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Michaela’s Blog

Arithmetic Average: Reflecting on Property 5

In my last post, I discussed whether an instructional programme could be created to teach a deeper concept of mean. I mentioned five different properties.

After a conversation with Kris Boulton about the blog post, he spotted something interesting about property 5.

Property 5: The average is only influenced by values other than the average.

At the end of my previous post, I explained property 5. It is because when we add the average value to the data set then the resulting sum is divisible by the mean. When we add a value that is not the previous average, the resulting sum is not divisible by the new denominator. However, the second sentence is incorrect.

Image 1

If I do not add the average value of the original data set, I can still have a resulting sum which is divisible by the new denominator. The third example shows this.

Example 1: Original data set with a mean of 5.

Example 2: Add the average value to the new data set – resulting sum is divisible by the new denominator. The mean has not changed.

Example 3: Not adding the average value to the new data set – resulting sum isstill divisible by the new denominator. The mean has changed.

Is there another way to communicate this property?

Image 2

This is an idea from Kris which he discussed as a potential second attempt to communicate property 5. It communicates property 5 nicely. It also has non examples too. If I do not add the average value then the mean is not 5.  Below is another example that can go along with the example above because the average value (19) is not a number in the original data set. Pupils are then seeing an example where the average value can be present in the data set, and where the average value will not be a number in the data set.

Image 3

 

I have added the average value to the original data set.

The mean has not changed.

 

 I added the average value twice to the original data set.

The mean has not changed.

 

I have not added the average value to the original data set.

The mean has changed.

I think both examples are important to use because they communicate the same property. The second example communicates the message in a more explicit fashion. The first example should be included because it is a nuanced example compared to the second. This may or may not be that important, but telling a pupil that the average value can be a number in the original data set is knowledge a pupil should know.

I wanted to convey an example where the procedure of dividing by the number of values was visible to pupils. However, at the same time, what is the most important aspect is pupils seeing that we have added the average value and the new average hasn’t changed. The division isn’t adding the value that I initially thought it was. The new examples communicate a quality of sameness that can clearly communicate the property effectively. Also, showing the features of the new data set which are permissible to be a positive example, and the non examples show the features of the data set which are not present to be an example for property 5.

Arithmetic Average: developing conceptual understanding

Many pupils are taught how to calculate the arithmetic average (the mean.)  They are also taught a shallow understanding of the concept: it goes along the lines of “the mean is a calculated ‘central’ value of a set of numbers.” Or pupils have a very loose understanding of ‘average’ as colloquially referred to as ‘on average’. This is a start, but can an instructional programme be created to teach a deeper concept of the mean, accessible even to young pupils (8-14)?

I believe it can.

If so, it will aid future understanding of the weighted average at GCSE level.

Furthermore, it will lay the groundwork to understand how the trapezium rule approximates the area beneath a curve, at A Level.

I’m going to cover:

  1. Properties of the mean
  2. Selecting which properties to teach
  3. How to teach them
  4. Why it’s important

Fundamental properties of the mean

In Strauss and Efraim (1988) seven properties of average are outlined and these properties were chosen to be discussed because “they are fundamental, and tap into three aspects of the concept”.

I’m going to discuss what I think are the five most important.

Property 1: The average is always located between the extreme values

The average value of a set of data cannot be smaller than the minimum value or larger than the maximum value of the said data set.

This seems like common sense to an adult but expertise induced blindness underestimates the difficulty for pupils to understand this property of the mean.

How can this be communicated to pupils?

By example, show that the average value is never smaller than the minimum value or larger than the maximum value.

average pic 1

Then, test pupils’ understanding of the property using the following type of question:

For each question the average value could be true or definitely false, state whether the average value is true or false for the corresponding data set:

average pic 2

The following true or false questions are testing a pupil’s understanding of the property rather than their procedural knowledge of calculating the mean.

Property 2: The average is representative of the values that were averaged.

More technically: “the average is the value that is closest to all of the others in the set of values that are being averaged.”

Therefore, the average value represents all the values within the data set.

This property ties in nicely with the third property.

Property 3: The sum of the deviations from the average is 0.

Each value in the data set is a certain distance away from the average value which is clearly understood from a visual example. If we total the distances between each value and the average value, that total will equal to 0. This helps pupils to visualise that the average value is central to all the values. This highlights that the concept of the average value is again the central value to values within the data set. Furthermore, this highlights that the average value represents all the values that were averaged (property 2).

How do we communicate this to pupils?

  1. Outline how to find the distance between each value and the average value.
  2. Emphasise that I do (value – average value) and that the distance can be negative, if the value is smaller than the average value. However, the pupils need to picture the distances.
  3. Total the distances, also tell pupils that we call this sum the sum of deviations

average pic 3

average pic 4

average pic 5

There are two questions that can be asked to develop understanding around property 3:

  1. Show for each data set that the sum of the deviations from the average is 0 (Qs 1 – 3 only)
  2. For each set of data, the average value is either true or false. Determine which of the following average values for the corresponding data set are true or false using your knowledge of the following property:

The sum of the deviations from the average is 0.

The first question is asking pupils to apply their knowledge of finding the sum of the deviations for the average value. The second question is asking pupils to apply this knowledge but then decide which of the following data sets has the correct average value.

average pic 6

Property 4: When one calculates the average, a value of 0, if it appears, must be taken into account.

Dylan Wiliam has an excellent hinge question that deals with this property very well.

average pic 7

Pupils often think they don’t have to include a 0 value when calculating the mean.

A good way to overcome this is to simply include it in one of your examples.

Seeing this in a concrete context also allows pupils to see why 0 must be included as a value in the data set.

image 5a

Concrete context:

Sarah collected some money from her three siblings to raise money for charity. Hannah donated £5. Adam donated £7. Clare donated £9. Sarah did not donate any money. What was the average amount of money each of her siblings and Sarah donated?

Property 5: The average is only influenced by values other than the average.

average pic 9

This is a really nice point to make about arithmetic average:

Adding a value to a data set which is equal to the average value of the current data set does not influence the new average.

Why not?

Because when we add a value that is the previous average, the resulting sum is divisible by the new denominator. How can we communicate this to pupils?

The worked examples above show how the mean does not change when you add the average value of the previous data set (going from example 1 – 2 and 2 – 3). The last example shows that the mean does change when a value which is not the average of the previous data set is added.

To communicate this I would ask pupils to determine if the number added would change the average value. For example, the question series is designed for pupils to do the following

  1. calculate the average for each data set
  2. Decide if the mean has changed by adding a value.
  3. Explain that we have added a value which is equal to the average value. Adding the average value does not influence the values being averaged.

average pic 10

At the end, I would explain how when we add the average value to the data set (5), we get a sum which is divisible by the denominator (20/4 = 5). When we add a value that is not the previous average, the resulting sum is not divisible by the new denominator.

average pic 11

In summary, these properties of arithmetic average can be taught effectively with the correct worked examples and problem exercises which communicate each property at one time. It gives pupils a spatial understanding of the mean which is above and beyond the procedural calculation of calculating the mean. I think it is highly powerful knowledge that can lend itself for more complex understanding of the mean when learning about the difference between simple mean and weighted average mean.

Reading all the Books – Warm Strict

08 May 2017, Posted by admin in Michaela's Blog

Warm – strict

I have written previously about Teach Like a Champion, a book I feel to be the most important contribution to pedagogy advice I have read. Although it is nearly impossible to pick which of the important techniques are the most vital, ‘warm strict’ is definitely up there: in fact, it may even be the foundation of a successful education.

The thinking behind ‘warm strict’ is that you should not be either the warm, friendly, kind teacher or the strict teacher: you need to be both. And not one after the other – it’s not Jekyll and Hyde – but both, at the precise same time.

So ‘warm’ and ‘strict’ are not mutually exclusive. In fact, at Michaela, we have found that the more strict we want to be, the more warm we have to be.

Anyone who has visited Michaela is immediately struck by the behaviour of the pupils. It is unusual, they say, to find classroom after classroom where 100% of pupils are focused for 100% of the time. Row upon row of eyes are fixed on their teacher, or on their exercise books. There is no staring out the window, no fiddling with a pen, no hanging back on their chairs.

But this does not happen by magic. Watch any Michaela lesson, and teachers areconstantly issuing corrections to pupils. These can take the form of reminders or demerits, and are swift and public. ‘Kevon, remember to keep those eyes glued to your page,’ might be issued to a year 7 who is still in terrible habits from primary school, who desperatelywants to focus on his work but just isn’t quite in the habit of it. ‘Shyma, that’s a demerit: if you focus 100% on your paragraph you know it will be the best you can do,’ might address a year 9 who is knowingly letting their eyes wander because they are seeking to distract others or themselves. It’s a judgement call, and one we don’t all always get right, but in general Michaela teachers are incredibly consistent in the messages they give the children. (We achieve that consistency through frequent observations – the topic of a future post.)

In my previous schools, I was also issuing constant corrections; the difference was my stress level. With a tough class, counting up those three warnings before issuing a sanction would lead to me delivering corrections with an emotional tone, conveying the stress I was feeling. Because the bar for behaviour is set so ludicrously high at Michaela, and pupils are never doing anything worse in lessons than turning around, whispering or fiddling with a pen, we can all take the time to explain every correction we give throughout the lesson. And we give corrections, reminders, demerits and even detentions with care and love: ‘that’s your second demerit, which is a detention – this will help you to remember to keep your focus so you will achieve your full potential.’

Not only within lessons, but also between lessons, Michaela teachers are seeking out opportunities for warm interactions with pupils. At break time, tutors circulate the hall their year group is based in, shaking hands, chatting about their weekend or their interests; we even have footage of pupils teaching their tutors how to dance. At lunchtime, we eat with our pupils; teachers will seek out kids they have had to sanction or have a difficult conversation with, and use that friendly interaction to reset the relationship in a more positive tone.

Because we are so strict, it is vital that every teacher greets every child with a smile and happy ‘good morning!’ prior to each lesson. Because we are so strict, we must smile and chat with the pupils on the playground, in the lunch hall, and even at the bus stop. Because we are so strict, we need to let our love show.

All truly excellent teachers love their pupils – that seems obvious to me. But if you want to be really, really strict you need to show them that love in every smiling interaction.

Lesson Observations at Michaela

At Michaela, lessons are not graded. Like many schools, we have moved away from the old paradigm of observing teachers once per half term in a high stakes, pre-planned, pay-determining observation, towards a new one.

We have a truly open door policy in which all staff are encouraged to observe each other regularly. We also have a number of visitors each day, so I regularly have people observing from the back of the room. Rather than feeling a bit unnerved by the presence of an observer, I now feel pretty relaxed, like it’s the norm. I don’t feel like I have to perform or change what I’m doing just because someone has walked in. Even when one of the school’s SLT pop in, I don’t feel the pressure to show off.

The other great thing about having an open door policy is that there’s no pressure to spend hours the night before planning the lesson or creating silly bits of paper that only benefit the observer (endless data sheets, colour-coded seating plans, 3 page lesson plan pro formas, and so on).

My most frequent classroom visitor, of course, is Olivia Dyer, my Head of Department. She pops in perhaps three or four times a week- perhaps for 10 minutes, perhaps for the whole lesson- and fires through a quick feedback email straight away.

The main purpose of lesson feedback at Michaela is twofold: first, it needs to be immediately actionable. I should be able to read the email in a couple of minutes over break, perhaps, and then implement it straight away in the next lesson. If I can’t get to the email until the end of the day, I want to improve quickly, I do try to read it as soon as possible. Secondly, lesson feedback aims to help all teachers hone their practice, and most importantly, becomemore ‘Michaela’.

At Michaela, we believe in ‘rowing together’. We value consistency and want all our pupils to get a similar experience across the board. Critics might argue that such an approach is too rigid, that it prevents teachers’ individualities from coming across in lessons. This is a misconception, I think. Teachers’ personalities are still able to shine through. When I go to see Mike Taylor or Hin-Tai Ting teach, I see many of the same features (kids encouraged to put their hands up, books given out in 10 seconds, teacher saying ‘321 SLANT’, etc.), but their respective personalities are different, and their relationships with the kids can flourish as a result. In many ways, the fact that we get teachers to align on the things the mechanics of the lesson means that teachers are free to be themselves and build relationships.

We also accept that there may be differences between departments. So Science lesson feedback will probably have different emphases to that of a History lesson for example. The differences might be quite small, such as ‘when explaining the experiment, hold the beaker higher up so the kids can see it at the back’- clearly, that sort of thing won’t come up in History, but tailored feedback on the teaching of historical paragraph writing will at some point. But because the general principles are the same across the board, subject-specific differences don’t undermine the overarching philosophy of consistency.

Observers provide helpful, concise, immediately actionable feedback that supports teachers to improve.

Here is an example of a lesson feedback email:

Hi Cassie

Thanks for having me in again today.

A couple of tiny, quick fix things!

  • Only ever write on the board in black, other colours are really hard to read
  • It’s always ‘3, 2, 1, SLANT’, not ‘3, 2, SLANT’

 Some bits of feedback to think about:

 1) Pace.  You need to ensure absolutely no downtime in the lesson.  You should aim for the kids to always be actively listening, tracking, thinking and working.  Figure out ways of logging merits, etc. quickly so it doesn’t interrupt the pace of the lesson.  I like to give the kids a piece of paper and get them to write down merits and demerits – that means you can just get on with teaching.

 2)  Sharpen the timing of sanctions, and always frame them positively.  “Sammy*, we SLANT perfectly so that we’re 100% focused at all times, so I’m giving you a demerit to remind you to SLANT really well at all times, thank you.”  You don’t always need to seek understanding from the kids, but if they have any kind of negative reaction, give them a second DM.  They’ll get the message!  Make sure if you’re giving a second DM that you tell the kids it’s a detention – I think you gave Sammy 2 DMs but didn’t say ‘that’s a detention’. Also, pre-empt further demerits by saying something like “remember guys, a demerit is half-way to a detention, and it’d be a real shame for you to miss your lunchbreak or go home later than your friends.”

 3) Radar. Your use of silent reminders and getting them to practise routines is great; now, you need to practise having your eyes on every child every minute.   This will take time to develop, but when you’re delivering from the front, try and scan the room as much as possible to pick up distractions. Stand up on your tiptoes and look round the room like a meerkat so the kids know you’re watching them. 

 Keen to learn more about Michaela? Read our book, Battle Hymn of the Tiger Teachers, available here.

If you’re keen to learn more about how we resource our lessons, why not join us for a Summer Project this year? More information here.

The Summer Project (#French)

In the past, my hands-down-favourite thing about teaching was being in the classroom, with the kids, bouncing off their energy and enthusiasm and teaching them French, and seeing them grow in knowledge and confidence and ability. That feeling only increased when I started at Michaela. The exceptional behaviour standards meant that we could have a real giggle, that my pupils would learn LOADS in an hour and their energy and enthusiasm would stem, in the main, from feeling clever.

But a new thing threatens to knock the act of teaching off its perch as my favourite thing about… well, teaching. And that is: resourcing.

We teach French very differently at Michaela, and in order to do that we make our own resources. Those resources are very dense – you can get loads out of a single text, or a single practice activity, and our emphasis on repetition and spaced practice means they get reused again and again. I have shared a couple of our knowledge organisers – one of those, along with other practice activities, normally lasts six weeks. We are constantly refining and developing the materials we use, but they are built on a solid foundation of the written word, lots of reading, lots of transparency and lots of beautiful language.

This Summer, from Monday 24th July to Friday 4th August, we are running a Summer Project to design and produce even more resources, with a particular emphasis on Key Stage 4. We – my colleagues and I – will be working for two weeks to design and produce the texts, activities, practice drills etc. that we will use with our pupils week in, week out, for years. I’m ridiculously excited!

Evangelical as we are about how we teach French, we want to share this experience with others, and give you the opportunity to come and work with us. We’ll be based at Michaela, and you will get an insight into the thought process behind our resources, a chance to work with Michaela staff to create them, and a chance to discuss the fun parts of pedagogy, like “what little story can we tell about the word ‘quelqu’un’ to help pupils remember it?” and “which idiom sounds more authentic here?”.

It would be ideal if you could do the full two weeks, so that you get the most out of it, but we will consider those interested in doing one of the two weeks. Expect it to challenge everything you previously thought about language teaching! If you’re interested in taking part, please send a brief email explaining why you’d like to be part of the project to me at jlund@mcsbrent.co.uk.

Staying stoical in school

Some two thousand years ago, a teacher, a playwright and an emperor asked:

What is the best way to live?

How can we deal with the difficult situations we face?

What does it take to improve our minds?

Their answers are the heart of Stoic philosophy. We in schools can use their insights on themind, on adversity and on practice to help our pupils shape their thought patterns.

Stoics.png

Here is a rough overview of what we teach our pupils about staying stoical at Michaela:

  1. Mind

“Some things are under our control, some are not. We are responsible for what is in our power to control: our mind and its perceptions. The chief task in life is simply this: to identify which externals are not under our control, and which are the choices we actually control. We control our opinion, choice, attachment, aversion. It is learning to separate the things that lie within our power from those that don’t.

“Whoever can be irritated – that person is a slave. No one can frustrate you without your cooperation; you are only hurt the moment you believe yourself to be. So when we are frustrated, angry or unhappy, never hold anyone except ourselves responsible. We forever compound our problems because we make them out to be worse than they actually are.”

Epictetus

“Life itself is only what you deem it. Your mind will take the shape of what you frequently hold in thought. Such as are your habitual thoughts, such will also be the character of your mind. Your anxieties are creatures of your own imagination, and you can rid yourself of them. Observe how disquiet is all of our own making. Troubles never come from another’s hand, but are creatures of our own creation.”

Marcus Aurelius

 

  1. Adversity

“Difficulties show a person’s character. So when trouble comes, think of it as training, strengthening, toughening. When a challenge confronts you, remember you are being matched with a stronger sparring partner, as would a physical trainerA boxer derives the greatest advantage from his sparring partner, training his patience and even temper. In adversity, be happy that what you have learned is being tested by real events. Philosophy is preparing ourselves for what may come. So, what should we say to every trial we face? This is what I’ve trained for, for this is my discipline!”

Epictetus

BoxingOlympic.png

“Like a boxer with a sparring partner – no protest or suspicion – act this way with all things in life. Instead of: “How unlucky I am, that this should have happened to me!” Say rather: “how lucky I am, that it has left me with no bitterness, undismayed: to endure this is not misfortune but good fortune.” The obstacle on the path becomes the way.”

Marcus Aurelius

“Seize your adversities head on. To bear trials with a calm mind robs misfortune of its strength and burden. Complain little: no condition is so bitter that a stable mind cannot find some consolation in it. Think your way through difficulties: harsh conditions weigh less on those who know how to bear them.”

Seneca

 

  1. Practice

“Study, practice and train if you want to be free: for as time passes we forget what we learned. The secret of happiness of the human mind is gratitude. Discipline yourself with thinking, exercises and reading: that’s the path to human freedom. If you’re succeeding, you see any moment as an opportunity to practise.”

Epictetus

“Train your mind through regular practice. Persevere in your efforts to acquire a sound understanding. One needs constant, daily practice … the pursuit of wisdom sets us free.”Seneca

“Make a habit of studying the maxims. Practise even when success looks hopeless.Discipline brings peace of mind.”

Marcus Aurelius

 

Stoicism in school

Children at school are learning how to deal with difficult emotions: frustration, worry, fear, cravings, temper, arguments, gossip, jealousy, squabbles, hurt, bitterness and more. Stoicism offers fortifying ways to think about these difficulties.

Teachers can …

  • show pupils what is always within their control: their thoughts, responses and reactions.
  • preemptively teach pupils how to anticipate and cope with adversity.
  • guide pupils to change their perceptions so that they complain, blame and resent less, and instead keep perspective, stay grateful, and are happy.

Pupils can learn how to…

  • let go of frustration: by realising that it is in the mind, so within our control, and by remembering that irritation is counter-productive and should be released and not dwelt on.
  • let go of worry: by remembering that the more we worry about things we can’t control, the worse we feel; the less we worry, the calmer and happier we feel.
  • cope with arguments: by avoiding criticising, complaining, blaming or resenting others, and instead feeling cheerful and grateful for what we have.

School leaders can convey the lives of the thinkers, and their thinking …

Epictetus was a Greco-Roman slave who earned his freedom and became a teacher.

Marcus Aurelius was a Roman emperor taught by philosophers who wrote personal diaries.

Lucius Seneca was a tutor and advisor to the emperor of Rome, who wrote letters and plays.

All three thinkers used analogies, mantras and writing: notebooks, diaries, letters or plays.

Three Analogies

Slavery: irritation enslaves us; wisdom frees us.

Illusions: anxieties are creatures of our own creation; we can get rid of them.

Boxing: we must train hard so we don’t get beaten!

Boxing2.png 

Interactions

At Michaela, we continually return to stoicism in our interactions with pupils. From their first week at Michaela, we teach them about its approach to life: that everyone experiences difficulties, but that we can overcome them. We have assemblies on it, we spend an hour lesson on it, we discuss it over family lunch, and we revisit it in form time. We teach them to anticipate the frustrations of life, and the mantra: ‘stay stoical!’

Here are six ways we see it come in handy: in detentions, in exams, in arguments, inpain, in sport and with their families.

  1. In detention … stay stoical!

Sometimes, pupils feel upset or resentful about being given a detention. In conversations with them, we remind them to stay calm, stay stoical, keep perspective, let go of anger, work out what they control, think about how they can build trust in future, and decide what they can do differently next time.

  1. Before, in and after exams … stay stoical!

Tests can be seen as stressful by children. We help them see that preparation, revision and overcoming procrastination is within their control. If they fail an exam, or get a poor result, staying stoical and not agitating about it, but instead focusing on what they can do differently, helps them for the next assessment ahead.

  1. In arguments … stay stoical!

When children get into arguments with their friends or fellow pupils, stoicism can help remind them to practise keeping a calm mind, ignoring gossip, vicious rumours, or insults, and staying positive rather than exacerbating mistrust and anger.

  1. When ill or struggling… stay stoical!

It is sometimes a struggle to come in to school if children have a cold or feel a little ill; persisting rather than giving up makes them feel proud and strong. One Year 7 pupil we teach lost a tooth in assembly, put it in his pocket and carried on listening. One was stung by a wasp at sport, and overcame the pain by keeping stoicism in mind. Another gets regular nosebleeds but proudly endures. After she broke her wrist just before exams, another taught herself to write with her left hand, and is now ambidextrous. The stoic mindset reduces our fragility: the volatility of the world can’t destabilise us. With stoicism, problems become opportunities to train our resilience.

     5. At sport … stay stoical!

Sport, by its competitive nature, is a time when tempers can run high. Stoicism reminds pupils not to overcelebrate and jeer at others when scoring a goal or winning a match; and not to despair or blame team-mates (or the ref!) having conceded a goal or losing a match. It prevents yellow cards turning into red cards and prevents fights breaking out. At Michaela, we call our football team The Stoics as a symbol of this mindset. If it’s cold and raining, we say: ‘great! character-building stuff: a chance to train our willpower!’

   6. With your families … stay stoical!

Children experience difficult times growing up with parents, siblings and cousins. If they are taught how to stay stoical at school, it helps them to overcome arguments, illness and adversity in their family. It can help them keep perspective, stay grateful and not take their brothers, sisters, mums and dads for granted, but appreciate them.

Teaching children to stay stoical when times are tough gives them a powerful perspective that helps them improve their resilience, their relationships, and ultimately, their lives. It even improves teachers’ lives, too!

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Stoicism, humility, space: how Michaela changes the people who work there

One of the things I hear a lot from my colleagues at Michaela is that working at the school has made them better people. Why is it that so many of us feel we have improved as humans through our collective endeavour to teach children? 

Stoicism

‘I’d never heard of stoicism before I worked here. Now I’m reading Epictetus with eleven year olds. It’s mad!’ (Michaela teacher)

We explicitly teach stoicism to the kids from day one of Bootcamp in Year 7. We teach them that adversity is there to test them, and the true test of character is how they choose to respond. They will, like everyone, experience difficulties in their lives: stoicism gives them tools to rise to those challenges.

By continually reminding pupils to ‘stay stoical’ – when they get a detention, when they cut their finger, when they have a cold, when they’re finding a topic or idea difficult, when they have six hours of exams a day in exam week – we are also internalising that message ourselves. It is really quite extraordinary how high staff attendance is at Michaela.

Stoicism is a life-changing philosophy. I’ve recently started reading The Daily Stoic, which is like a Bible for perseverance and perspective. When difficult things used to happen in my life, I would go to pieces. I would cry, or feel anger, or feel that it just wasn’t fair. But now, I remember what we teach our kids: Nelson Mandela spent 27 years wrongfully imprisoned; Victor Frankl endured the miseries of a concentration camp. Nothing that can happen in my life will come close to the suffering they endured, but through our own endurance we can set examples for others.

In August, for example, I slipped a disc in my neck. I was in absolute agony for months, and felt very sorry for myself continually. But when I was tempted to indulge in considering how ‘unfair’ my lot was, I had only to look at the shining example of my step-mother, whose M.S. may make her daily life incredibly difficult, but will not take the smile and positivity from her. She is a force of nature, and a wonderful human to be around. She is an example for me to live up to: if I feel pain, it is nothing compared to what she feels every day; if she can endure it, I can follow her example.

 

Humility

Before I worked at Michaela, I loved being told I was great, and hated being told how to improve. I was, in short, pretty arrogant. But through our culture of candour, and through a culture of continual improvement and continual feedback, through working at Michaela you become more humble. We are all always improving, and come to actually look forward to our candid conversations as we know they will genuinely help us improve.

It is great to work somewhere free of blame. In about my first month of working at Michaela, Katharine asked me to present something at a staff meeting. I came up with a handout I know would have worked well at any of my previous schools and talked people through it. It didn’t seem to go as well as I had anticipated, and I wasn’t sure why.

The next day, Katharine asked me to see her. ‘It didn’t work,’ she said. ‘You’re telling people “why,” but they know why – they need to know “how.”’ Does that sound harsh? It wasn’t harsh in the delivery – it was delivered without blame, without recrimination, in the spirit of sharing information: this didn’t work, so next time do this.

Being humble means you learn more: you don’t write off the first year teacher’s advice because you are more experienced, you don’t discount the ideas of others in the school because of any misguided notion of ‘rank.’ You listen to everyone, and you learn more than you could have ever expected.

 

Space

I do wonder if all schools could be like Michaela in terms of the ethos and atmosphere for staff. Certainly, our workload is very intense – days are packed from 7:30am to 4pm – but we also have evenings and weekends and holidays free to see friends, to see family; to read, to write, to think. I’ve never had so many colleagues go to the theatre mid-week, or go away for the entirety of a half term or long holiday without the slightest qualms.

And every day is zen: silent corridors, quiet classrooms, children behaving beautifully is all conducive to feeling happy to take on difficult advice, and finding it easier to deal with emotional or physical problems. In times of crisis, Michaela is a really lovely place to be.

In my previous jobs, I absolutely loved what I was doing, but I was often exhausted: I would cancel plans at the last minute because I could not bear to leave the sofa at the weekend, I wouldn’t see close friends for months on end, I wouldn’t book long holidays because I knew I would find the workload unmanageable when I returned.

Now, I still love what I’m doing, but it’s not all of my life – it’s part of my life. But that part of my life impacts on everything else, and I find my friendships deepen, and my relationships with my family soften, because I see them more often, and I am more present with them.

Changing my mind on grammar schools

When I was 16, after having completed my GCSE examinations, myself and a few friends travelled to London for a few days of sightseeing. We visited the Palace of Westminster on our last day and out of the corner of my eye, I spotted our local MP. As any group of precocious and politically aware grammar school boys would, some of us decided to go over and greet our Member of Parliament.

  At the time, our MP was Graham Brady, who had just resigned from his position as Shadow Spokesperson for Europe in light of David Cameron’s pronounced objection to any further expansion of grammar schools in England. Myself, and one other friend of mine congratulated him on his principled stand. My constituency at the time was in the heart of Trafford – one of fifteen selective local education authorities still left in England. Our school was one of the remaining 164 grammar schools in England after the Crosland scythe in the 1970s (As he so eloquently put it:“If it’s the last thing I do, I’m going to destroy every f****** grammar school in England. And Wales and Northern Ireland”). At the time, we thought Brady was absolutely right to defend grammar schools, particularly as our school and many others in Trafford were so oversubscribed and successful in the national league tables. Our school was also a Catholic grammar, which meant that it selected on religious grounds, as well as academic. This made it somewhat more interesting, as catchment selection was much further down the pecking order than most grammar schools where they are often stuffed full of middle class kids whose parents didn’t want to stump up the private school subs but were willing to move into the school’s catchment area – selection by stealth, and wealth if you will. As a result, our kids had a real mix of middle class and less affluent pupils, many of whom went on to Russell Group and ancient universities.

Drill and Thrill

This is a summary of the presentation from Maths Conf 9, held in Bristol on 11/3/2017. Thanks to everyone who came and who asked questions!

drill and thrill 1

What is a drill?

A drill is narrow. It should be focused on a single thing, such as:

  • Decision-making
    • Which fraction ‘rule’ to use for a mix of fraction operations (i.e. choose the rule, don’t complete the operation)
    • Do I need to borrow? Write ‘B’ above each calculation where this is the case
    • Will the answer be positive or negative? Write + or -, nothing more (a mix such as -2-6, 9-12, -4+8, etc)
  • Speed
    • Times tables
    • Expanding single brackets
    • Simplifying indices
    • Multiplying and dividing directed numbers
  • Improving accuracy (and fine motor skills!)
    • Multiplying and dividing by powers of 10 (e.g. practising simply ‘moving the point’ correctly!)
    • Rounding (underlining to the correct digit, circling the correct digit)
  • Recognising and deciding
    • A drill to with a mix of questions that are either rounding OR multiplying/dividing by powers of 10 (confusing for a small number of pupils!)
    • Do I need an LCD? A mix of questions: some multiplication, some fractions which already have an LCD, some fractions without
  • Improving muscle memory (automation of multiple steps)
    • Completing the square
    • Calculating the gradient of straight line
    • Rationalising the denominator

 

Why Drill?

  • Many operations require a level of deep understanding that overwhelms pupils. We need to build proficiency in every exception; first separately, then together. As teachers our expertise and knowledge can blind us how challenging this is for pupils. We a fluent in exceptions in how we speak; we must help pupils become fluent in the exceptions of maths (which is, in this respect, much like a language).
  • It’s important to recognise that progress doesn’t happen in a lesson, but over time. It isn’t seen in their books from that lesson, but in long-term memory and the speed of subsequent connections…drills are an investment in their long-term memory!
  • Drills allow you to build motivation, as they can manufacture the sense of having lots of success
  • Drills offer quick wins: automaticity, confidence, buy-in
  • In the long-term, drills strengthen vital links that allow maths to feel less laborious and confusing.

Here are three examples that Hin-Tai Ting used over several months with 7 Zeus (fourth quartile group in Y7). He has described the design process in fascinating detail here.

drill and thrill 2

Let’s focus on the first column. In the first example, pupils are completing a simple procedure, focusing on a single decision (i.e. what happens when multiplying by 10). This is focusing on accuracy and motor skills, and automatising the many ‘weird’ things that seem to happen with the decimal point… 

drill and thrill 3

In the second example, we can see that they now have mastered ‘moving the decimal point’ and are focusing on fluency with moving 1/2/3 decimal places in either direction. 

drill and thrill 4

7Zeus are now very competent with multiplying and dividing by powers of 10. This drill is now focused on fluency with varied representations: using powers and decimals (e.g. recalling what happens when multiplying by 0.01). 

What can be drilled? And what should be?

  • The aim is effortlessness. If it feels effortless for you, as a maths teacher, you want it to feel effortless for them.
  • Focus on:
    • High leverage (topics that reap benefits across the curriculum, such as fraction-decimal conversions)
    • High frequency (topics that you KNOW they will need in many exam questions, such as rounding)
    • High complexity (topics that have mutiple and confusing steps that need to be chunked and automated, such as adding and subtracting fractions)
    • Error prone (topics where they know roughly what they should do, but tend to mess up, such as multiplying and dividing by powers of 10)
    • Confusion prone (topics where pupils are easily confused and tend to eventually guess, such as adding and subtracting directed numbers)
    • System 1 override! (topics where their first instinct is often wrong, such as dividing fractions and index laws)

drill and thrill 5

The drill above is an example of a decision-making drill: pupils need only to decide if the answer will be positive or negative. 

drill and thrill 6

This is an example of a speed drill: the focus is on getting some (very weak!) pupils to be faster and more accurate with very simple mental calculations, both to move them away from finger counting and to improve accuracy in column addition and subtraction. Each day was +-2/3/4/5/6/7/8/9, cycling back until all were speedy at all of them. They seemed to really love it, and it was very quick each lesson. 

Ones that didn’t work, or don’t suit

  • Some topics are too complex to be suitable (e.g. metric conversions)
  • Some Too simple (multiplying proper fractions – it quickly becomes a times tables exercise, and doesn’t make them better at multiplying fractions)
  • Questions that allow them to go on autopilot (20 values multiplied by 0.1…they’ll quickly switch to ‘divide by 10’ in their minds, and will not have strengthened their recall of what happens when multiplying by 0.1)
  • More fiddly does not mean more challenging (making the numbers longer or more annoying is just….more annoying)
  • Progressively harder questions (that’s normal work!)
  • Varied questions (Corbett Maths and Numeracy Ninjas are both AMAZING but they are revision and varied practice, not drills)
  • Questions that make you stop and go ‘hmmm’…these are part of a (nice:) normal lesson
  • Take more than ~15 seconds per question…possibly even more than 5 seconds per question, although it depends. If it takes too long, urgency will be lost and it will feel flat.

drill and thrill 7.png

This is a drill I used for ~a month last year with 7Poseidon (first quartile). Some worked well, but some were a nightmare. Writing those numbers in base 2 is not suitable as a drill unless you are the Rainman! And they inevitably dawdled when they got to x2.5 – it would have been better as part of mixed practice each day, with space for working, not as mental maths. 

Rolling out: when and how 

  • Drills are not a teaching tool! They are for automating procedures/connections already in place.
  • During a teaching sequence: to practice a specific and isolated decision (e.g. What is the LCD?)
  • AFTER the content has been grasped and foundations are in place: to improve speed and accuracy (and confidence)
  • To get more ‘bang for your buck’ (and check they are ready), you can complete it orally (Line 1…Sarah…Line 2…Thomas…Line 3…Abdi…etc), then in writing (In your books…go!)
  • WATCH OUT: Practice makes permanent! If they are not secure with the content, they will be practising and automating getting it wrong. This is the nightmare situation! To avoid this, check the whole class on whiteboards first, possibly many times (to allow for false positives) and for the first few days you do a new drill (they’ve slept since the last one so may well have forgotten………)
  • Narrate the why (To build up our confidence, To improve your accuracy, So we can test ourselves and push ourselves to improve, To see how much we can improve as a team)

Joy Factor

This may be hard to understand if you weren’t there! You’ll have to visit our school to see the kids in action…

  • Make it quick and short: race per column, or even per 10 (think: spinning! It’s unbearable to push for 3 minutes, but manageable if the trainer breaks it into 30second bursts)
  • Raise the tension: music and timers (Youtube’s ‘tension music’ is surprisingly good!)
  • Raise the stakes: Check for cheaters! Just before it starts, they ‘check for cheaters’ (peering at each others’ like little meerkats; if a cheater is ‘caught’ writing before the ‘go!’ they have their hands up (as if they were a robber in a cartoon(!)) for 5 seconds before they get to join in)
  • Feel like a team: All have pens poised, the teacher calls “Ready!” [everyone bangs the table with their other hand] “Get set!” [two bangs] “Go!” [everyone writes furiosly, and there is a crisp start to raise excitement]
  • Celebrate together:
    • Mexican Wave (“Mexican wave if you got…10 or more! 15 or more!”;
    • “Show me one hand if you managed ______! Show me both hands if you managed _______, wave your hands like crazy if you managed _____!”;
    • “If you managed to do [really difficult thing], then 3,2,1….” [pupils who succeeded go ‘yussssss’ and a fist pump together] [less mad than it sounds, they seem to love it]
  • Celebrate individuals: top rockstar wears ‘rock glasses’, Queen of Quadratics gets a crown
  • Make improvement visible: tick (Mark column 1), target (write a target for the next one), repeat (do column 2…possibly give them more time, but don’t tell them!), improve (give yourself a pat on the back if you improved).
  • Patterns in the answers: If you are a masochist with time to spare, pupils LOVE if there is a pattern in the answers (e.g. Fibonacci fractions)

Design advice

  • Use Excel for speed/fluency questions
  • Even if written ‘by hand’ on your computer, use Excel to shuffle them and reuse day-by-day
  • Think about what the pupils will need to produce…this affects spacing
  • Make sure rows and columns have names!
  • Write on or read from? A drill sheet can be reused many times if they have to write in their books, using a ruler to go down the rows and keep track.
  • Make it re-usable (lists of numbers, some 1-20, some integers, some decimals, some fractions), then give different instructions, depending on the class
    • Add 1
    • halve
    • double
    • add 10
    • write as words
    • round to 2s.f.
    • find lower bound if they were rounded to the nearest 10
    • partition
    • x or ÷ by 10

Risks 

  • Very mixed ability groups can lead to dead time for the quickest
    • UKMT question on board
    • ‘tough nut’ at end of column
    • pattern in answers
  • Privileges speed: it is essential that pupils understand that being fast is a route to fluency, and NOT the same thing as being good at maths. I talk about athletes doing drills for fitness and speed. Being a quick runner won’t guarantee you are a good footballer, but being slow and unfit guarantees you won’t be (i.e. the causality goes in one direction)
  • Speeding leads to errors (pre-emptive crossness! Talk about how annoyed you’ll be if it is a sloppy rush, and how they won’t improve. List physical signs of care you expect to see – underlining, decimal point moved, etc)
  • Inaccurate marking (focus on improvement when you narrate, do random checks, narrate ‘lying to yourself’)
  • Practising making mistakes: as discussed, you MUST check they are competent before they do a drill for speed/accuracy
  • No deep thought – that’s the point! Deep thought is for the rest of the lesson.
  • Workload and proliferation of paper: use Excel, reuse sheets

Lastly…

  • Get in touch with what works and what doesn’t!
  • Obviously photos and videos of kids feeling proud will make my day… 🙂
  • Easiest way is @danicquinn
  • Or come say hi and see our kids! We’re in Wembley: MCS Brent

And to make use of Hin-Tai’s super drills for Y7, go to http://tes.com/teaching-resource/maths-drills-generator-rounding-halving-doubling-multiplying-by-powers-of-10-11536042

Mathsconf9 – Automatising Procedural Knowledge for A Level study

Here is a bit of context behind what inspired me to deliver this workshop. I was tutoring a friend’s daughter in the summer holidays (2016) before she started learning AS Mathematics after achieving a A in her maths GCSE. My friend’s daughter is intelligent, enjoys studying mathematics and is able to learn new concepts very quickly but she struggled to make the correct decisions to complete each sub-procedure in long calculations. She would struggle to multiply an integer and a fraction, because she didn’t know what decision to make i.e. write the integer as a fraction over one.  So I spent a lot of time creating questions within drills and deliberate practice exercises for her to gain the procedural knowledge to a degree of fluency where she wasn’t struggling to complete different sub-procedures.

I am going to look at one high level procedures that I mentioned in my workshop; multiplying fractions.

Multiplying fractions.

Historically, when teaching how to multiply fractions we teach how to multiply fractions where there are two terms within the equation. However, there are several different problem types that can be taught:

  • Multiplying two terms
  • Multiplying more than two terms
  • Multiplying terms that share factors that can be cross-simplified
  • Multiplying a fraction and an integer
  • Multiplying a mixed number and a fraction
  • Multiplying mixed numbers
  • Multiplying decimals (that can be converted into fractions) and fractions.
  • Multiplying an improper fraction with and/or proper fraction

fraction 1

When teaching my tutee index notation she struggled to attempt the following problem because she wasn’t able to identify how to multiply an integer with a fraction. I told her explicitly how to multiply 2 and a third quickly, because that is what I do, and all pupils do this if and when they spot the pattern (but not applicable when the numerator of the fraction is not 1).

fraction 2

fraction 3Later on in another session, my tutee struggled with the following question because she didn’t realise that a fraction can be written as the product of an integer (numerator of the fraction – 2) and a fraction (unit fraction with the same denominator – 1/3 ) e.g. 2/3 equals to the product of 2 and 1/3.

fraction 4Furthermore, this piece of procedural knowledge is necessary to attempt the following question:

fraction 5And again, when a question gets more complex like the one below because the base number is a mixed number, rewriting 5/2 as a product of 5 and ½ is a necessary step to evaluate the question.

fraction 6However, what the take-away here is is that multiplying fractions is a high leverage procedure. A part of the procedure is to rewrite a fraction as a product of an integer and a fractions. So how can pupils learn this  to successfully automatise their procedural knowledge of multiplying fractions? This can be achieved through an understanding of multiple representations of the following possible problem types that arise when evaluating the base number/term held to a power which is a fraction.

fraction 7

The emphasis of exploring different problem types available within the topic of multiplying fractions, and recognising that a fraction is a product of an integer (numerator of the fraction) and a fraction (unit fraction using the same denominator), and then seeing the multiple representations of this enables a level of proficiency to allow successful learning of new content. My tutee wasn’t really learning how to simplify indices because she was focusing on learning how to complete each sub-procedure. I think this proficiency needs to be achieved in KS3.

In my next post, I shall give another example – halving!