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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…

https://www.tes.com/jobs/vacancy/french-teacher-london-527575

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:

CONNECTING MATH CONCEPTS

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.

Keppel

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.

Summary

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.

dali

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?’

sisyphus-image-01c

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.

Summary

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.

What Makes Us Happy

At Michaela, we aren’t only focused on our kids working hard, learning loads and achieving great grades. We also want them to be happy. We think long and hard about how we can foster our kids’ happiness, and, having read wisdom literature, there are three strands to building a happier life that we focus on: gratitude, personal responsibility and duty.

Gratitude

In a TED talk, David Steindel-Rast says: ‘It is not happiness that makes us grateful. It’s gratefulness that makes us happy.’ As adults, we have come to find that it is better to give than to receive. Wisdom literature tells us that the more grateful we are, the happier we are; that is why all religions set aside prayers to thank God for what we are given. In secular life, we know that counting our blessings helps us to focus on what we haverather than what we lack, something an increasingly materialistic society wishes us to dwell on.

At Michaela, we know that kids are not naturally grateful – in fact, it is often the opposite. They are surrounded by images of acquisition with constant advertising and MTV-style programmes of the rich and famous, not to mention social media allowing the kids to see the extraordinarily luxurious lives of their idols.

To foster gratitude, we have daily appreciations at lunchtime. The kids take it in turns to say who they are grateful to that day, and for what. We choose ten to twenty children each day to stand up and say, in front of the whole lunchroom (some 180 children and adults) who they are grateful to and why. You can see an example of this in action in this video at around 1:20.

A couple of times a term, tutors also allow pupils the opportunity to write postcards. The pupils are encouraged to write their thanks to a teacher or member of staff who has really helped them. For teachers, there is nothing better than finding a handful of postcards in our pigeon holes with messages of gratitude. We teachers also remember to be grateful to our wonderful kids: every week, every teacher writes three to ten postcards to members of their classes who have made a real effort or really impressed us that week.

Personal responsibility

When we believe events to be outside our control, we find life endlessly frustrating. We are stymied in our efforts by continually thinking: if only this wasn’t the case! If only this had happened instead! It is a frustrating way to live, because we cannot control the actions of others. If we focus on how others behave, we will often find ourselves at a dead end, unable to reach our goals.

If, on the other hand, we focus on our own locus of control, we feel much more content. If we are unhappy with how something has gone, instead of blaming the wider world, we instead focus our attention on the things we could have done differently.

So, for example, if a pupil finds themself in a detention they perceive as unfair, we have a conversation with them to talk them through to taking responsibility.

‘But I was whispering because the boy next to me was whispering to me first! Why did I get the detention but he didn’t?’ This reaction leads to anger and resentment. The children feel unhappy that a punishment seems to have been given unfairly. It is hard for kids to understand that teachers are never omniscient, and we can’t catch every infraction. We turn this conversation around by first emphasising that we have strict rules and detentions so that the children can have a peaceful and calm environment to learn in. We then refocus the child on what they can do differently. What can they do next time to change their consequences? Not talk back. If someone whispers to them, they cannot control that person’s actions but they can control their own, and how they respond. Taking responsibility is always better than blaming others. As adults, we know that the best people take the blame and are honest when they get something wrong. As adults, we feel happier when we know we can change things for the future. We want our kids to learn that lesson too.

Duty

Modern Western society encourages us to think about our dreams: what do we dream of doing with our lives? What do we dream of achieving? What do we dream of having? Yet focusing on ourselves and what we hope to get, acquire, achieve is not the route to happiness. Inevitably, things will happen in all of our lives to prevent us from ever enjoying the ideal life.

If, instead, we focus on our duty, we can find a surer route to happiness. Duty means finding your contribution. It is not what you can get from society that we should focus on, but what you can give to society.

This was a lesson I learned over many years. I was always very ambitious and very self-focused, never satisfied with what I was achieving, and never just enjoying life for what it was. In taking a demotion to join Michaela, I hugely struggled with my ambitious side. Yet over time I realised that if I focused on my duty instead of my ego, I would be happier. My ego wanted to feel important; but my duty was to contribute to what I firmly believe is a ground-breaking, paradigm-shifting school. To focus on position is frustrating; to focus on what you can contribute to a movement is exciting.

Similarly, we often talk to our kids about legacy: what legacy do you want to leave in the world? What will your contribution be? It can’t be about how much money they want to earn, or how many accolades they wish to be given. Instead, if they focus on what they can give and contribute they will lead happier, more fulfilling lives.

Three Assessment Butterflies

Winston Churchill once said ‘success is stumbling from failure to failure without losing enthusiasm.’

Looking back now on assessment in our first year at Michaela, I can now see what I was blind to then: we stumbled and blundered. What mistakes did we make, and how did we stumble?

We spent hours marking. We spent ages inputting data. And we didn’t design assessments cumulatively.

  1. Marking

First mistake: we spent exorbitant amounts of time in the first year marking, in particular marking English and History essays and paragraphs. We wrote comments, we set targets, we tried individualised icons, we corrected misspellings, we corrected grammatical errors, we judged and scored written accuracy, we wrote and shared rubrics with pupils. We spent hours every week on this. Over the year, we must have spent hundreds of hours on it.

The hidden pitfall of marking is opportunity cost. Every hour that a teacher spends marking is an hour they can’t spend on renewable resourcing: resourcing that endures for years. Marking a book is useful for one pupil once only: creating a knowledge organiser is useful for every pupil (and every teacher) that ever uses it at again. Marking is a hornet. Hornets are high-effort, low-impact; butterflies are high-impact, low-effort. Knowledge organisers are a butterfly; marking is a hornet. We had been blind to just how badly the hornet’s nest of marking was stinging us. So we cut marking altogether and now no longer mark at all.

MarkingOrganisers.png

  1. Data

Our second mistake: we spent far too much time in the first few years on data input. We typed in multiple scores for pupils that we didn’t use. Preoccupied by progress, we thought we needed as many numbers as we could get our hands on. But the simplistic equation of ‘more data, better progress’ didn’t hold up under scrutiny. Every teacher typed in multiple scores for each assessment, which were then collated so we could analyse the breakdowns. We were deluged in data, but thirsting for insight. There was far too much data to possibly act on. My muddled thinking left us mired in mediocrity, and we had invested 100s of hours for little long-term impact.

What we realised is this: data must serve teachers, rather than teachers serving data. Our axiom now is that we must only collect data that we use. There’s no point in drowning in data, or killing ourselves to input data that we don’t use.

  1. Design

Our third mistake was this: we had forgotten about forgetting. We designed end-of-unit assessments that tested what pupils had only just learned, and then congratulated ourselves when they did well whilst it was very fresh in the memory. We had pupils write essays just after they had finished the unit. We coached them to superb performances – but they were performances that they would not be able to repeat on that text in English or that period of History even a few weeks later. Certainly, months later, they wouldn’t stand a chance. Just as if you asked me to retake for Physics GCSE tomorrow, I would flunk it badly, so just one year on, our pupils would flunk the exact assessment that they had aced one year earlier.

Looking back with hindsight, these three mistakes – on marking, data and design – helped us realise our two great blind spots in assessment: workload and memory. We didn’t design our assessments with pupils’ memory and teachers’ workload in mind.

We were creating unnecessary and unhelpful workload for teachers that prevented them focusing on what matters most. Marking and data were meant to improve teaching and assessment, but assessment and teaching and had ended up being inhibited by them.

We were forgetting just how much our pupils were forgetting. Forgetting is a huge problem amongst pupils and a huge blind spot in teaching. If pupils have forgotten the Shakespeare play they were studying last year, can they really be said to have learned it properly? What if they can’t remember the causes or course of the war they studied last year in history? Learning is for nothing if it’s all forgotten.

The Battle of the Bridge

Assessment is the bridge between teaching and learning. There’s always a teaching-learning gap. Just because we’ve taught it, it doesn’t mean pupils have learned it. The best teachers close the teaching-learning gap so that their pupils learn – and remember rather than forget – what they are being taught. We’ve found the idea of assessment as a bridge to be a useful analogy for curriculum and exam design. Once you see assessment as a bridge, you can begin to ask new questions that generate new insights: what principles in teaching are equivalent to the laws of physics that underpin the engineering and construction of the bridge? How can we design and create a bridge that is built to endure? How can we create an assessment model that bridges the teaching-learning gap?

We’ve found 3 assessment solutions that have exciting potential. Here are the reasons I’m excited about them:

They have absolutely no cost.

They are low-effort for staff to create.

They have high impact on pupils’ learning.

They are not tech-dependent at all.

They are based on decades of scientific research.

They can be immediately implemented by any teacher on Monday morning.

They have stood the test of time at Michaela over the last three years.

I anticipate we’ll still be using them in three, six and even ten years’ time, and beyond.

In short: no cost, low effort, high impact, research-based, long-term solutions.

Three of the most effective assessment tools we’ve found for closing the teaching-learning gap are daily recaps, weekly quizzes and knowledge exams.

Over 100 years of scientific research evidence suggests that the testing effect has powerful impact on remembering and forgetting. If pupils are to remember and learn what we teach them in the subject curriculum, assessment must be cumulative and revisit curriculum content. The teaching-learning gap gets worse if pupils forget what they’ve learned. As cognitive science has shown, ‘if nothing has been retained in long-term memory, nothing has been learned’. Assessment, by ensuring pupils revisit what they’re learning, can help ensure they remember it.

Pupils forget very swiftly. We use daily recaps, weekly quizzes and biannual knowledge exams to boost pupils’ long-term memory retention and prevent forgetting.

  1. Daily recaps

Daily recaps are a butterfly: low-effort, high-impact. Departments create recap questions for every single lesson. Every single lesson starts with a recap. They are easy to resource. They consolidate pupils’ learning so they don’t forget. Every day they spend up to 20 minutes in each lesson applying what they’ve learned before. In English, for example, we spend those 20 minutes on grammar recaps, spelling recaps, vocabulary recaps, literature recaps (with questions on characters, themes, plots, devices and context). We do recaps on the unit they have been studying over the last few weeks. We do recaps on the previous unit and previous year’s units. This daily habit builds very strong retention and motivation: pupils feel motivated because they see how much they are remembering and how much more they are learning than ever before. All recaps are open questions, and weaker forms might be given clues. The recaps are always written; they are no-stakes, without any data being collected; they give instant feedback, as they are swiftly marked, corrected and improved by pupils themselves. We’ve ask pupils after: ‘hands up who got 4 out of 5? Hands up who got 5 out of 5, 100%?’ Pupils achieving 100% feel successful and motivated to work hard to revise.

 

  1. Weekly Quizzes

Weekly quizzes are a butterfly: low-effort on workload, high-impact on learning. Departments create quiz questions for every week in the school year. Every week there is a quiz in every subject. They are easy to resource. They challenge and test pupils’ understanding. They are mastery tests, where most pupils should be able to achieve a strong result.

We have dramatically, decisively simplified how teachers score them. Instead of marking every single question laboriously, teachers simply sort them into piles. They make swift judgement calls about whether each pupil’s quiz is a pass, excellent, or fail. Each judgement is a simple scan of the pupil’s quiz paper and a decision as to which of the three piles it should be in. Accuracy isn’t perfect, but nor does it need to be: there are diminishing returns to perfecting accuracy.

The data is then inputted in 30 seconds into a beautifully simple tracker. Any pupil failing often is red-flagged, so teachers can focus in lessons on pupils who are struggling. And that is the only data point that our teachers have to keep in mind: which pupils are struggling most?

 

  1. Knowledge Exams

Knowledge exams are another butterfly – high impact, low effort. What I love about our knowledge exams is that they are cumulative, so that pupils revise and remember what they’ve learned. We have exam weeks twice yearly, in January and July (not half-termly). We set GCSE-style exams for depth, and we set knowledge exams to test a much fuller breadth of the knowledge pupils have learned. Knowledge exams are 35-question exams that take 60 minutes to complete. They are beautifully simple: they are organised onto 1 sheet of A4 paper, and they can be answered by pupils on one double-sided piece of A4. The breadth we can achieve with these exams is staggering. By Year 9, we have 3 knowledge exams in History, Religion, Science and English alone; they organise 35 questions on what pupils learned in Year 7 and 35 questions on what pupils learned in Year 8, centred on those years’ knowledge organisers. Twice a year, pupils are challenged to revise and remember what they’ve learned over all the years they spent in secondary school. This means they answer 12 knowledge exams – over 400 questions in total across 4 subjects. I am willing to bet that many of our teachers could not beat even our Year 7 pupils on these exams across all subjects! Imagine more than 24 sides of A4 packed with answers from every pupil in the school. The humble knowledge exam is a great catcher of knowledge.

As for marking them? We simply sort them into three piles: excellent, pass and fail. We don’t even record the marks. Teachers just note the names of pupils who failed multiple knowledge exams so we know who’s struggled.

Knowledge exams solve the breadth-depth tradeoff in exams. They give pupils maximum practice with minimum marking burden on teachers.

3AssmntButterflies.png

Simplicity must cut through assessment complexity. We should practise what we preach on cognitive overload for teachers as well as pupils. Assessment resources must be renewable, replicable, sustainable, scalable, enduring, long-term.

And the impact of recaps, quizzes and knowledge exams? Well, it’s very early days yet, but we’ve had some (very weak) Y8 or Y9 pupils miss an entire term though unavoidable long-term illness, only to return fully remembering what they’ve been taught the previous term and previous year. It’s an early indicator that the assessment strategy is bridging the teaching-learning gap and overcoming the savage forgetting curve. The real test of its impact will be GCSE results in 2019, A-level results in 2021 and University access and graduation beyond.

Blind, still

The two blind spots we’ve discovered – memory and workload – provide us with ways of interrogating our teaching and assessment practice:

  • How much are pupils remembering?
  • Where are they forgetting?
  • Where are teachers overloaded?

And I still think that we at Michaela can do more and find better ways of creating assessments with memory and workload in mind. I’m sure our pupils are not yet remembering as much as we’d like them to. I had a conversation with Jonny Porter, our Head of Humanities, just this week, about ramping up the previous-unit daily recaps we do. In this sense, even at Michaela we still feel blind on the blind spot of memory – pupils are still forgetting some of what we are teaching, and we want them to remember what they are learning for the very long-term. Our ambition is that they have learned what we’ve taught for years to come: for five, ten, twenty years.

Every day, teachers and pupils at Michaela see Churchill’s words on the wall: ‘success is never final; failure never fatal; it’s the courage that counts.’ It takes courage to radically simplify assessment – and courage to continually confront our workload and memory blind spots.

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.

Teach Knowledge

What can you do when you inherit a clueless year 10 or 11 class? Teach them the test. It’s something that teachers who join struggling schools know well. Principals who are drafted in to turn around failing schools are not fools to throw their resources at year 11 intervention, a.k.a., teaching to the test. What is the alternative?

But kids at private schools and grammar schools don’t do better on these tests because they were drilled better in exam technique. They don’t even do better because their teachers are better paid, or better qualified, or their schools have bigger, better buildings. They do better on the tests because they have deep subject knowledge, built up incrementally over a great number of years, often beginning in the cradle with a loving parent’s reading aloud each night.

In state schools, we have, for too long, been teaching skills and neglecting knowledge. In English, we have taught any novel, or any poem, thinking that the thing that is important is the ‘skill’: of reading, of inferring, of analysing. And yet, novel finished, what have the children learned? Daniel Willingham says that memory is ‘the residue of thought.’ The problem with skills-based lessons is that they don’t require thinking about anything you can commit to memory. Nothing is learned because nothing is being remembered. Over years and years of skills-based teaching, children aren’t actually learning anything. They are simply practising some skills in a near vacuum.

And yet, when it comes to the exams, we all know what to do: we teach them the test. We don’t like knowledge, but we’ll drill children in quotations and PEE and techniques used in key poems. We’ll drill kids in how long to spend on each question and how many marks are available. We’ll drill kids on the key words in each question (‘bafflingly, when AQA says “structure”, what they actually mean is…’). And then we will complain that we have to teach to the test.

I say ‘we,’ because I am equally culpable. Before joining Michaela, I could not see an alternative way of teaching English. Surely it was all about the skills! Who cared whenOliver Twist was written or what the characters’ names were? The kids could look that stuff up! What mattered was their ideas about the text!

We hugely underestimate how vital knowledge is. Skills-teachers across the land cannot work out why their kids cannot improve their inferences, cannot improve their analysis. Why can’t their ideas about the text just be a bit, well, better?

The kids’ ideas can’t be better because they don’t know enough. We don’t think it matters whether they learn chronology, but we forget that it is not obvious to children that Dickens is a Victorian. It is not obvious to children that Shakespeare is an Elizabethan. It is not obvious to children that the Elizabethans pre-date the Victorians. They simply do not knowthis.

The children who grow up being taught facts and knowledge will thrive in their national exams. They will use all their background knowledge and cultural literacy to deliver deft insights in glorious prose, and sweep up the top grades with ease. The children taught through skills will improve slowly, painfully, and nowhere near fast enough to compete. They will endure two years of teaching to the test and lose any love of learning they might have gleaned in the previous years.

Is there another way? Of course: teach a knowledge-based curriculum from the very start. Stop giving the rich kids a head start.

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:

UKMT 1UKMT 2UKMT 3UKMT 4UKMT 5UKMT 6UKMT 7UKMT 8UKMT 9UKMT 10