Michaela Community School | The latest news at Michaela School
Michaela School Wembley London
268
paged,page-template,page-template-blog-template3,page-template-blog-template3-php,page,page-id-268,paged-2,page-paged-2,tribe-no-js,ajax_leftright,page_not_loaded,

Teach Knowledge

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

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

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

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

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

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

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

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

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

UK Maths Challenge

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

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

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

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

Procedural knowledge:

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

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

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

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

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.

Assessment in a Knowledge Curriculum

I have written and spoken at length about simplification. In short, I have come to believe that a knowledge curriculum simplifies everything we do as teachers. Rather than considering engagement, entertainment, or pupil interest, a knowledge curriculum relentlessly and ruthlessly prioritises kids learning stuff in the most effective way: that is, reading it, writing about it, and being quizzed on it.

In my past life, here are some ways I assessed pupil learning:

  • Painstakingly marked their books with lengthy written targets for improvement that pupils responded to
  • As above, but for essays and assessments
  • Used spurious National Curriculum levels to denote the level the child appeared to be writing at
  • Developed an assessment ladder based on vague descriptors provided by GCSE exam boards to denote how far a child was from the GCSE expectations
  • Had pupils complete multiple choice exams which, having sweated over making, I would then have to mark
  • Had pupils swap books with one another to write insightful comments such as: ‘good work. Next time, write more’
  • Asked pupils to tell their partner what they know about a topic
  • Asked pupils to write a mind-map of what they know about a topic
  • Asked pupils to make a presentation of what they know about a topic

Not only are the above techniques unnecessarily complicated, they almost never gave me any useful information about what my kids could do.

At Michaela, we ask the kids questions constantly. Every lesson begins with two to five practice drills. In English, this would consist of two or more of the following:

  • A spelling test
  • A vocabulary test
  • A grammar drill
  • A gap-fill on a poem the pupils are memorizing
  • Knowledge questions on a previous unit
  • Knowledge questions on the current unit

We then read some material, and ask the pupils questions to ensure they have understood. The pupils then answer some questions about the material. We then go over the questions as a whole class, and pupils edit their responses using the whole-class feedback. For a lengthier piece of writing, I would use a half-page of feedback as outlined in my post ‘Giving Feedback the Michaela Way.’

For our bi-annual exams, pupils write an essay or, in subjects like Science or Maths, complete an exam paper that tests their ability to apply their knowledge. They also complete two to five ‘knowledge exams,’ which are simply open answer questions about everything they have learned that year. (Example questions from English could be: ‘What is a simile? When did Queen Elizabeth die? When was Macbeth first performed and where?’) We don’t painstakingly mark every paper – instead we sort them swiftly into three piles: A, B and C. A quick glance can tell us how a pupil has done – lots of gaps is a C, a sample glance at a number of correct answers and all questions attempted with a well-worked extension an A; everything in the middle a B.

The reason we can assess so simply is that in a knowledge curriculum there is a correct answer. There are, though we love to deny it, right and wrong things to say about literature. At Michaela, we are explicit about this. When I asked for pupil inferences about Curley’s wife in Of Mice and Men in my previous school, I remember asking them what the colour red could symbolise. Their answer, ‘jam,’ was simply wrong. What we do at Michaela is to codify the knowledge we want the pupils to learn, teach that knowledge, and then relentlessly test that knowledge.

Simple.

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.

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.