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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.

Until I Know Better – Maths Muddle

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

Maths Muddle: alternate segment, same segment, alternate angles

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

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

angles 1.png

angles 2.png      angles 3.png

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

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

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

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

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

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

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

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

#Mathsconf10: Worksheet-Making Extravaganza

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

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

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

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

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

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

  • Arithmetic complexity
  • Visual complexity
  • Multiple steps
  • Decoding

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

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

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

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

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

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

Image 8

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

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

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

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

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

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

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

Image 14

Image 16

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

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

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

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

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

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

UK Maths Challenge

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

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

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

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

Procedural knowledge:

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

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

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

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

Never Let Me Go

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

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

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

enlargements - modelled example

The tasks for the pupils to undertake following this example:

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

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

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

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

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

 

Here are some counter-arguments:

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

 

What we should be doing instead:

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

Year 7 History at Michaela

One of the great things about working at Michaela is watching the rapid transformation of our Year 7 pupils. From their baptism of fire induction in the first week, through to the end of the summer term, our youngest cohort improves hugely in terms of habits, behaviour and confidence.

The same is true when teaching them history. I was really impressed by their latest essays, particularly from the lowest ability groups, that I thought it would be useful for me to blog about how we got there.

We are so lucky in our department, in that we are given the space and time to talk about what to teach and how to make sure it sticks. Unlike the monstrous regimes of data meetings, school wide marking policies, or discussing the latest diktat from on high induced by ineffectual accountability regimes, we are actually given the space to talk about what to teach our Year 7s.

That process of constantly thinking has led us to where we are now. We have lots to learn, and we are by no means there yet. My colleague Jonny has outlined some of this thinking in his blog, in particular the focus we have on the ‘horizontal’ or ‘domain’ specific vocabulary needed to make sense of a historical era or epoch and the ‘vertical’ vocabulary which is needed to analyse a specific historical debate, issue or question. Some might call the latter ‘conceptual’ vocabulary – tools which enable pupils to discern change and continuity, causation, significance or interpretative elements of the past. I think that can be quite limiting. While ‘concepts’ are vital components of the historian’s tool kit, it is important to remember that any historical question is rooted in a specific domain of knowledge, even if there are conceptual similarities across time. The causes of the enlightenment will need a radically different conceptual toolkit to the causes of the First World War, despite both questions being causation focused, for example. We would therefore expand ‘vertical’ vocabulary to include all of the thematic and conceptual terms needed to answer a specific question and to understand a particular period.  We try to choose questions which allow pupils to master a broad domain of history, without the tapered and conceptual enquiry questions which often mean pupils only remember knowledge which is needed for that specific enquiry. Our Year 7s answer the following questions: ‘What led to the development of Egyptian/Greek/ Roman civilisation?’ ‘What led to the development of the English nation by 1066?’  & ‘What were the main challenges to the king’s power in late medieval England?’

How do we get Year 7 writing analytically?

Writing analytically is, of course, not a generic skill, but an ability to bring relevant horizontal and vertical (conceptual) vocabulary to bear on a specific question, which draws upon the wider context of the relevant domain. Many make the mistake of assuming this ‘analysis’ is divorced from the wider domain of knowledge. This wider domain and the vocabulary to make sense of it needs to be explicitly instructed. In essence, we do the following:

  1. Use of Knowledge Organisers to embed key dates, people and events as well as analytical vocabulary needed to analyse a specific question.
  2. Lesson content which challenges all pupils, with extra content used to enable the high ability pupils to draw upon more examples in their essays.
  3. Pre-empting common misconceptions and more practice for the weaker pupils.
  4. Constant repetition and drilling of the key horizontal and vertical vocabulary within lessons.

Pupils need to be constantly exposed to good historical writing. We are doing them a disservice by assuming that even with enough knowledge, pupils will be able to conjure up a well written and historically sound essay. It is fair to say, that most of the main historical ‘points of clash’ in a debate have already been discerned. A pupil in Year 7 is not going to break any new ground in terms of historical scholarship. We, the experts, need to guide our pupils onto the right path of analysis (for that specific topic). Critics may say that this is too prescriptive, or that we should just teach them how to ‘think critically’, and analysis will follow naturally.

I think we need to be teaching them the main arguments for an essay as explicitly as we teach the dates, people, and events needed to construct the essay.

Patently, as the pupils acquire more knowledge and with our guidance they will come to realise that there is not necessarily a clear cut answer to historical questions. Our latest Year 7 essay was “What led to the development of the English nation by 1066?” This is clearly a question about causation. In most history classrooms, teachers will be getting the pupils to sift through card sorts, evaluating and synthesising the relative impact of different factors. There is nothing disingenuous about getting kids to evaluate the relative impact of causes on an event. However, the questions we need to ask are:

  1. Is this how historians really conduct their analysis? Do they approach every question with the same critical toolkit, or does each question require specific domain knowledge in order to answer that question and formulate that critical toolkit?
  2. Can our time in the classroom be better spent embedding knowledge and the structure of analytical writing, or getting them to discover the answers themselves – particularly in Year 7?

By the end of year 7, some of our pupils are able to prioritise causes and sequence them both chronologically and in order or importance. By Year 9, given the strong foundation they already have, we are able to develop their vertical vocabulary to include the ‘weight’ of different causes (latent causes, triggers, underlying causes, accelerators and pre-conditions).

Every time pupils write a paragraph, we give short and sharp feedback which the pupils then act upon immediately. Because the behaviour is so good, it also gives us time to go around and give verbal feedback to each pupil in the lesson itself. Their paragraph structure has remained the same since their first unit, with more examples and complex conclusions built in as time goes on. In the future we can flex and adapt this structure to meet the complex demands of historical questions later at KS3. In their final year 7 unit, we try and flex this foundational structure even more by looking at a question which incorporates change and continuity as well as causation. Below are some examples of the final essays they wrote for their Anglo-Saxon history unit. All essays were completed in 50 minutes entirely from memory. These are a selection from our high, middle and low ability groups.

Essay 1 – High Ability

M1

M2

Strengths: Excellent sense of period. She clearly understands the ideas of peace and conflict as pre-conditions for nation-building in the English context. It is also really pleasing to see clear references to her knowledge of religion, which is taught alongside history as a Humanities subject (see use of ‘monotheistic’ and her knowledge of the ‘Ten Commandments’. She also makes a link between King Alfred and King Hammarubi, from when she studied Ancient Mesopotamia.

Essay 2 – Middle Ability

AG1

AG2

Strengths: Correct chronological sequencing.  Huge amounts of specific details recalled to strengthen her analysis. Eg. Correct dates of the Synod of Whitby, Bede’s Ecclesiastical History of the English people. This pupil has an excellent sense of period, with the essay drawing upon ideas from 597AD-1066, and with clear references to the development of the ‘heptarchy’ through comments about the development of a nascent identity through religion and clarity on the role of translation from Latin to West Saxon English as a vehicle for unification. She also makes links between her bigger concepts, by linking Alfred the Great’s penchant for translating great works and his leadership of Wessex during the Viking invasions.

Room for improvement: There is an implication that conflict was the key factor, but it is not explicit. As I suggested earlier, this is not necessarily a problem at this stage – particularly for such a broad question. These analytical structures will be taught in the subsequent late medieval unit.

Essay 3 (two paragraphs) – Middle Ability

P2

Strengths: The strengths of a knowledge-rich curriculum are really highlighted with this pupil’s paragraph on conflict. Here the pupil makes reference to the Great Heathen Army, the Battle of Brunanburgh and even the campaigns of Edward the Elder and Aethelflaed. He also remembers the historical significance of Aethelflaed as one of the key women in English history. He also uses the correct Old English diphthong: Æ.

Room for improvement: He is evidently thinking in a thematic, rather than chronological way. It would have been much better had he had prioritised and ordered the disparate elements of warfare in a logical way.

Essay 4 (three paragraphs)- Low Ability

J1

Strengths: Excellent clarity and prose. Use of various synonyms to highlight ‘importance’. Good links made to his religion lessons (monotheistic, Ten Commandments).

Room for improvement: Clearer explanation of why Offa’s coinage enabled him to project his authority and create a shared sense of loyalty, rather than just focusing on the economic element.

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.

Edu Dyertribe – Our Mistakes in Science

10 May 2017, Posted by admin in Latest News

OUR MISTAKES IN SCIENCE

As is the case in every school, the Michaela Science department has made its fair share of mistakes. After a few years of experimentation, trial and error, we have learned many lessons about what to do – and what not to do – in the science laboratory. Like all teachers, we want to provide our pupils with the best possible experience, and enable them to learn as much about the subject as possible. We also want to instil a lifelong love of science, and support lots of our pupils to use science in their future careers.

Fundamentally, our belief is that the best way to support pupils’ curiosity in science is through knowledge. By giving them the building blocks of scientific knowledge, and by supporting them to remember it throughout their time in school not only ensures they will have secure understanding of the world around them, but will provide them with the foundation for greater creativity in science.

But whilst our fundamental philosophy hasn’t changed over the last three years, we have of course reshaped and adjusted where necessary.

Booklet Annotation

Rather than using traditional textbooks or piecemeal worksheets as we did early on, we print each pupil a copy of our co-planned department textbook. Textbooks contain recap questions, explanations, diagrams and questions, and pupils can annotate examples with any further clarification given to them by the teacher.

Standard Lesson Format

In the first year, with only one year group, I was the only science teacher in the school. This had lots of benefits: I was able to spend my time focusing on the pupils and getting the curriculum ready for future year groups. In terms of lessons, I relished the flexibility. I was able to spend longer on certain topics, often blending from one lesson into the next. But as new staff joined the department, I realised that wasn’t helping them. To support them better, I had to decide on what to cover in each lesson, and make that clear to them in the textbooks.

Regular Exam Practice

It doesn’t matter how well taught pupils are in the fundamentals; if they don’t practice exam technique frequently enough, they will struggle in assessments. I have dedicated more time to this over the years, and now all of our pupils answer a 25 mark exam paper every week.

Practicals

Untangling practical skills from ‘Bloom’s Taxonomy’ skills took a while. Now, before every practice, pupils are explicitly taught practical skills such as identifying variables, plotting graphs, choosing graph scales, completing risk assessments, writing conclusions and evaluations.

Drilling

We’ve switched from verbal to written drills over the last year, and I wouldn’t go back. This has really helped to increase the amount of practice each pupil gets, and gives me clear, instant feedback that I can focus on immediately as well as in subsequent lessons.

Quizzes

In the first year, we spent hours each week planning multiple choice quizzes. At one point, I was even differentiating these for each class, which was extremely time consuming, and didn’t seem to have much impact on learning. Now, we simply test content taught that week in a simple format. The quizzes increase in complexity each week, meaning that there is enough stretch built into the unit for the most able pupils.

We haven’t perfected it yet, but learning from our mistakes has helped to department to grow, supporting pupils and staff to achieve their very best.

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.

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.