to be SENCO at Michaela. As a school that does things differently, and that aims to achieve the highest possible outcomes for every child, I often feel as if we are fighting against a tide of negative defeatists who genuinely do no think that some children will ever be able to achieve our aims.

Tell a child that phonics won’t work for him because of his ‘needs’, and you disempower him.

Tell a child that starting a fight in the playground or swearing at a teacher was not his choice, and you undermine him.

Tell a child that he can’t concentrate for more than a few minutes, and you erode any belief he has in his potential future success.

Of course children may struggle to meet expectations (of learning and/or behaviour) at first. That’s the nature of the beast. But it doesn’t mean that, simply by virtue of the fact that the child has a label attached to them, that they will never be able to meet these standards. It just means they haven’t learnt the right behaviours yet. And the more that people come in to the school and tell me we’re wrong to do what we do, the harder it is for the child to overcome these barriers.

So if you are in my position- either as a SENCO fighting against a tide of the most despicable orthodoxy, or a classroom teacher or a Head of Department or a Literacy Lead just desperately trying to get your weakest or most vulnerable pupils to learn and behave, hear this. There is no limit to what a child can achieve if you focus on getting the behaviours you want. If he’s struggling to learn, create conditions for high amounts of practice and drill, drill, drill him until he gets it. If he’s struggling to behave, teach him how to behave with love and respect, and do not relent. As soon as you excuse his behaviour or his underachievement, you let him down because you are implicitly telling him that he cannot do it.

And he can. He really, really can.

Until I Know Better – Maths Muddle

03 Jul 2017, Posted by admin in Michaela's Blog

Maths Muddle: alternate segment, same segment, alternate angles

I had an insight into Y9 minds this week, realising that two (maybe three) things I thought were obviously different look really similar to them.

It came about with similarity proofs, and realising that, to them, it is really hard to tell when angles are the same because the initial diagram looks like similar (sorry) diagrams that don’t have the same properties.

angles 1.png

angles 2.png      angles 3.png

I’ve spent some time now with them practising what is the same, and what is different, for A and D in particular, but also contrasting with B and E to avoid them thinking we can never have alternate angles (i.e. Z-angles) when angles are subtending the same arc.

I realised an additional point of confusion is that there are three things that sound similar, and that adds to the difficulty in remembering which is which, and when it is ‘allowed’:

  • alternate angles are equal
  • the angle in the alternate segment is equal (to the angle between a tangent and the chord)
  • angles in the same segment are equal

It made me realise it isn’t enough to recognise the diagram, but also to be able to state clearly the conditions for the relationship:

  • alternate angles: looking for a Z-shape and parallel lines (too often we don’t say that bit because we assume it is obvious)
  • angles in the alternate segment: there needs to be a circle (!!!), a tangent, a chord, and an angle in the alternate segment subtending that chord
  • angles in the same segment: again, there needs to be a circle (!), and the angles need to be subtending the same arc

Focusing on being able to recall and identify these ‘necessary conditions’ has helped, although I can see it is really adding to the pressure for them, as things that appeared simple now feel much more technical and challenging. So, it feels like a hit with buy-in but will pay off once they are competent with it as they will be so much more fluent and confident.

[C and F haven’t been as big an issue, but were a while ago when learning the difference between angles subtending the same arc in the same segment, and angles in the centre compared to those at the circumference (i.e. using only the conventional ‘arrowhead’ for angles at the centre leaves them thinking that it can never have the ‘wonky bowtie’ look that they associate with angles on the same arc).]

Hopefully you can learn from my mistake and avoid this confusion for your kiddos!

#Mathsconf10: Worksheet-Making Extravaganza

On Saturday, I spoke at #mathconf10 in Dagenham. The workshop was titled “Worksheet-making Extravaganza!” Indeed it was that.

When I started teaching I regularly went onto TES to download resources for my pupils. Usually, I would be annoyed that I couldn’t find a good enough worksheet because the worksheet I wanted would never be available. I then realised that I needed to start creating my own. However, in the process I realised that I found making problem types for different topics to be difficult because my subject knowledge wasn’t up to scratch. Now, I am very good at maths but that doesn’t mean that I am good at making a worksheet of questions for pupils to do and learn from. Then after spending a summer between my first and second year of teaching where I made different resources combined with my own experience of working at Michaela Community School I finally realised how to make my own worksheet. Here are the two things that make a good worksheet:

  • High quality content
  • High quality structure (deliberate practice)

What do I mean by high quality content? I am referring to the questions that are made to test a pupils’ understanding of the concept or procedure that has been taught. What is the starting point of gathering this content? If you are planning a lesson on how to simplify fractions then list out all the problem types from the easiest to the most difficult one. More problem types can be found by looking at different textbooks, asking colleagues and reading blogs etc. Let’s look at the problem types that can be included when planning a lesson on simplifying fractions, simplify a:

  • proper fraction
  • (an) improper fractionImage 1
  • fraction that simplifies to a unit fraction
  • fraction that simplifies to an integer
  • mixed number
  • fraction with large numbers (still divisible by a common factor)
  • fraction that simplifies to 1

It is all well and good gathering a bunch of problem types but I have deliberately chosen the ones I have listed. This is because these problem types present four features that make the content of worksheet high quality:

  • Arithmetic complexity
  • Visual complexity
  • Multiple steps
  • Decoding

Arithmetic complexity basically means including more difficult numbers or large numbers in your questions. Can a pupil simplify one hundred and eleven-thirds? Also, it means including questions which have decimals and fractions, and creating questions where the answer is a decimal or fraction as well.

Image 1aVisual complexity refers to creating questions which look a bit scary. Again, this can entail including large numbers to make a pupil think. For example, in terms of calculating the area of a square you can write a question where the length is a large two digit number or a decimal.Image 2Another example is to only label one length of a square, more than two lengths, four length etc. A question like this is testing whether a pupil can calculate the area of a square from squaring one length. Another very simple approach to make this type of question visually complex is to have the image at a certain orientation.Image 3Multiple steps is a feature in a question where you have increased the number of steps between the question and the answer. This can include having calculations in the numerator and denominator before a pupil is asked to simplify a fraction.

Image 5Decoding is a feature of a question where its set up is testing whether a pupil can rearrange or manipulate their new found information of the concept or procedure that has been taught. For example, providing pupils with questions which have incomplete answers. Or asking if the following equations are true or false?

Image 4Image 6An exercise on a worksheet can look something like this. Question (g) is testing a pupil’s misconception where dividing a number by itself is not 0 but 1. Similarly, question (k) is testing a system 1 mistake where a pupil is not fully thinking about the question, they simply see that the half of 18 is 9 therefore the answer is 9.

Image 7Can I apply these four features of high quality content when planning a series of lessons where I want pupils to learn how to add and subtract fractions with like denominators? Again, start with listing out the potential problem types. Adding or subtracting:

  • Two proper fractions where the result is less than 1
  • A proper and an improper fraction
  • Two proper fractions where the result is greater than 1
  • Two mixed numbers where the sum of the proper fractions is less than 1
  • Two mixed numbers where the sum of the proper fractions is greater than 1
  • Two mixed numbers where the sum of the proper fractions equals to 1

Image 8

I can make these type of question arithmetically complex by having large numbers in the numerator and the denominator. I can include a question when I subtract two proper fractions and the result is negative.

Image 9Questions can be visually complex by including more than two terms, or a string of fractions including fractions, mixed numbers and different operations.

Image 11Similarly, making questions where pupils are asked to add or subtract a proper fraction with a mixed number can be visually complex too.

Image 10Questions where pupils have to rearrange their information of adding and subtracting fractions with like denominators can include true or false question. The first question below is conflating adding a numerator and multiplying the denominator.

Image 7bSimilarly, it is a really powerful form of testing pupils’ knowledge by including questions where the answer is incomplete. Can the pupils identify what the numerator is? Can pupils identify that 1 can be written as 10/10 and that 2 can be written as 20/10.

Image 12The following questions demonstrate the feature of questions having multiple steps between the question and the answer. Yes, you can easily write the answer, but if you wanted a pupil to write their answer as an improper fraction, you would want them to write the integer as a fraction.

Image 13Lastly, can we apply the following features to creating high quality content for the topic of powers and roots with fractions. Here are the problem types:

Image 14

Image 16

Image 17The mixed number examples are tailored so pupils are able to create an improper fraction where the numerator is still less than 15. Why? Simply because the kids at Michaela have committed the first 15 square numbers to memory as well as the first 10 cube numbers, but if I made the fraction more difficult where the numerator was greater than 21 then I would be making the question difficult for the wrong reasons. If the fraction was four and one-fifth then the numerator would be 21, and squaring 21 isn’t valuable. What is valuable is spotting the first step of writing the mixed number as a fraction, and then squaring the fraction. The mixed number examples demonstrate questions being visually complex as well as arithmetically complex.

Image 15The following question is demonstrating questions which have multiple steps to the answer, where calculations in the numerator and the denominator need to be simplified before squaring.  Again, calculations were designed where the numerator and denominator values would be less than 15.

Image 18The same form of thinking in applying the three features (arithmetic complexity, visual complexity and multiple steps) can be applied when fractions are being square rooted or cube rooted. However, ensuring that the numerator and denominator values are numbers that can be square or cube rooted.

What is new is including negative fractions when they are being rooted by an odd number. Furthermore, visual complexity as well as multiple steps to be performed to get the answer make these questions challenging and rigorous.

In summary, the four features which make a worksheet consist of high quality content is creating all problem types where they are arithmetically complex, visually complex, require decoding of current information from instruction and multiple steps to be performed to result in the answer.

I am still trying to wrap my head around creating a worksheet with high quality structure which will be my next blog post.

UK Maths Challenge

On Friday, I spent a couple of hours with a pupil in Y8 who has been selected to go through to the Junior Kangaroo challenge. Over half term, he was given past papers to attempt. We went through his answers and we discussed the questions he found difficult to do. In the process of going through some questions I realised how important knowing specific factual and procedural mathematical knowledge is crucial to a pupil’s success in the UK Maths challenge.

I am a huge fan of the UK Maths challenge. Why? Simply because each question is created to test ‘powerful’ knowledge. Daisy Christodoulou and Michaela Young refer to ‘powerful’ knowledge as certain forms of knowledge which allows humanity (or in this context specifically pupils) to advance in some way to communicate more accurately. This blog will include questions which explore the ‘powerful’ knowledge that the Maths challenge tests, and how teaching such knowledge allows pupils to be successfully in the  challenge.

Here is a list of the topics that I am referring to as ‘powerful knowledge’ that allows pupils to become incredibly flexible in applying their knowledge to such questions:

  • Divisibility Tests
  • Prime factor
  • Identifying Square numbers
  • Angle facts (vertically opposite, interior angles of regular polygons)
  • Common Fraction/Decimal/Percentage facts 3/8 à375 à 37.5%
  • Double the radius for the diameter, Halving the diameter for the radius
  • Formulas for the area of a rectangle, triangle, trapezium
  • Algebraic expressions for the area and perimeter of a rectangle/square e.g. 2(a + b)

Procedural knowledge:

  • Decimal multiplication
  • Order of Operations (GEMS and BIDMAS)
  • Applying exponents and roots
  • Four Operations applied when using fractions
  • Doubling and halving
  • Formula manipulation
  • Forming and Solving Expressions

Each UKMT question requires pupils to have basic knowledge facts at the fore front of their mind e.g. a square number has an odd number of factors, the first 6 rows of pascal’s triangle etc. More importantly, it is the interleaving of different concepts in each question which is why I think the questions are intelligently designed.

If pupils are quick in identifying the completing basic procedures within the following topics they can apply that knowledge to a wide selection of UKMT questions. Here are a few examples of questions which evidence this. The questions below also demonstrate how pupils’ knowledge of multiple concepts are being tested too:


Never Let Me Go

At which stage in a topic do you stop modelling examples? I’m going to go out on a limb and suggest the answer is probably….too soon. I’m going to go further and suggest that the rationale you have for doing so is probably wrong.

Here are examples of conversations I’ve had, or resources I’ve seen shared, where what was suggested sounded superficially sensible until a moment’s reflection made me think they had it completely backwards.

  1. (paraphrasing) I’ve prepared the lesson on multiplying proper fractions, and set them an extension homework on doing it with mixed numbers to really stretch them, since it’s set 1. 
  2. (paraphrasing) I’m going to model how to expand and simplify (a+2)(a+3) and (b+3)^2 and (a-5)(a-7). They need to be able to do it with coefficients in front of the variable as well, and when the order or variables are different, such as a (3+a)(5-b), or three terms in one of the brackets, but I don’t want to show them too much so I’ll put those ones in the worksheet. 
  3. This sequence of tasks in a lesson on TES (I’m not going to say the creator as it is obviously an act of generosity on their part to have shared and I am not seeking to ridicule them). First the example they planned for sharing with the class:

enlargements - modelled example

The tasks for the pupils to undertake following this example:

enlargements - worksheet part Aenlargements - worksheet part Benlargements - worksheet part C

I don’t know the author for the TES resources, but I can certainly attest that the first two were intelligent colleagues who were making those judgements as a result of serious thought.

They are decisions I would have taken earlier in my career as they fit with not wanting to spoon-feed pupils and I wanting them to enjoy a few head-scratching moments. Both are laudable aims and are important considerations. It is boring for pupils if they never have to think, and it makes it less likely that they’ll remember what you teach them.

With the benefit of (some) experience, the examples above now strike me as badly misguided. They reflect a common problem in maths instruction: we model examples that are too easy, then leave pupils to ‘discover’ methods for the harder ones. I think this has several causes:

  1. Modelling seems like a boring thing to do. They are just sitting and (hopefully) listening. Pupils will get bored and restless.
  2. Modelling takes ages, especially if you’re checking they understand as you go and are asking questions to get them to tell you the next step, and so on. If you modelled every example, they’d never get any practice, surely!
  3. We want to feel that they’re getting a chance to figure things out for themselves and really enjoy maths (i.e. feel like mathematicians, trying out strategies and reflecting on them).
  4. It feels a bit cold and exam-factory-ish to explicitly model every form a procedure or problem can take. Where is the room for surprise?
  5. They only need a basic grasp of it, they’ll get a chance to look at the harder ones again in Y9/10/11, right?
  6. Homework should be interesting and new. Researching and learning about more advanced or complex versions of a problem or procedure is way more interesting than more practice of what they’ve already done.
  7. (I suspect this is truer than we’d like to admit) The difficult examples are really difficult to model. We’re not sure how we know how to do them, they’re just….you just can see it, right?


Here are some counter-arguments:

  1. Modelling isn’t thrilling, but it is the most efficient and effective way to get pupils to be able to do procedures. Procedural mimicry isn’t the goal of our instruction, but it is a crucial foundation. It is also the bulk of what we are doing in secondary. It is almost impossible to get a (typical) child to understand a concept when they have no procedural fluency with the topic.
  2. Modelling wouldn’t take so long if you didn’t ask so many flipping questions during it. We do so much talking that isn’t simple, concise narration during an example that pupils – understandably – think procedures are much longer and more complex than they actually are. Think about how long it takes you to solve ‘4 + 5a = 17’ if you aren’t showing it to anyone. I assume 5-20 seconds (since you’re a teacher…), maybe 30 if you show every line of working. They should see this! Taking several minutes to model one example sets them up to think it is fine to think they can spend more than 2 minutes per question on questions that are, frankly, trivial.
  3. Letting them have head-scratching moments is fine, but it should be in response to a question that synthesises what they have been taught, and doesn’t require them to invent new knowledge (well, new to them). Leaving the hardest content for them to try on their own is just crazy: it disadvantages the weakest pupils and those who haven’t had good teachers on this topic in the past, it makes ‘giving it a go’ seem incredibly hard and adds to the perception that teachers are just refusing to tell them how to do things to be annoying. They should try hard problems on their own, but you should have first equipped them with requisite techniques and knowledge.
  4. You should still show them the elements of surprise in maths. When showing them how to form algebraic proofs, we first look at the propositions with real numbers and form conjectures (e.g. for four consecutive integers, what is the relationship between the product of the 1st and 3rd and the square of the 2nd? Does this always happen? How will we prove it?). They should get hooked in, and make conjectures, and experience the pleasing smugness of having applied what they already know…BUT YOU SHOULD ALSO SHOW THEM, REALLY CLEARLY. Ten may have figured it out before you finished modelling it, but twenty didn’t, and they’re relying on you. Well-designed questions in their independent work will still force them to think and synthesise what they know and test methods, but won’t ask them to stab in the dark with no sense of if they are right or wrong. We tend to overestimate how much pupils can tell if their work is correct or  not.
  5. Lucky Y9/10/11 teacher. This increases the gap they need to bridge as they approach the end of Y11, and also means that they have simplistic ideas around a topic (and probably many misconceptions). They need to see the complex versions, and grapple with them, to really grasp a topic. Once you have taught them to reflect in the line y = x, reflecting a simple shape in a given line seems like a breeze. Often the fastest route to mastering the basics is tackling the tough stuff (at least, that seems to be true for my weaker classes!).
  6. What a way to make homework seem annoying and pointless. If they struggled in class, there is no way they will be able to do the homework. If they got it in class, they MIGHT be able to do the homework…but probably only if there is an older person in the house who can help them, or a really good CorbettMaths video on exactly that procedure (so it will only work for procedural work anyway). I’m all for letting the most able fly ahead, but it shouldn’t be so cynical as this. I cannot see the logic of presenting the most challenging work at the point of lowest support (i.e. you’re not there, and they are tired and probably rushing it). It also makes you seem like a rubbish teacher: “I can’t do it because she didn’t teach it to me.” They wouldn’t be wrong for thinking this.
  7.  Explaining how to enlarge a fiddly shape by a negative scale factor is annoying and difficult. Proving congruence with annoying parallel lines is really difficult to do, let alone explain in a clear way. I just ‘see’ the answer to those long locus questions and feel like if they just read the question more carefully then maybe…maybe they’d just kind of realise? If we struggle to explain it, and are relying on our instincts to know what to do, then goodness help them trying to do it with no guidance or instruction from us.


What we should be doing instead:

  1. Think of all the variations a problem or procedure can have, and plan examples that cover all these eventualities.
  2. Pair every modelled example with a highly similar one for pupils to do. This holds them accountable for listening to, and thinking about, what you are showing. It also gives you feedback on if that small segment of instruction made sense to them.
  3. Make examples challenging. Yes, start with (a+4)(a+6) to help them see the simplicity of a grid. Unless you haven’t secured the basics with multiplying and simplifying, you should be able to then show (2a+5)(4+3b) and then an example with three terms in one of the brackets. I was struck by this in a talk at MathsConf last year, where teachers who had visited Shanghai were showing the kind of examples that teachers there used with pupils. They were doing LOADS of examples, but briskly and covering a wide range of applications, which meant that pupils were in a better position to see the common threads and critical differences, and not jump to incorrect conclusions. I shudder to think how many pupils in the UK think that the terms in ‘the diagonal’ in a grid for double brackets must be the ones ‘that go together’, having only seen examples in the form (x+a)(x+b).
  4. Just model it! Make it quick, make it concise. Something that works well for my weakest pupils is to have a silent model (i.e. they see the physical motions to enlarge a shape, or expand two brackets, or do the same to both sides), then one which I narrate (no questions or clarifications), then one where I check that they can mimic (i.e. they do a very similar one on whiteboards so I can see if they can recall and follow the steps – this is much more important than them being able to articulate the steps…at least at the early stage).
  5. Start at the basics, but move as fast as you can to the harder content, and you will reap rewards. They will also ask much more interesting questions and feel smart. You’re also not saddling their future teacher with too much to get through.
  6. Just use homework to practise and consolidate things they already know how to do. Set homework where you can expect 100% of pupils can get close to 100% (provided they are putting effort in). Otherwise you are setting up a grey area where you can’t tell if non-compliance is your fault or theirs, and you have lost the homework battle. You also save class time by not practising things where they need your help less.
  7. The harder it is to explain, the greater the imperative that we spend time together planning and refining (and ideally practising) those explanations. The return on planning and rehearsing complex examples is huge: it will illuminate the key features for simpler cases, help you see the crucial things they must do from the start, and also streamline your language for easier questions. Pupils always rate highly a maths teacher who can explain well, and lose faith in those who can’t. If you can’t explain it well, they’ll think the problem is that they are stupid. That is a tragedy.