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Posted on March 13, 2016 by Naveen Rizvi

Sequence of Instruction

School visited: Uncommon Collegiate Charter High School

Class observed: AP Grade 11 Statistics (UK equivalent Year 13)

Theme of post: Pedagogy and Curriculum

This is the second blog of a series to complement a workshop I delivered at #Mathsconf6  on Mathematics Pedagogy at USA Charter Schools, following my visiting a selection of Uncommon and North Star academy schools in New York and New Jersey. During the workshop, I discussed how the common denominator in each maths lesson I observed was the level of academic rigour. I am still trying to find a way to define this term, because it is overused and poorly defined.

Nevertheless, I believe I can state that the teaching and learning delivered by such extraordinary teachers in the classrooms I observed displayed features of academic rigour. These features of academic rigour are evident in the structure of a teacher’s pedagogy and instruction:

Academic rigour featurse

In my workshop, I discussed the four main elements that all teachers up and down the country can implement into their teaching practice with minimal effort and maximum impact. I will discuss each of these four elements in turn, by way of one per blog post; today I will discuss the sequence of instruction.

I was observing a lesson, Grade 11 Statistics, and the teacher had organised the lesson so that he was trying to have pupils factorise cubic expressions where one of the expressions in the factorised form could be further factorised into two expressions because one group  was a difference of two squares. (Figure 1) The sequence of instruction to explain to pupils how they can factorise such cubic expressions is what made his lesson so powerful.

Fig1SIFigure 1 – Sequence of Instruction in terms of expressions used in teaching

Here, I entered the classroom, and the teacher had displayed the following algebraic expressions that he wanted pupils to factorise. (Figure 2) Before the pupils had even started to factorise he spent some time identifying the common features which determined this expression to be a difference of two squares. It is known as the difference of two squares because we have even powers, square numbers as coefficients and one positive term and one negative term.

Fig2SIFigure 2 – Difference of two squares expressions + common features of this type of expression

What I found really fascinating was the examples that he had initially chosen in order to begin teaching pupils how to factorise such expressions. Previously, whilst teaching, I would begin teaching the topic by using the following examples and non-examples to state what the common features of an expression known as a difference of two squares actually looked like, and therefore what it would not look like. Therefore, I utilised simply examples and non-examples. (Figure 3)


Figure 3 – Examples and Non examples used in the initial stages of teaching how to identify an expression known as a difference of two squares

The way the teacher started was better, because the expressions he had used made the common features more obvious to pupils. So the initial examples of a difference of two squares that I would have started to teach from are actually more nuanced and special; the coefficient of x2 being 1 is hidden, and instead of having two terms where the variables both have even powers we have a constant which is a square number.

Second, he then introduced the consistent algorithm he wanted pupils to use to go from the expanded form to the factorised form of the expression. He was very systematic (Figure 4 – consistent algorithm):

Step 1: square root both terms;

Step 2: divide both exponents by 2 and

Step 3: Write your answer as such.

He did this with a selection of examples, as you can see below:


Figure 4 – Consistent algorithm to factorise an expression known as a difference of two squares

Following this, he then introduced the notion that the pupils would now learn how to factorise a cubic expression. At this point in the lesson, I wasn’t fully sure with where he was going (until we reached the end of the lesson). He stated that when we factorise a cubic expression as such we need to factorise the first two terms and the last two terms. He structured it for his pupils because he pre-empted a common mistake that pupils would have made unless he structured it for them. We are factorising the first two terms by dividing both terms by 1, whereas with the last two terms we are going to factorise by -1, because -1 is the greatest common factor of -8p and -24. (Figure 5)  Therefore, the first step of the algorithm is not line 2: it is line 3. This was so powerful because by pre-empting this misconception, and by being very explicit with this instruction and guidance, it made the lesson run more smoothly because pupils were more independent to get on with the practice set of questions – because they weren’t making this mistake.


Figure 5 – Structuring the first step of the algorithm

He then carried on to lay out the consistent algorithm between the expanded form and the factorised form.

Step 1: factorise the first two terms, and the last two terms. (be careful when factorising the last two terms)

Step 2: Now factorise the greatest common factor for both terms (ensure that you are factorising the GCF of the coefficients as well as the variables) However, he posed the question one more time: “What is the greatest common factor  of 16p^3 and 48p^2?” Students then noticed that we are no longer factorising by identifying the greatest common factor of the coefficients but also of the variables of each term. (Figure 6)


Figure 6 – Step 2 – Step 4 demonstrated

Step 3: factorise the greatest common binomial of the expression:

Again, the algorithm was consistent. Furthermore, he said factorise “the greatest common binomial”. This level of technical language being used in the classroom is fantastic, and is something I certainly intend to implement in my teaching practice.

Step 4: “Are we done with this problem?” he asked this question with the intent of having students identify that the first expression in parenthesis is not fully factorised – which pupils identified.

Here he paused and said to his pupils “Now you are thinking that we are finished because we have two groups that can then be expanded and simplified to form the expanded form which we start with, but we are not finished. What is the greatest common factor of the first group?” However, he posed the question one more time “What is the greatest common factor of 16p^2 and 8 ?”.Pupils then identified this to be 8. He then proceeded to do a selection of examples to complement the first example, and to also reiterate the consistent algorithm. Figure 7 

Fig7aSI Fig7bSI

Figure 7 – Further examples of factorising cubic expressions presented in class using a consistent algorithm

Now, this was the pinnacle of the lesson. This was the point where the teacher introduced the next expression to factorise but in doing so he was combining the procedural knowledge of factorising a cubic expression where one of the groups could be factorised because it was a difference of two squares. The sequence of instruction was incredible. Let’s have a look.

Again, he used the consistent algorithm he showed in the previous examples:

Step 1: factorise the first two terms, and the last two terms. Pre-empt that we may be factorising the last two terms by 1 or -1.

Step 2: factorise the greatest common factor of both expressions:

Step 3: factorise the greatest common binomial of the expression.

He asked “can you factorise the greatest common factor of the second binomial? Hands up – what is the greatest common factor of the second binomial?” Pupils confidently answered – “1”. The teacher responded “Excellent, you can display it like this (teacher writes it on the board) and can you do the same on your mini-whiteboards please.” Again, he pre-empted another misconception that the pupils could have potentially made. (Figure 8)


Figure 8 – Structuring a key step in the algorithm where pupils are more likely to make avoidable mistakes though explicit instruction.

Figure 9 – Introducing an expression where one group is an expression known as a difference of two squares

Step 4: factorise the greatest common binomial, which he got above (Figure 9):

He asked pupils to pause. He then asked pupils to look at the problem on the board (the one above). “Can you raise your hand when you can identify what type of binomial expression we have for the first binomial .” Ten hands were raised within the 5 seconds. In total there were 22 hands raised 10 seconds later. Now, after 30 seconds in total following the initial posing of the question, the teacher had an overall 28 out of 32 of pupils’ hands raised. The teacher selects a pupil to state the answer: “Sir, the first binomial is an expression which is a difference of two squares.”

The sequence of instruction here is incredibly effective. What I saw in real time, and what I have outlined for you, is not necessarily profound or extraordinary – on the contrary, it is very simple. It can be performed by all teachers.

It was evident from this lesson that the teacher had put a significant amount of thought into the sequence of his instruction to his pupils. It was well-crafted and sequenced so that all pupils could see the re-iterated key points. What are the characteristics of a problem which is a difference of two squares? What are the characteristics of factorising a non-linear expression? What is the greatest common factor of this expression – ensuring the greatest common factor including coefficients and variables?

I am still working on trying to collate a few different examples in my teaching practice of minimally different examples, and how I have interleaved concepts via sequence of instruction. I shall get back to you on this. Watch this space. Any questions then please do not hesitate to contact me or DM on twitter.

Posted on March 12, 2016 by Joe Kirby

Year 8 on Macbeth

Y8 Macbeth Witches.png

Y8 Lady Macbeth.png

Y8 Lady Macbeth 2.png

Y8 Macbeth Conclusion.png

These are some paragraphs of Year 8 pupils writing about ‘Macbeth’, on the witches, Lady Macbeth, and an essay conclusion comparing Macbeth to the play they studied in Year 7,‘Julius Caesar’.

Snapping and sharing pupil paragraphs is a good opportunity for us to reflect on what is working well about our English teaching and what we want to help pupils improve.

Strengths: Vocabulary and Connections

One strength of pupils in English at Michaela is the vocabulary we have developed in them. Pupils are using ambitious words like ‘eponymous’, ‘pejorative’, ‘despotic’, ‘ambivalent’, ‘susceptible’, ‘diabolical’ ‘paradoxical’, ‘contradictory’ ‘asymmetric’, ‘tyrannical’, ‘protagonist’, ‘conspiracy’ and ‘patriarchy’, and largely using them accurately. They are using technical vocabulary, such as ‘soliloquy’, ‘antithesis’, ‘pathetic fallacy’ and ‘chiasmus’, and largely using it accurately. Sometimes, though, they are misspelled, as in ‘purnicious’, and sometimes, they make mistakes, such as seeing an oxymoron where none exists!

Pupils are beginning to make some striking connections between, for instance, the context and the play: for instance, on the regicidal 1605 Gunpowder Plot, or James I’s 1603 accession and his belief that he was descended from Banquo. Contextual connections are a real strength of our knowledge-led approach to the curriculum. They annotate and improve their own paragraphs, as you can see in green and blue pen. 

Improvements: Le Mot Juste, and Perceptive Insight

At this stage, I find pupils struggle to find ‘le mot juste’. One has written: ‘This emphasises the witches’ cloudy, disturbed and tempestuous mindset, bleeding into Macbeth’s mindset.’ I’m not sure ‘cloudy’ is the best adjective this pupil could have chosen! Another has written in a contrast between Macbeth and Caesar: ‘we cannot make these characters equal, however, because Caesar was a victim of conspiracy and assassination, Macbeth was part of conspiracies and murders.’ I find the phrasing of ‘make these characters equal’ a little clumsy, and the sentence is missing a ‘whereas’. Articulate, sophisticated syntax is an area for improvement.

Increasingly, what I’d like to improve about our teaching is this: sharing concrete examples of perceptive insight – moments in their analysis that make the reader of the essay see the play in a different light. I will try to do so in another blogpost.

By the time our pupils finish Year 9, they will have studied five Shakespeare plays. Year 10 begins with a synoptic Shakespeare unit where they compare villains and stagecraft, themes and plots from Julius Caesar, Romeo and Juliet, Macbeth, The Merchant of Venice and Othello, before they start on The Tempest, Shakespeare’s last play, for their GCSE course. By then, their intertextual connections, comparisons and contrasts should be increasingly insightful. It is an exciting prospect!

Posted on March 12, 2016 by Jessica Lund

Something we do – Reading

At Michaela, pupils read a lot – hundreds and thousands of words in every lesson.  The same is true in French. This is a snip of our Year 7 booklet:Capture

As you can see, it’s full to bursting with words.  Long passages of French with parallel translations into English.  And not just normal English – dodgy English that enables pupils to see the precise connections between French and English words.

“Yesterday, I am goned to the pool after having done my homeworks.”

“Hier,           je suis allé   à la piscine  après avoir fait     mes devoirs.”

Pupils read with their rulers under each line (the better to see and track exactly what they’re reading.  They are questioned constantly to see if they are making the connections between the English and the French.  If they struggle to see and retain a link, we make it even easier for them:

“Quelqu’un.  Someone.  Quelque means ‘some’ and ‘un’ means ‘one’.  Obviously you can’t have two vowels fighting it out in the middle so the ‘e’ becomes an apostrophe.  Look at the spelling – QUeen ELizabeth is off visiting another QUeen who has a feather ‘ in her hat.  And look at what she’s riding: a UNicycle.  And if you can’t remember that, it’s 6 letters then 2 letters.  Count them.  If you remember that it’s 6’2, you’ll know how to spell it when you need to.”

As one pupil reads at length, pupils track what they are reading for themselves.  A beautiful thing happens: pupils who aren’t reading out loud are silently mouthing the words so that when their turn comes, they remember the pronunciation.  They listen intently to their peers, and if they make a (very rare) mistake, hands shoot up all over the room to correct it.  Pupils are now in the habit of self-correcting as they read.

So, pupils are always looking at French words, and having their attention drawn both to the syntax of the sentences and the shape and spelling of individual words.  They are always hearing French words: if it’s not the teacher reading out loud, modelling excellent pronunciation (always with a bit of a dramatic flourish to emphasise key vowel sounds and accents) then it’s their peers practising.

Their peers are confident in their reading.  Why?  Because a) we do it so much, b) they are positively encouraged all the time (“PLUS FORT! Lion voices!”), c) we make it easy for them with two key techniques:

  1. We put dots under silent letters.  Pupils don’t pronounce them.  They know they’re there, and that they need to be written, but they don’t say them.  If they, on the rarest of occasions, do, their peers fall about laughing and they quickly correct themselves with a grin.  We don’t dot them forever: by February, year 7 read competently without them.  They become used to the fact that ‘évidemment, les lettres -nt à la fin sont muettes, Mademoiselle’, and they tell me, too.  Yesterday a pupil saw the word ‘jettent’ for the first time – it didn’t even occur to him to say the -nt at the end.
  2.  We underline the common vowel and letter combinations that differ from English. ‘in’, ‘en’ and ‘an’ are common ones that pupils might slip up on – we are constantly reinforcing their pronunciation by drawing parallels to other words. Lapin, sapin, main, maintenant, dingue, zinzin, radin, malin, calin, intéressant, intelligent.  Eu – peu, eux, peux, veut, neuf, malheureusement, chaleureusement.  They become used to sounding these out correctly.  As a result, ‘new’ texts are never really new – they’re simply a reordering of things they’ve seen hundreds of times before.

When I was training, I was told that reading was a presentation/input activity, not a production/output activity.  N’importe quoi!  At Michaela, it’s both – we read out loud, all the time, sometimes for entire lessons.  Because we foster confidence, pupils love it, and they are starting to really act with it.

Having recently started to read ‘Ticked Off’ by Harry Fletcher-Wood, I thought I’d create a ticklist of things that our pupils do when reading.  Here it is:

Do they read lots, at least a few hundred words per lesson? Do they understand, and are they able to recall, what they read? Do they read high quality, interesting, real-world French? Do they regularly read mark-winning structures, including PROFS (past, reasons, opinions, future, subjunctive)?  Do they read out loud with confidence and pizazz?  Do they notice vowel combinations and patterns that aid with pronunciation, recall and accurate spelling? Do they quickly correct themselves, and others, when they make a daft mistake? Do they absolutely love reading?

I would absolutely love to get more people visiting Michaela to see what we do – both in the French department and across the whole school.  If you want to know the answers to these questions, come and visit any time – have a tour, have lunch, watch a lesson, talk to us. It’s brilliant. Email me at jlund@mcsbrent.co.uk. It’d be lovely to see you.

Posted on February 21, 2016 by Bodil Isaksen

What’s the point of knowledge if you don’t use it?

Production. Outcome. A result. Project-based learning seems intently focused on this. Ron Berger’s An Ethic of Excellence has a lot going for it: its attention to disciplined, persistent work is admirable, but it, too, falls into this trap. Learning is only seen as valuable if something is donewith it.

In a similar vein, Debra Kidd said of School 21:

What is the point of learning if no-one hears you? What is the point of academic success if you can’t interview successfully? What is the point of knowledge if you don’t use it to change the world?

This view of learning just doesn’t resonate for me. There is intrinsic value in learning: in making your mind a more interesting place to be; in the wonder and awe that new information can bring. We may be unlikely to use, in a practical sense, much of what we learnt in school, at university and in life, but I certainly feel privileged to know it. Why do David Attenborough documentaries get so many viewers? Why do long reads get millions of clicks? Why do books still sell in an age of Kim Kardashian apps and Candy Crush? Because learning is wonderful, whether or not it is shared or used.

As for interview skills, my mind is immediately drawn to my first week of university, at the matriculation dinner. By the third course, the computer science students had ceased engaging in conversation and broken out their laptops. A lot of them would be the first to concede they would not interview well (how do you know a computer scientist’s an extrovert? They look at YOUR shoes when they’re talking.). But that damned well did not render their academic success pointless.

It’s not just ivory towered academics who benefit from knowledge in the absence of interview skills. A plumber probably won’t need to know much Shakespeare on the job, but knowing which plays he wrote and some of the fabulous lines would mean he feels just as entitled to go and see a play at The Globe as anyone else – and why not? The works of the Bard are worth knowing because of their greatness, regardless of whether you use them for practical purposes.

Corporations will inevitably concentrate on production and utility: I expect the CBI will be grumbling about skills for work on the Today programme every few months until I retire. But we teachers need not fall into that trap. We can recognise and celebrate all knowledge: the useful and the useless, the easily accessed and the esoteric, the applicable and the arcane.

Knowledge is an inherent good. Let’s stop obsessing over what we can produce with it, and simply laud learning in its purest form.

Posted on March 9, 2016 by Dani Quinn

Masses of Maths: what should pupils learn by rote?

Should maths be learned by rote?

Some of the most egregious pedagogy is born when the answer to that question is ‘100% yes’ or ‘100% no’.

“100% yes” conjures up – perhaps rightly – an image of maths as a joyless subject whereby pupils are learning algorithms without meaning. Although it can feel like an easy way to teach, pupils are unlikely to succeed with equations such as  = 17 if the approach to linear equations has simply been ‘change side, change sign’ and practise only the simplest problem types (e.g. 4a + 3 = 23). Automaticity with times tables, simple written calculation and being able to regurgitate the order of operations is of limited help if the pupils aren’t taught how to think flexibly (i.e. if they can’t see the deep structure of a question).


“100% no” is also problematic. Typecast as the progressive approach to maths, it is founded on exploring maths as a way to develop deep understanding (and an assumption that fluency and confidence arise from there). It is championed by academics such as Jo Boaler and many teachers (and maths consultants…), and the heart of much debate. This approach argues that relational facts needn’t – and shouldn’t – be taught as such and certainly don’t need to be explicitly memorised.

Relational facts are those that can be derived from a smaller field of arbitrary conventions (such as ‘angles in a straight line sum to 180o is derived from the convention that angles around a point sum to 360o) or easily understood and recalled relationships (e.g. I can calculate 3 x 8 by doubling a relationship I do recall – 3 x 4 = 12 – to get 3 x 8 = 24).

There is clear merit in an approach that builds relational understanding1. It is an important part of building the storage strength of concepts2 (how well a concept or fact connects to other memories and concepts) but, used alone, it ignores what is happening in pupils’ brains as they work.

Simplistically put: as pupils work on a new problem or idea, their working memory is gradually being ‘used up’ until there is little capacity for additional processing. Take this problem:

0.8 + 0.4 x 52 ÷ 0.01

A pupil has to think about all of the following:

  • The order of operations (that they should complete the multiplication and division first AND that, within that, that they should work from left to right)
  • What the notation []means
  • The value of 52
  • A strategy to multiply an integer by 0.4
  • A strategy to divide by 0.01
  • How to add 0.8 to the answer

That is a lot to think about! If trying to think about each idea from scratch, their working memory will soon overload, making the calculation seem more complex than it is.

In comparison, the problem is much simpler for a pupil who confidently knows the following facts by heart:

  • 52= 25
  • ÷0.01 = x100
  • To multiply an integer by a decimal, I can ignore the place value at first and adjust afterwards
  • 4 x 25 = 100
  • 4 x 25 = 4 x 2.5
  • 4 x 2.5 = 10

…will see this instead:

0.8 + 0.4 x 52 ÷ 0.01 = 0.8 + (0.4 x 25 x 100) = 0.8 + (4 x 2.5 x 100)

A much less daunting calculation, and one where much less tricky processing or self-doubting thought has taken place.


What does a knowledge grid have to do with it?

In the Michaela maths department, we aim to identify all the facts and relationships that can be codified as a single nugget of knowledge (or set of clear steps) that will reduce pressure on pupils’ working memories. This frees them up to tackle more complex and interesting problems and allows them to feel confident in their reasoning and solutions.

This does NOT mean teaching without understanding. It is the opposite: we aim for pupils to understand why something works, or is the way it is, and then to be so confident of that fact or relationship that they can recall and use it with minimal effort and worry.

The purpose of a knowledge grid – explained in detail by Joe Kirby – is to set out what these facts and relationships are, and to support pupils in learning them by heart.

Take indices, which the Y7 pupils have just learned about:

indices knowledge grid

This sets out what we expect pupils to know by heart if they are going to be able to tackle more complex or interesting problems involving indices (e.g. What is the final digit of 10100+999+598?). Know by heart that ax a= ab+c doesn’t replace knowing why this relationship is true. But, knowing it by heart – and practising explaining why it is true – frees pupils up to tackle problems like ‘evaluate 2x 52x 2x 53′.

Here is the grid for Y8 pupils at the outset of learning to solve linear equations:

masses of maths 3

Here is an example for Y8s learning to substitute and use formulae:

masses of maths 2.PNG

Sometimes it is solely a collection of relationships, such as the grid Y7 are about to work from:

masses of maths 4

(shading in grey typically indicates ‘optional’ knowledge, in that it is possible to be successful in maths without knowing those facts by heart…at least not at their stage!).

A useful rule of thumb is: if we, as maths teachers, know these facts by heart because they help us work more efficiently and confidently, then the pupils should know it by heart too.

How is it used?

In lessons, the knowledge grid lays out the agreed definition and procedures that we want to share with pupils. The constraint of the definition means we teach to a higher technical standard, ensuring that we stick to language like ‘eliminate this operation’ (instead of saying ‘get rid of the 4’ in a bid to make the maths feel more accessible). Knowing that the pupils must understand and use a phrase like ‘isolate the unknown’ forces us to explain it with greater clarity, check they understand it precisely, and then use it constantly.

In most lessons, pupils are quizzed on the terms and facts in the knowledge grids. This can be cold calling (asking questions and picking students), checking everyone’s answer on mini-whiteboards, or giving a 1-minute quiz in books (e.g. “write the formula for the area of each of these shapes” or “rewrite each of these as a multiplication: ÷0.5, ÷0.1, ÷0.25, ÷0.125, ÷0.01, ÷0.2”).

Once a week, pupils ‘self-quiz’ at home on the definitions and facts the teacher has set for that week. Typically, this is 10-15 facts/definitions. Pupils first practise saying the facts to themselves, then cover the right-hand side and write the definitions based on the prompts on the left-hand side, and then correct their errors in green. They continue this until a page is filled. It is possible to game it by mindlessly copying, but it becomes obvious if they’re doing so because…

Once a week, pupils take a formal, but low-stakes, written quiz, of which half will be a knowledge grid test (the other half tests their ability to apply procedures and try unfamiliar problems).

The levels of scaffolding vary; these are the knowledge grid sections Y8 took recently:

masses of maths 5.PNG

masses of maths 6.PNG

Pitfalls We Fell Into

An easy temptation is to produce a ‘revision mat’ full of facts, examples, diagrams and mnemonics. Although this is close to a knowledge grid, it isn’t as useful. It must be REALLY EASY to test yourself from a knowledge grid without ‘accidentally’ seeing the answer, or having prompts. It must be really clear what they should know by heart (the definitions and terms and facts) and what is just useful for jogging their memories (examples, where appropriate).

Another easy error is to go overboard with how much you try to codify and write down. If you, as teachers, struggle to articulate the definition or steps for something, it probably isn’t useful or suitable. Make steps for a strategy (e.g. solving equations) as generalised as possible so that pupils aren’t learning multiple minimally different steps and becoming muddled and frustrated. The more generalised the steps, the more they can be used to illuminate the common features of varied problems (and thus help pupils see the underlying structure).

Pitfalls We’re Still Trying to Avoid

We are still struggling to decide which aspects of algebraic simplification can be listed as facts: here is the start of a debate I was having in my head this morning for updating the facts in the ‘expressions and simplification’ grid:

masses of maths 7.PNG

Any that are included are there because pupils had become faster by recalling them as facts (as opposed to working them out) or their work was slowed because they weren’t confident when simplifying a fundamentally identical expression.


I hope it goes without saying that we would love to know what you think and if you have tried anything similar. Do you have facts and rules, besides those set out in examination specifications, that make a big difference to your pupils when learned by heart?


Whether this fascinates or enrages you, get in touch and come see the pupils (and grids…!) in action. You’ll have a great time 🙂


1: See Skemp, R.R (1977) Relational Understanding and Instrumental Understanding, Mathematics Teaching, 77: 20-6

2: See https://www.youtube.com/watch?v=1FQoGUCgb5w for Bjork discussing research in this field.

3: https://pragmaticreform.wordpress.com/2015/03/28/knowledge-organisers/


Posted on March 6, 2016 by Jonathan Porter

The Golden Mean Part 2: Sample and Domain

In my last blog I spoke about the ‘Golden Mean’. This is the Aristotelian principle that what is virtuous is always between two states: one of absolute excess, the other of absolute deficiency. I suggested that this could be quite a helpful way of thinking of teaching and curriculum design. I said that over the next few posts I would try and explain how this insight has informed my thinking on teaching and assessment in KS3 history. Here are three questions that came to mind:

  1. How do I strike the right balance between the sample (the end of unit test) and the domain (in this case, the period of history in question)?
  2. How do I help my pupils remember what they need to remember for the assessment, as well as helping them to remember what I think is just important for them to remember?
  3. And how do I guide them towards an answer without answering for them?

Sample and domain

 In my last blog, I explained why I think the sample/domain issue is one that particularly afflicts the teaching of history at KS3. The more our assessment system leverages the assessment, the more tempting it is for the teacher to narrow the domain and teach to the test. This is particularly so in schools like my own where pupils, even in Year 7, do all assessments without notes and from memory. The alternative, however, is arguably just as bad: unpredictable tests sprung on new arrivals to the school, which help us to norm-reference one pupil against another, but potentially crush a child’s confidence in the process.

For this reason, I am in favour of giving younger pupils the question in advance. But only if the question itself avoids the sort of ‘narrowing’ I’ve described. An example of this sort of narrowing might be a very specific question about the significance of the Magna Carta or the ‘greatness’ of Alfred the Great rather than a broader question that aims to assess whether pupils have retained a good knowledge and understanding of the period in question. Alfred the Great is perfectly interesting, but I don’t want my pupils’ appreciation of Anglo-Saxon history totally dictated by what could become both a semantic and substantive discussion of his ‘greatness’.

In part, as was put to me, this is just the age-old question of breadth vs. depth. In the case of history, we often choose to sacrifice breadth over depth. With a limited amount of time, we cannot possibly cover everything that happened in, say, Anglo-Saxon or medieval England. We also want to ‘scale-switch’ and to help our pupils see continuity and change in the short, medium and long-term. I think this is true.

But I think what we’re also seeing is some of the effects of the ‘skills-centred’ attitude to the teaching of history. That is ‘We don’t need to worry about teaching a broad sweep of Anglo-Saxon history because it is learning to think about causality/significance/continuity/change that is most important.’ I think this is a mistake. Why? Because I think KS3 gives us a great opportunity to put in place the broad domain knowledge that we need for long-term expertise. In the long run, enquiry questions that are too narrow have the potential to erode the domain at exactly the point in a child’s development when, in my view, the domain should be most broad. With that in mind…

All enquiry questions are equal, but some are more equal than others.

 I think we can find a Golden Mean between the sample and the domain – between breadth and depth. I think we can have enquiry questions that ask our pupils to consider second-order concepts while also giving them the best possible opportunity to learn – and remember – as much as they can of the period in question.

Here is the question that we settled on for our unit of work on medieval England:

‘What was the most significant challenge to the King’s power in medieval England?’

 I expect a good response to include a paragraph on each of the three challengers (the Church, the nobles and the peasants) and top answers to include a fourth paragraph in conclusion. The pupils have 45-50 minutes and all pupils write from memory without notes or sentence starters. We try and achieve this through regular self-quizzing of both the upcoming ‘sample’ (What was the most significant challenge to the King’s power in medieval England?) and the broader domain (the chronology of medieval England). You can see that I’ve tried to place an equal emphasis on this on the pupils’ knowledge grid, which they use in weekly self-quizzing as well as in class.


My hope is that this strikes a ‘golden’ balance between the sample (right-hand side) and the domain (left-hand side). The pupils learn about, and have to remember, important events in medieval English history (The Battle of Hastings, the Anarchy and the Battle of Agincourt), which are not needed to do well in the end-of-unit assessment. At the same time, I hope I’ve also supported all pupils to learn the specific information that they’ll need to do well in the assessment.

Next time, I’ll write a little bit more about how we use these grids, low-stakes testing and songs to try and help pupils automate people, dates and concepts so that their working memory is focussed on the most challenging parts of the assessment.

Posted on March 6, 2016 by Naveen Rizvi

Minimally different examples

School Visited: Uncommon Collegiate Charter High School

Class Observed: Algebra 1 Grade 9 Statistics (UK equivalent Year 10)

Theme of post: Pedagogy and Curriculum

This is the first blog post in a series complementing a workshop I delivered at Mathsconf6 entitled “Mathematics Pedagogy at USA Charter Schools”. Here I spoke of my experiences whilst visiting a selection of Uncommon and North Star academy schools in New York and New Jersey.

I discussed how the common denominator in each maths lesson I observed was the level of academic rigour in the teaching delivered to pupils. I am still trying to find a way to define this term, because it is overused and poorly defined. However, I do believe I may be able identify the evident features of academic rigour in the structure and delivery of each teacher’s pedagogy and instruction:

Academic rigour featurse

In my workshop, I discussed four examples of academic rigour that different teachers up and down the country can implement with minimal effort and maximum impact. Each blog post will discuss one example at a time. Today, this blog will address using minimally different examples during instruction.


Minimally different examples are examples to explain different concepts where the difference in the algorithm between one concept and the other can be distinguished as one additional step. This was one example I witnessed in one of the classrooms I was in.


Minimally different examples 1

The teacher presented the first expression to factorise, where the coefficient ofa in the quadratic expression is 1. This was a recap exercise where she stated the values for the coefficients, and then she determined the two values of which the product gives us the value of c, and the sum gives us the value of b. She wrote the expression in its factorised form.

She then introduced the next expression to factorise. She stated that it looks visually very different to the first one, but that there is only one difference between the algorithm to factorise the first and second expression. She defined the minimal difference between the two as one additional step – which was at the start of the algorithm. She asked her pupils, “What is the greatest common factor (GCF) of all the terms in the expression?” Pupils spotted the GCF to be 3p, which she then factorised. She said “look, we now have an expression in the parenthesis which looks similar to the first expression we factorised.” The minimal difference between factorising the first and second expression was classified as one additional step. Traditionally, in my training, I would have seen the algorithms to factorise both the expressions as isolated and fragmented. Instead, they are connected; pupils were empowered to factorise more complex expressions from prior knowledge of the algorithm to factorise the first expression. Minimally different examples are a good strategy and example of academic rigour in the classroom because they help pupils to understand knowledge and concepts which are initially complex and ambiguous. Here, the difference between the two examples is classified and determined as one additional step in the algorithm. The possibilities are endless.

I plan to show different examples of how I have used the idea of minimally different examples in my teaching, sometime this week.

Posted on March 5, 2016 by Jo Facer

Reading Reconsidered

Teach Like a Champion,’ by Doug Lemov, changed my teaching profoundly: it was the most practical and helpful piece of writing I had ever encountered, and transformed my classroom practice, giving me specific aspects to hone and improve.

When I heard that Lemov had been an English teacher, it didn’t surprise me – in particular, in TLAC 2.0 there are several techniques which are especially useful for the English teacher. When I heard he was co-authoring a book on reading, I had very high hopes. ‘Reading Reconsidered,’ written by Lemov with Erica Woolway and Colleen Driggs, does not disappoint. With a nod to the poetic importance of literature (‘this book is about the enduring power of reading to shape and develop minds’), again, we have a manual for practice; specific things that teachers can do, day in, day out, to read effectively with pupils.

‘Reading Reconsidered’’s opening gambit is that text selection is key: in pages referencing Hirsch and Arnold, canon and cultural capital, the writers note: ‘part of the value of reading is to be able to read and talk about important books that almost everyone else has read.’ The great conversation of literature, intertextuality, ‘works only when pupils have read some texts in common.’ The writers extol the value of a common reading curriculum for all pupils, and warns us to select our texts carefully, noting: ‘a typical pupil might read and intentionally study forty or fifty books in English classes’ over their time in school and ‘these few books form the foundation of their knowledge of how literature works.’ If only there were as many as fifty texts! Such a sentence puts the demands of the new GCSE English Literature, with its four texts (I include the poetry cluster as a ‘text’) over two years into frightening perspective. To those who argue that canonical texts are unreadable by youngsters, the authors respond: we just need to get better at teaching them.

The writers go on to isolate the ‘five plagues of the beginning reader’, looking at five challenges all readers face in encountering tricky texts, and how we can overcome these in our everyday practice. One example is using ‘pre-complex texts’ to prime pupils for the canon, such as children’s classics like ‘The Secret Garden’ which use challenging syntax but have child-friendly story arcs.

The chapters on close reading are a must-read primer for all English teachers, going meticulously through how we should read closely in class, supplemented with specific questions designed to unlock meaning in complex passages. One key take-away for me was: teachers! Prepare to close-read! Annotate your text! It sounds blindingly obvious, but I know I’ve been guilty of sauntering into class, blank copy in hand, hoping for the best. Yet what more important preparation can there be for a lesson than our own annotation?

The most revolutionary chapter for me in ‘Reading Reconsidered’ was that on non-fiction. It made me recognise how vital it is for pupils to read non-fiction alongside fiction to assist with their comprehension and to enable really excellent analysis: ‘reading secondary nonfiction texts in combination with a primary text increases the absorption rate of pupils reading that text’; ‘when texts are paired, the absorption rate of both texts goes up.’ Overall: don’t teach non-fiction as a separate unit, but rather interweave non-fiction texts into your teaching of literature, either with short, contextual glosses or in-depth historical study of the time period in question to deepen analysis.

Though reading is this book’s chief subject, the authors do not neglect writing: ‘we are suggesting that pupils [should] write with more frequency and consistency as part of their daily work of responding to texts’. They recommend intervening at the point of writing to help pupils improve (no mention of lengthy, burdensome and delayed marking), explaining: ‘great teaching begins at the moment learning breaks down.’ ‘Writing,’ in this guide, also encompasses annotation, and again there is detailed advice for modelling these, with the goal of pupils eventually annotating autonomously.

Again, though the goal is for pupils to read independently, we need to be aware that if they do this poorly they are ‘inscribing errors’ (and of course we know from Lemov himself that ‘practice makes permanent’). It is vital that pupils read aloud, as well as listening to great reading being modelled for them. In considering ‘accountable independent reading,’ the writers give such guidance as using short sections with a specified focus, or scaffolding pupil comprehension by using questions.

Although the focus of the books is practical, with advice to be found on the specifics of vocabulary instruction and the dynamics of a classroom discussion, the underpinning voice here is one deeply concerned with children loving reading and doing it effectively. The voice of the parent in each writer is heard most clearly in the book’s dedication: ‘to our kids, with whom we have 16,000 more nights to read – not nearly enough.’ Foundational to this book is a personal and deep love of reading, for all the right reasons.



Posted on February 28, 2016 by Naveen Rizvi

Act the anger, feel the warmth

I always believed that I had high academic and behaviour expectations of my pupils whilst working at my placement school in South Manchester. I was known to be strict at times because of my high expectations. I would give sanctions out to pupils who would not be giving me their 100% attention. I would insist my pupils SLANT whilst I was teaching to avoid fiddling. I would teach from the board and go through worked examples. My pupils knew how I wanted them to behave in my classroom. My pupils knew how I would behave if they were compliant or defiant. However, when I arrived at Michaela, my expectations were seen as low, and they were, and here I explain why.

At Michaela, we have 14 teachers and 6 teaching fellows where 9 members of staff are founding staff. I was observing a founding member of staff guiding and monitoring pupils to ensure that they were transitioning around school corridors in silence. I saw founding members ensuring that all pupils during assembly and class were slanting, if they weren’t then a teacher would give non verbal cues where a teacher would slant themselves and pupils would follow. I saw members of staff giving pupils demerits for not tracking them whilst they were speaking. At the same time, I saw founding members hysterically laughing with pupils during break time and lunch time when staff made witty comments about Mr Smith having luscious long locks of hair sarcastically (…Mr Smith has some hair). Pupils would be playing basketball or ping pong with teachers during lunch time. I would have pupils talking to me in the morning to greet me “Good morning Miss Rizvi, how was your evening?”

Now I look at pupils around the school and think now I know why Michaela pupils were so kind and polite. Let’s remember pupils did not come to secondary school as polite, kind or obedient as they currently are now. Pupils were trained and moulded to embody Michaela values. During our initial behaviour bootcamp, teachers taught pupils how to sit up straight in class. We taught pupils how to address a teacher. We would over exaggerate our behaviour when pupils made mistakes. For example,

“Excuse me Mr Rubbie, how dare you walk past me and not respond ‘Good Morning’ when I have wished you a pleasant morning. I am always polite to you so I expect you to always be polite to me,”

and “Zakye I am incredibly disappointed in your disingenuous apology and in you rolling your eyes when I was speaking to you! I am utterly horrified over how disrespectful you are to your teacher who works so hard for you.”

We would overexaggerate when pupils demonstrated exemplary behaviour:

“I am so proud of you Yasmine for scoring 100% in your science quiz this week, I would like to give an appreciation to Yasmine for her excellent self quizzing which resulted in her scoring 100%. We all want to be as successful as Yasmine. 1, 2 *two claps*,”

Olivia Dyer, Head of Science and Founding member of staff, said to whilst giving me feedback that “we act the anger, and feel the warmth.” Being angry and negative is emotionally draining for a teacher so we act when we over exaggerate and over justify why talking in the corridors is unacceptable.  Furthermore, we are wholeheartedly loving when a child is successful and we celebrate it to make that child feel valued, and to have surrounding pupils identify the kind of behaviour they need to demonstrate to be identified as an exemplary Michaela pupil.

This gives pupils clarity on how to behave and how not to behave. We do not give leeway to different extents of behaviour. If a pupil is doing something that they are not supposed to do then we would pick them up on it, no matter how subtle. At my previous school, I would give demerits because pupils would not be tracking me whilst I was teaching. If a pupil was rude even in the most subtle way such as smirking or curling their hands I would go ballistic. Surrounding teachers thought I was crazy for it.

At Michaela, teachers have sky-high expectations of pupils and our behaviour management is consistent because we all use the same language for rewarding and sanctioning pupils.

At my previous school, there was minimal consistency in rewarding and sanctioning pupils. We did not all have the same dialogue for sanctioning or rewarding students. If I ever pulled a child aside in the corridor for being too loud or demonstrating inappropriate and unprofessional behaviour then I would hear the response “Well [insert a member of SLT’s name here] did not say anything down the corridor when I was behaving this way.” It was exhausting. I would be the teacher who was crazy or too strict because of the high expectations I did have of the children. I would be unapologetic for giving a child a detention for not having a pen with them. I had one pupil who always forgot to bring in their exam prep folder, so for two weeks I proceeded to call the pupil’s house every morning at 7:15am to ensure that their daughter would bring their folder to school so they could be more prepared for my lesson.

At Michaela, all teacher’s have unapologetic and uncompromising high expectations of each and every pupil. Even more so our expectations are even higher for the pupils with the most tragic circumstances. We cannot change a child’s tragic circumstances at home but we can control their learning circumstances at school. If anything, it is more important to have high expectations for pupils who have those tragic circumstances so they have all the potential to escape their situation using education as the engine for such mobility. I know that the sanction I give to a student for a particular issue in science would also be sanctioned three floors downstairs in MFL by another member of staff. We are incredibly consistent. And students know that too…isn’t that the dream in any school? Students do not push their limits with us.

For example, we expect all pupils to bring a pen to school, and if they do not have one then we provide them with the opportunity to rectify the issue in a way which does not impact others around them. We have a stationary shop open between 7:30 – 7:50 am just before schools starts. The onus is on the children to meet our high expectations and we are explicit with what our expectations are. We are all consistent with the dialogue we have around school when we interact with children which contributes to a consistent and strong school ethos. As teachers we have bought into what is considered as Michaela standards. We have children bought into Michaela standards too.

There are many reasons behind the success of Michaela Community School and one of them is the uncompromising high expectations that all teachers have of each and every pupil our school serves. Children respect our standards because they understand that we want the best for them. Furthermore, to be the best pupil then pupils need to meet those high standards. My standards were considered high at my placement school where teachers were inconsistent with the dialogue they used and the reasons for sanctioning pupils or rewarding pupils. At Michaela, I realised my standards were not high enough. I had to raise my expectations, and I still am, but now I could not imagine reducing them. I would be failing myself as a teacher and the pupils that I teach. High expectations are underprioritised at majority of schools where at Michaela Community School it is part of the golden triangle of success.

We know that what we are doing having our sky-high expectations is right when a pupil who I have given detention to, which I have accidently forgotten to log, goes to detention and informs Mr Miernik that he has detention and that he is here to serve it even though Mr Miernik has informed him that he hasn’t got one. Or when I get an appreciation during Family lunch “I would like to thank Ms Rizvi for coming into school today and teaching us even though she isn’t feeling 100%” or when I get a postcard from a pupil expressing their gratitude.

Our Michaela high expectations in sanctioning and rewarding pupils comes from a place where we want our pupils to be the best possible student and human being. If you want to see us Michaela teachers and pupils in action then do come and visit us during a school day. Our doors are open to all. Come and have lunch with us.

Posted on February 27, 2016 by Olivia Dyer

Selling Science

Astronomy is my hard sell of physical science. Think of astronomy as the confectionary placed at the point of purchase in a well-known chain of high street stationary shops. How did the checkout boy, Alex, know I wanted a super sized bag of sour sweets at 10 am on a Saturday morning? I want the children to LOVE physics. I want them thinking, “How did Miss Dyer know I loved physics so much?!”. Which is why their introduction to secondary school physical science is astronomy. This is an eight-week unit designed with the number one purpose to blow their minds. The reason I chose to teach them about astronomy before electronics or mechanics is because it is truly fascinating. When I speak to pupils from other schools or my grown up friends about their experience of ‘physics’ at school, they tell me that they are or were taught so badly that they gave up, thinking that physics was not for them. If not for them, then who is physics for?

Caroline Herschel is the first scientific enquirer that pupils are introduced to in this unit. She was a pockmarked, four-foot three-inch woman whose family assumed that she would never marry and felt it was best for her to train to be a housemaid due to a childhood bout of typhus. In fact, Caroline Herschel beat the odds to receive many honours for her scientific achievements. Together with Mary Somerville, she was first woman to receive honorary membership of the Royal Society in 1835. Children are designed to leave the lesson where they learn about Caroline Herschel thinking, “Yes, physics is definitely for me”; “If Caroline Herschel can do it, why can’t I?”; “Nebulae are amazing!”. The astronomy unit taught at Michaela goes into far more depth than any other astronomy unit that I have ever taught at any other secondary school.

In this post, I have included an excerpt of the astronomy textbook that I have written and teach from, to give an idea of the content. The devil is in the detail. You cannot expect a pupil to love astronomy merely by learning the order of the eight planets and about the phases of the Moon. No, let’s not patronize the children that we are expected to teach. At Michaela, pupils learn about different models of the Solar System and references can be made to Ptolemy and his view of the Universe, thanks to Jonathan Porter’s history curriculum that includes Ancient Greece. Pupils learn that although Georges Lemaître first conceived the Big Bang theory, that the phrase ‘big bang’ was coined, ironically, by Fred Hoyle – an astronomer who disagreed with this theory. It is the links that can be made between science and other disciplines and the small pieces of information that are not usually found in science curricula, that children really love.

A recent visitor to Michaela asked a group of five year eight pupils what their career aspirations were. The responses were: experimental physicist, astronomer, ambivalent (!), mathematician and pilot. A year after being taught astronomy, some pupils actively want to pursue a career in the physical sciences. Using astronomy to sell the physical sciences? I say give it a go.