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Teaching mathematical grammar

Consider these questions:

  1. 3361 × 100 = 336100
  2. 3.361 × 100 = 336.1
  3. 33.61 × 100 = 3361
  4. 0.3361 × 100 = 33.61
  5. 336.1 × 100 = 33610

To a maths teacher, or a pupil who achieved old-money KS2 level 5 in maths, these are utterly trivial questions that require barely any conscious thought.

Yet to any pupil who finds maths difficult, these questions can be bewildering.

On the surface they are all the same process – ×100 – yet the process performed on each number looks radically different. Take question 1: 3361 × 100. Simple, right? Put an extra two zeros at the end of the number, and it’s right. But apply that process to questions 2, 3, and 4, and you’ll get it totally wrong. Question 2: 3.361 × 100. Similarly simple: all you have to do is move the decimal place two places to the right . But in question 3, the decimal point has disappeared altogether in the answer – where’s it gone?! Then over to question 4 and 5, and two weird things happen: whilst the decimal points have moved as expected, a zero has disappeared from the left of the number, whereas in question 5, a zero has appeared on the right of the number.

To illustrate: have a look at these mistakes, and see if you can spot the inferences made:

  1. If 3361 × 100 = 336100, then: 3.361 × 100 = 3.36100
  2. If 33.61 × 100 = 3361, then 0.3361 × 100 = 03361
  3. If 3.361 × 100 = 336.1, then 336.1 × 100 = 3361 .

Of course pupils find it bewildering.

I’ve started to think that this topic is a great example of mathematical grammar, and as such, requires a specific form of teaching for pupils to learn it well. What do I mean by this? And how can we help pupils get it?

When I first saw my pupils making errors when multiplying and dividing by powers of 10, my first instinct was to gently cajole them: ‘look this is easy, just move the decimal point this many times’. Yet implicit in my answer would be any number of things: for example, I’d remove unnecessary placeholder zeros, or fill in placeholder zeros, or make decimal points disappear. What was I doing? Over-simplifying and over-generalising, at the same time. In fact, in my head, I was doing a massive number of things – I wasn’t ‘just’ moving a decimal point. Implicit in my personal sense of ease was an expert understanding of the number system, of place value, and of number-system-notation conventions. None of that is easy or straightforward! Yet it seemed easy to me, because of ‘the curse of knowledge’. What we know is utterly transparent to our minds, such that we don’t even realise that we know it or that our reasoning is built upon it.

Something similar seems to happen in learning grammar. Daisy Christodolou’s blog on grammar provides very useful parallels.

How does one explain to a pupil like this where to use full stops, or apostrophes? In her own words:

We have to start with the basics… We need to get them to appreciate, first of all, what a sentence is. And to understand what a sentence is, you have to know what a verb is and what a subject is.  How do we get pupils to know what a verb and a subject are? These are not simple concepts…

If you look at the sheer amount of stuff that is necessary just for a basic understanding of a simple sentence, you will see firstly that you need to spend quite a lot of class time on this, and secondly that pupils will need to memorise a lot of the rules and processes. They will need to memorise the rules and processes in order to free up space in working memory so that when they come to write their own simple sentences, the process is as automatic as possible. They also need to memorise the rules and processes because grammar goes on to get a lot more complicated than this, and they need the basic rules well established in long-term memory so that the more advanced rules can build on these without taking up too much space in working memory.

Daisy’s view, in short: the basic understanding required for foundational topics, like sentences, is huge; and so it requires lots of class time devoted to memorising certain rules, procedures and concepts.

Katie Ashford makes the same point about apostrophes in a brilliant blog here:

When teaching something as seemingly straightforward as the humble apostrophe, we can underestimate the amount of knowledge required to really understand it. In order to use an apostrophe correctly, pupils need to understand five complex, overlapping rules:

  1. Singular and plural nouns not ending in ‘s’: show possession by adding apostrophes.
  1. Singular and plural nouns ending in ‘s’: show possession by adding an apostrophe (and sometimes an extra ‘s’ at the end).
  1. Plural nouns that don’t possess anything do not require an apostrophe.
  1. Pronouns of possession do not require an apostrophe.
  1. Contracted verbs/nouns: show omission by adding an apostrophe in place of the missing letters.

To teachers, it seems trivial knowing where to put full stops or apostrophes… yet when you start to try and explain it, it suddenly starts to seem rather reliant on deep, embedded understandings.

The same issue is true for teaching × 100. Let’s make it concrete: here are some rules, procedures, and concepts pupils need to know to answer expertly questions 1-5 on × 100.

  1. The convention of leaving blank the place value columns which are of greater value than the value of a number (i.e. not writing zeros on the left of numbers greater than zero). E.g. to write 36, instead of 0036.
  2. The convention of filling zeros into place value columns in order to express certain numbers greater than 10 and less than 0.1 (i.e. to write zeros on the right of numbers and on the left of numbers). E.g. to know the necessity of the zeros in 3600 and 0.00036.
  3. The equality of value between 5 and 5.0 and 5.00 etc, and the convention of writing integers without decimal points and placeholder zeros, and at the same time the usefulness of re-expressing integers with these decimal points and placeholder zeros.
  4. The convention of having invisible decimal points on the right of integers, e.g. not writing ‘5.’ but ‘5’ instead.
    1. (For that matter, the definition of integers!)
  5. Multiplication by powers of 10 means moving decimal points sometimes, or putting extra zeros other times, and the essential equivalence between these concepts, because of the above conventions. (only now are we getting onto the actual understanding of the procedure!)
  6. (When it comes to dividing by powers of 10) The convention of expressing numbers less than 1 as beginning with 0.x, as opposed to just .x

Now, I haven’t expressed these particularly well, but that’s partly my point. These are conventions and understandings that are so deeply embedded and so intuitively known by maths-experts that they are very difficult to explain well. In fact, I think they are nearly impossible to ‘explain’, in the sense of ‘here’s a 2-minute explanation on this convention that will give you everything you need to know!’ To the novice pupil, they seem so arbitrary at best, and contradictory at worse. It takes a very sophisticated understanding to differentiate between conventions 1 and 2, which are essentially telling you to do opposite things at opposite times. Sometimes we need to put an extra 0 in front of .36; other times we have to remove the extra zero in front of 036. Why? Well… give it your best shot explaining it if you think I’m wrong.

divide2bpower

A lesson preview from TES – note, there’s nothing ‘simple’ about what’s happening. Firstly, where did those zeros come from? Secondly, where did the decimal come from? Thirdly, we never see numbers written like .8 – what’s going on?!

This stands in contrast to the popular view on ‘teaching for understanding‘, which is all the rage. Don’t get me wrong: I’m all for helping pupils understand. Yet in my view, topics that are more grammatical – topics that rely heavily upon subtleties of notations and conventions – are difficult to understand in the abstractEither your explanation will over-simplify but then it won’t convey all that’s needed for understanding; or it will convey the full breadth of a topic but totally overwhelm most of your pupils.

It may help to define mathematical grammar. In my view, a grammatical topic is one where subtleties in convention and notation can lead to serious confusions and apparent contradictions, despite deep logic underlying those conventions and notations. This is exactly what we see in teaching multiplying and dividing by powers of 10. (For what it’s worth, I’m less interested in the ontological question of whether there is such a thing as mathematical grammar. My interest is focused on the extent that grammar is useful as a pedagogical concept in maths, for categorising certain topics and implying a best way to teach them.)

So, to link it back to English grammar: my pedagogical-functional definition of grammar shares similarities with the learning of English grammar. For example: these two examples of full stops, and the definitions given, seem to contradict:

1. to mark the end of a sentence that is a complete statement

2. to mark the end of a group of words that don’t form a conventional sentence, so as to emphasize a statement – ‘I keep reliving that moment. Over and over again.’

Of course, they don’t contradict. Yet I don’t think any primary English teacher will race to teach and explain usage number 2 whilst also teaching usage number 1. Similarly I remember learning these golden rules in year 3: never ever start a sentence with ‘and’ or ‘because’. And they were good rules at the time! (Irony intended). But after you’ve gotten to an age where you’ve read thousands of stories of increasingly sophisticated language, and received enough of the appropriate guidance, the two usages will begin to fall into place, and you’ll reach a level of understanding where using full stops is as natural as breathing.

So. The common issue with grammatical topics: just like the usage of full stops, they are easy to explain & understand poorly and very difficult to explain and understand comprehensively. There’s too much underlying logic that pulls in different directions. So, I’m increasingly convinced that with grammatical topics in maths, just as with language, understanding must follow grammatical fluency and familiarity. Our understanding of them requires a mass of examples, together with a mass of practice, cohering together, over a stretch of time. Now, I’m not saying that this is sufficient for understanding place value, but that it’s undoubtedly necessary for pupils who struggle with maths.

Let me illustrate this with some topics. Aside from × 100, here are some other topics which strike me as grammatical:

Rounding – a notoriously difficult subject to make stick – why does 0.446 to the nearest hundredth round to 0.45, whereas 446 to the nearest hundred rounds to 400? Why doesn’t the former become 0.450? Why doesn’t the latter become 4? These questions are hard to explain, and a good explanation doesn’t usually help pupils get it right! Again – some of the similar conventions we saw with × 100 play havoc with rounding.

Simplifying algebra – after all, the rules of simplification are like the grammar of another language! 3a + 2a = 5a, yet 3a × 2a = 6a²? A tiny change in notation (+ to a ×) leads to a radically different process; there’s a deep logic, but it’s hard for pupils initially to understand. Then there’s all the conventions of invisible multiplication signs. I think simplifying algebra is a particularly good example of a grammatical topic, since in my experience, pupils are absolutely fine within lessons, within each topic; yet when it comes to tests and mixed exercises, all manner of confusion erupts. Again, there’s a very deep and rigorous logic to these processes, but I’ve found it’s far more helpful to tell confused pupils initially ‘this is how you add terms, and this is how you multiply them – just remember the two processes’.

Rearranging equations: again, a topic that seems so natural to maths teachers, yet so alien to most pupils. At first there seems to be a clear logic (and of course, like all of algebra, the logic is rigorous and wonderful) you can teach – ‘look at the order of operations performed on the variable, then reverse them!’ (although even that is difficult for most). But then you try and build up to funky questions like this:rearrangeAnd then all explanations are off, and it’s basically just ‘okay, watch how I do it, now try five, learn this pattern…’

A great example of this is seen in this brilliant post on Math with Bad Drawings:

Simplifying algebraic fractions: algebra fractions

If you’ve tried it, you’ll know it’s a pretty difficult task to explain why you cancel both fours in the numerator of the first fraction, whereas you cancel only onefour in the numerator of the second fraction. Of course there’s a deep logic, and there’s probably a decent paragraph length explanation – but I highly doubt any teacher tries to teach the patterns of simplification through an ‘explanation for understanding’. Again, a subtle difference in notation, and all of a sudden previously acquired heuristics are totally unsuccessful and highly misleading. As shown in how prevalent errors like these are:

error

Negative numbers: I think this one goes without saying! I’ve written on it previously; I think this a prime candidate because of that fatal and poorly used heuristic: ‘if the signs are the same it’s +, if they’re different it’s -‘ – that falls into the grammatical trap of over-simplifying a topic and thence leading pupils astray.

So, onto the golden question: how should we best teach mathematical grammar?

1. Before teaching a topic, reflect on how ‘grammatical’ it is.

Recall my definition: can you explain the concept/method/process clearly and succinctly in such a way that covers all the eventualities and forestalls confusions with similar-looking processes? Teaching percentages of amounts: yes. The topics above? Not so much. Too many apparent-exceptions, strange conventions, surface similarities with different topics.

If a topic seems grammatical, break it down into every single possible strange case, as I did with the questions right at the start. The best way is to treat each case as a different thing to teach and practice explicitly. For example, in basic simplifying algebra, I’d want to split up 4a + 5a separately from 4a + a, and recognise they’re quite separate cases. After all, in my experience, when I’ve taught both problem types together before getting pupils to practise, a proportion of the class will inevitably get it wrong because ‘oh sir I forgot about the invisible coefficient of 1’; they’re familiar with the key concept, but they just weren’t familiar enough with it.

Furthermore, when you realise that a certain topic is particularly grammatical, plan to spend more time than you might normally assume is necessary. Remember, grammatical topics are particularly prone to the ‘curse of knowledge’ – experts find them so easy and logical and straightforward that we don’t realise how much time novices need to get proficient in them. We’re happy leaving a lot of time for trigonometry, for instance, but perhaps we don’t even spend any time in KS3 explicitly teaching multiplying by powers of 10. (One of the reasons I love teaching at Michaela is how so much time is given to these topics: I got to spend the best part of a month on base 10, and our year 8s have spent half a year on algebra without even reaching equations yet!)

2. Start slow, build up.

Compare how Daisy suggests teaching grammar:

I would begin with very simple and patterned sentences like ‘I walk to the shops’, ‘I eat lots of chocolate’ and get pupils to identify ALL of the parts of speech in them. I would start with the following five parts: verbs, nouns, adjectives, articles and prepositions, in that order…. If you look at the sheer amount of stuff that is necessary just for a basic understanding of a simple sentence, you will see firstly that you need to spend quite a lot of class time on this, and secondly that pupils will need to memorise a lot of the rules and processes.

Try and strike the tricky balance of nailing just one concept, but practicing it in an intelligent way that requires thinking which doesn’t conflict with the concept you’re trying to get across. (This is for numerous reasons: 1) two birds with one stone; 2)desirable difficulties help us learn; 3) non-conflicting difficulties can prevent boredom in doing the task.) For example, if I want my pupils to embed the idea that the decimal point in integers is invisible, and on the right of the number, a suitable set of questions might take the form of:

  1. 36 ÷ 10
  2. 482 ÷ 10
  3. 57643 ÷ 10
  4. 4382 ÷ 100
  5. 381 ÷ 100
  6. 67524 ÷ 1000
  7. 38925 ÷ 1000

There’s some thinking involved in the different divisors; but throughout, pupils are practising and internalising the invisible decimal point. Notice that at this point, I haven’t introduced the idea of filling in placeholder zeros; I want the focus to be entirely upon one key new concept. I might then make it a bit harder, interleave questions like so, to practice embedding the concept backwards and forwards i.e. the decimal point can be made invisible if it ends up on the right of an integer:

  1. 36 ÷ 10
  2. 4.8 × 10
  3. 482 ÷ 10
  4. 1.9 × 10
  5. 57643 ÷ 10
  6. 4382 ÷ 100
  7. 6.22 × 100
  8. 381 ÷ 100
  9. 6.44 × 100
  10. 67524 ÷ 1000
  11. 7.536 × 1000
  12. 38925 ÷ 1000

Once this is in place, move onto a different task, to embed a different concept, and leave this one temporarily behind.  For example, I might move onto the idea of filling in placeholder zeros when decimal points are ‘moved’ beyond the visible number:

  1. 336.1 × 100 =
  2. 33.61 × 1000 =
  3. 24.7 × 100 =
  4. 1.91 × 1000 =
  5. 246.1 × 100 =

After they’ve got some fluency with this, mix the question types with the previously learnt process – and prepare for some confusion! E.g.

  1. 336.1 × 100 =
  2. 4.8 × 10
  3. 33.61 × 1000 =
  4. 3.26 × 100 =
  5. 24.7 × 100 =
  6. 7.536 × 1000 =

In my experience many hands will be raised – ‘sir do I have to fill in the zeros in question 2?’ to which a simple reassurance usually suffices: ‘no, this one was like the ones we did 15 minutes ago! Where should the point end up? Do we have empty cups that need filling? [‘No sir.’] Exactly.’

This might all take place in one lesson, or over week, depending on the class. But each time, follow these steps:

  1. Practice each new concept separately
  2. Gradually interleave with previously drilled concepts
  3. Move onto a new concept, and practice that separately;
  4. Gradually interleave with previously drilled concepts.
  5. Repeat!

3. Practice, practice, practice – over time.

Remember, the things they’re learning still seem to pull at odds, so over time, some of it will fade, and some of it will stick. As a result, if it all isn’t rigorously revisited, one pupil will be able to do 4.6  × 1000 = 4600, but then answer 4.6 ÷ 1000 = 0046, too.

A good way to revisit is to drill over time. Randomisable Excel spreadsheets are your friend here. Make it fun! Make it competitive! Time pupils and see if they can beat their class score. They will love their increasing success.

Here’s a drill I made just after my low-prior-attaining year 7s had gotten the hang of some basics of × 10. We practised this every day, or every other day, for a couple of weeks. Notice the different question types and all the different cases involved in multiplying and dividing by 10 – place value is really complex!

explosion 1

(Incidentally I love saying things like ‘GET YOUR EXPLOSIONS ON THE DESK!’ or ‘PUT YOUR EXPLOSIONS AWAY!’. I think the class enjoy it too.)

4. Prepare to fly

The great thing about getting it right with grammatical topics is that they are usually foundational to many other topics. And once pupils start getting mastery in the basics, their potential for growth is exponential. A nice illustration with the MENTAL MATHS EXPLOSION that those same year 7s, several months later, are currently doing:

explosion 2

It’s so much harder! But you quickly realise that once the foundations are firm, things like standard form, × 0.1, etc. are so easy to incorporate. In a future iteration of these Mental Maths Explosions I might spend 15 minutes of class time on multiplying by negative powers of 10, just to spice things up a bit – but since everything else is in place, they’ll be absolutely fine with that.

For what it’s worth, the first drill above was done in December, and we started this drill only three months later, in February – off the back of about 2 weeks’ explicit curriculum time, and 2 minutes a day thereafter. I was pleased to see that in our mid-term exams, one of the best answered questions by my class was the following:

Screen Shot 2017-03-11 at 13.03.26

Once the launch pad is in place, prepare for lift off.

_________________________________

Postscript: on moving the decimal point…

I’ve received numerous tweets and comments on how the decimal point doesn’t move. My wonderful boss (another great plus of working at Michaela!) has written a great reply to that point here. Interestingly, like her, I heard numerous times and saw numerous demonstrations in my PGCE year on how the decimal point doesn’t move. In fact I even copied wholesale a teaching approach I saw during one session: I printed out digits and a decimal point, 1 each on a piece of A4 paper, made pupils stand up, and slid them up and down whilst the decimal point stood the same. From that experience – and how it offered my pupils precisely zero benefit in learning any of the issues & conventions I discussed above – I quickly learnt never to spend time on it again. In my experience, the key roadblocks to success in this topic have nothing to do with whether the decimal point or the digits move; instead, they are all related to the highly confusing conventions I mentioned above, which stand regardless of whether you move the decimal points or the digits.

One might then argue that it’s best to teach ‘move the digits!’ because it’s true. My main response to that is that moving the decimal point is far easier and less cognitively overloading. That’s why numerate adults – even those who know that ‘the decimal point doesn’t move’ – nonetheless overwhelmingly move the decimal point when they are working out a divide by 1000 question.

But more related to teaching: many of my lowest-attaining pupils, the ones who struggle the most and who didn’t understand this topic at primary, have real trouble just moving the decimal point – their difficulties in spatial awareness makes them do things like this:

NC

Can you imagine how difficult these pupils must find the concept of ‘keep the decimal point there, and move all the digits along!’ – so much is moving at the same time! Especially as there isn’t any working out that can help them, unlike the arrows for the decimal point. I’ve seen these pupils gain real confidence in moving the decimal point and I am loathe to give them a much harder, more cognitively demanding process that some of them will find far too difficult.

For what it’s worth, our textbook also makes clear the concept that the decimal point doesn’t really move. But on a procedural level, we stick with moving the decimal point all the way, as a handy shortcut.

But please do read Dani’s post for numerous extra explanations on why ‘the decimal point doesn’t actually move’ is not particularly helpful pedagogically.

If you enjoy this sort of thinking into maths pedagogy, you might enjoy working at Michaela! Contact me on Twitter [at]HinTai_Ting for more details.

Cheng Reaction – Six Months In

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

Six Months In

Six months ago, I started working at Michaela. As a Wembley native, growing up just around the corner and attending a local state school, it has been an experience and a pleasure to teach pupils in the community and to see them thrive. Since starting here, I have questioned almost all my pre-existing beliefs about teaching, and have come to see that the Michaela way works extraordinarily well. In this reflection, I note down three aspects of the Michaela way that I wish I had known about before I first entered the classroom.

1) Department Planning

I mention here that we do not use PowerPoints at Michaela. Instead, Heads of Department co-ordinate the creation of all our resources, which we co-plan and share. There are so many benefits to this that it is hard to list them all, but here are a few that spring to mind:

  • Improve teachers’ subject knowledge: Every teacher has their own specialism, particularly in a subject like science where one might be an expert in physics but only have a basic grasp of biology. By pooling all our knowledge into our resources, we ensure that every child receives the best possible explanations and examples in every lesson. 
  • Consistency: Not only does our approach to planning support high quality teaching, but it ensures that every pupil receives the same standard of teaching in every lesson. 
  • Greater Depth. The time we spend thinking about our resources means that we are able to teach a curriculum that spans far beyond the GCSE requirements. Our pupils learn great topics such as the history of science and the biographies of significant scientists, including Herschel, Mendeleev and Dunlop, and hear about the latest discoveries (such as the recent categorisation of giraffes into one of four species). 
  • Renewable – Every moment spent on creating these resources is renewable. Rather than re-creating new stuff every lesson, we plan with reusability in mind. Each year we evaluate and improve on the content we have produced, and use it again the following year. 
  • Literacy – Literacy is deeply embedded in all science lessons. Pupils are exposed to approximately 1,000 words every science lesson. Commonly misspelt words such as temperature, neutron and flagellum are flagged up before pupils begin their writing. We explain the root of naming compounds and scientific words (such as chlor = green or aer = oxygen). Pupils are able to express the correct keywords in a sentence (e.g. weight and mass).

2) “3,2,1 SLANT”

SLANT is a behaviour tool used across the school: every teacher makes use of this strategy. SLANT stands for Sit up straight, Listen, Answer questions, Never interrupt and Track the teacher.Click here to see how often it is used in a lesson.

Below are some reasons why SLANT is important in class.

  • Calling the class together – It is a simple way of getting the pupils’ attention without the need for waving hands or raised voices.
  • Concentration – I find pupils listen attentively because they are tracking the teacher; it serves as a visual cue to be attentive and not daydream in class.
  • Great for posture – “strong spine, strong mind”. Slouching is bad for your back! Being encouraged to sit up straight could prevent serious posture problems later in life!
  • Prevents fiddling – Fiddling is distracting to pupils and others around them. SLANT prevents such distractions, allowing more focus on learning.

3) Feedback

I remember in my previous school, I used to enjoy marking…the first ten books! Then it became a chore that would often take around three or four hours per class. As Joe Kirby explains here, marking is a burden on teacher workload and has minimal impact on pupils. Instead, our pupils receive tons of useful, instantly actionable feedback that enables them to continually improve.

  • Green Penning – Pupils mark their own responses to questions using green pen. This way, pupils recognise and understand correct answers, learn from their mistakes, and –most importantly- gain ownership and responsibility over their work.

img_0404

  • Whole class feedback – Every two weeks I read through pupils’ books to spot misconceptions. I do not write long comments or “well done, you have understood xyz correct”. I do not set any targets or DIRT time. Instead, on a piece of paper, I write down any merits, demerits and any misconceptions I see. The following lesson, I spend around 5 minutes re-teaching or clarifying as necessary. The visualiser is also a great tool to display pupil work. As a class we discuss strengths and suggests improvements – such as misspelt words, use of keywords and how to improve the quality of the content.
  • Extended Prep – Year nine pupils receive examination questions as weekly homework. Teachers check these through briefly to get a general sense of how pupils have done before giving them back. Pupils then have an opportunity to improve their written answers.
  • Quizzes – Pupils are given a weekly quiz and receive instant feedback. In addition to this, pupils are given one practice exam every week, ensuring that their skills and knowledge are put to good use! More about quizzeshere.
  • Exams – There are two high stakes exams in the year (and two mock exams as well). On each occasion, pupils receive whole class feedback on content and exam technique.

Whilst none of this is especially radical or new, the striking simplicity of the Michaela approach enables teachers to teach and pupils to learn. The combination of these easy-to-use strategies reduces workload for teachers whilst simultaneously increasing impact on pupil progress.

If you would like to visit Michaela and see us in action, email info@mcsbrent.co.uk.

If you have excellent physics knowledge and would like to get paid resourcing for us, email odyer@mcsbrent.co.uk.

We’ve also released the Tiger Teachers book which outlines many of the strategies that we use at Michaela.

cover1

Deciding the first step – Different decisions for different problem types

MARCH 4, 2017

Deciding the first step is a type of question where pupils are only asked to decide explicitly which step to perform. A pupil reviews the problem type and identifies what steps he/she needs to take. After acquiring knowledge of a particular step then automatically the end of each step triggers the start of the next one. I will be looking at some questions which test whether pupils can decide on the correct step to perform, and compare them to more commonly asked questions which make the decisions for pupils.

Compare these three questions being asked by a teacher to a set of pupils

Decision Blog

What is 4 times 2?

What two numbers do I multiply first?

What is the first step?

The correct answer to all the above questions are the same. However, the first question being asked makes the decision for the pupil because they are told that they are to multiply, and which numbers to multiply. The second question is testing whether the pupils know that we multiply the numerators, and then multiply the denominators, but the decision is being made for the pupils. It is possible that a pupil will give a wrong answer by stating two numbers that we do not multiply (4 x 11). The final question is specifically asking a question to test if a pupil knows what step to perform, compared to the first two question.

Compare these two question being asked by a teacher to a set of pupils

Decision Blog 2

What do I cross-simplify first?

What is the first step?

Again, the first question makes the decision for the pupils compared to the second question. The second question is testing if a pupil can recall the first step in answering this question. This problem type is different from the one above because you can cross simplify. And I would consider cross-simplifying the first step to the procedure but simplifying the product of both fractions can happen after a pupil has multiplied the two fractions.

Compare these two questions being asked by a teacher to a set of pupils:

Decision blog 3

What is the first fraction as an improper fraction?

What is the first step?

Again, the first question makes the decision for the pupils. Furthermore, the first step of the previous two problem types cannot be applied here because there is a necessary step before both fractions can be multiplied.

The decisions pupils have to make to attempt the following calculations, vary:

Decision Blog 4

The following question – “What is the first step?” not only helps pupils to learn explicitly which step to perform first but it also allows pupils to distinguish between different problem types. This is because a pupil is then attaching their knowledge of what decision to take depending on the make-up of the question. They are acquiring surface knowledge of the problem type – If I see mixed numbers when I am multiplying fractions then I must convert them to Improper fractions first.

This form of questioning allows pupils to develop mathematical reasoning around different procedural calculations. For example, a pupil recognises that to multiply mixed numbers we must convert them into improper fractions because we cannot multiply 1 with 1 and 4/5 with 2/11 because 1 and 4/5 is not 1 x 4/5 but 1 + 4/5 which equals to 9/5. This also consolidates pupils existing knowledge of mixed numbers and their equivalent forms as improper fractions.

Another example which is interesting is making decisions about the first step when comparing two negative fractions. Here are the different problem types and the first step to each one. The question posed for all the problems below is “Which is greater?”

Decision Blog 6

Even though the problems may look the same to pupils, through teaching pupils explicit decision making around the problem types the pupils are doing two things. They are identifying the first step they need to take, and they are attaching their knowledge of the first step by identifying the features of each problem type.

For the most competent pupils their mathematical reasoning will make the features of each problem type and the first step they need to take seem obvious. For pupils who’s mathematical understanding isn’t as fluid they greatly benefit from being asked decision isolation questions, because they are identifying the features of each problem type which makes them distinct from each other, and thereby helping them know what is the first step they need to take.

Creativity? Just footsteps in the snow
footsteps-in-the-snow

learn (v.) – Old English ‘leornian’, ‘to get knowledge, to be cultivated’. From the Proto-Germanic, ‘liznojan’, ‘to follow a track’.

Robert Macfarlane seems to be the closest we have in Britain to an Innuit: not only does he actively seek out cold, featureless, uninhabited space, but he also has about twenty different words for snow. Reading his books feels like following him in a blizzard of fresh vocabulary. Recently, in his Old Ways, I’ve learnt to avoid three different types of bog: boglach (general boggy areas), blar (flat areas of a moor that can be boggy) and, the most dangerous of all, the breunlach (sucking bog that is disguised by the alluringly bright green grass that covers it). All good words to know, I thought. Even in Willesden Green.

But it was the end of his book – his acknowledgements – I was really taken with. He refers to Henry James’ novel The Golden Bough and, specifically, James’ second edition. In it James contributed a foreword in which he reflected on the process of self-revision. James conjures up the figure of the first writer as a walker who has left tracks in the snow of the page, and the revising writer as a tracker or hunter. Macfarlane notes:

‘It is significant that James is interested not in how we might perfectly repeat an earlier print-trail, but in how re-walking (re-writing) is an act whose creativity is founded in its discrepancies: by seeking to follow the track of an earlier walker or writer, one inevitably ‘break[s] the surface in other places’. One does not leave, in the language of tracking, a ‘clean register’ (placing one’s feet without disparity in the footprints of another, matching without excess or deficiency, as an image in cut paper is applied as a sharp shadow upon the wall). James sees our misprints – the false steps and ‘disparities’ that we make as we track – to be creative acts.’

He goes on to say that he has ‘inevitably followed in the footsteps of many predecessors in terms of writing as well as of walking, and to that end wish[es] to acknowledge the earlier print-trails that have shown [him] the way and provoked ‘deviations and differences’.’ He then lists the writers (and musicians) that have ‘shown [him] the way’ – everyone from Byron to Brahms, from Nabokov to Laura Marling.

Increasingly, I understand that that is that creativity really is: the deviations and discrepancies formed as we follow a path that has been trodden before (and for) us. MacFarlane – English tutor and Fellow of Emmanuel College, Cambridge – understands this better than anyone. His deeply creative use of language – the path finding he fashions out of words – is rooted (routed?) in the trails set by the writers that have blazed before him. It’s why our most creative authors have often memorised quotations and poems and prose, or spent their formative years gorging themselves on lists of vocabulary and the most challenging texts in the canon. They’ve spent hours and hours internalising the vocabulary, the syntax and the rhythm of the greatest writers who have ever lived.

Why is it that my pupils choose to use the word ‘hegemony’ to describe the dominance of the Catholic Church in medieval England? Have they conjured it out of the ether? No. They use it because I’ve taught it to them explicitly and we’ve practised using it in myriad different contexts, again, and again, and again.

George Monbiot, although a successful writer and journalist himself, doesn’t recognise this. Partly, this is due to his own expert induced blindness: he cannot see that the explicit instruction he received at his (private) school was what gave him the literary advantage he now so effectively employs. Partly, it is because he’s never taught in a school himself. And so when he snatches at inscrutable futurology it’s because, from the cloistered seclusion of King’s Place, it all seems very seductive, very modern, very progressive, very Guardian. And it is seductive; but that doesn’t make it true.

If we want our pupils to be creative then we need to show them the way. The evidence, which Monbiot is either unaware of or wilfully ignores, is clear: explicit instruction and deliberate practice works. It’s not fancy, it’s not fashionable and it’s not even that difficult. But it’s the reason why Shakespeare – a man whose secondary educationconsisted of hours of imitation, memorisation and drill – grew up to be the most creative wordsmith in the English language. We know that if you want to get good at something – anything – you need to practise that thing over and over again.

I wish I’d known this when I started teaching. I wouldn’t have wasted my pupils’ time desperately trying to extract answers in ‘engaging’ starter activities that they have no possible way of knowing. At Michaela, we call this ‘guess what’s in my head’. It’s one of the more stubborn pedagogical ticks our new teachers tend to arrive with.

I wish I’d been made to read MacFarlane. If I had, I’d have realised much sooner that learning is like following footsteps in the snow. It’s no coincidence that the etymology of ‘to learn’ is at root – and at route – ‘to follow a track’. Our responsibility as teachers is to illuminate the track and show our pupils the way. Without our instruction, how can they know where they’re going? They’re destined to stumble around in circles, trudging long distances but going nowhere. No, we must be clear. It’s by looking to those who have gone before us that we learn. And as we follow we will misstep and misprint. These ‘deviations and discrepancies’ will be our creative acts.

This was originally written for the campaign group, ‘Parents and Teachers for Excellence’.

Algebraic Circle Theorems – Pt 2

Last week, I explored different number based circle theorem problems that can test (a) a pupil’s ability to identify the circle theorem being tested and (b) problem types where a pupil has to find multiple unknown angles using their circle theorem knowledge as well knowledge of basic angle facts.

In this blog, I’m displaying a few different problems within the topic of circle theorems where each angle is labelled as a variable or a term. I am interleaving lots of different knowledge:

  • Forming and simplifying algebraic expressions
  • Forming algebraic equations
  • Equating an algebraic expression to the correct circle theorem angle fact
  • Equating two algebraic expressions which represent equivalent angles and solving for the value of the unknown. Furthermore, using the value of the unknown to find the size of the angle represented by the algebraic expression.

I have interleaved fractional coefficients into a couple of questions to add some arithmetic complexity to the questions. Enjoy!

A triangle made by radii form an Isosceles triangle

Image 0The angle in a semi-circle is a right angle

Image 1
The opposite angles in a cyclic quadrilateral add up to 180 degrees (the angles are supplementary)

Image 2Angles subtended by an arc in the same segment of a circle are equal

The questions for this circle theorem differ in nature from the problem types shown above. Here you are equating two algebraic expressions which represent equivalent angles. We are no longer forming a linear expression and equating it to an angle fact like 180o.

Image 3

The angle subtended at the centre of a circle is twice the angle subtended at the circumference

In these problems types the key mistake that a pupil may make is equating the angle subtended at the centre to the angle subtended at the circumference without doubling the angle at the circumference. This can be pre-empted by asking pupils a key question of “What is the first step?” The answer I would be looking for after going through a few worked examples would be “you need to double the angle subtended at the circumference.”

Image 4Tangents to circles – From any point outside a circle just two tangents to the circle can be drawn and they are of equal length (Two tangent theorem)

Image4a

Alternate Segment Theorem – The angle between a tangent and a chord through the point of contact is equal to the angle subtended by the chord in the alternate segment.

Image 5

I would be keen to hear any thoughts or feedback. Please don’t hesitate to email me on naveenfrizvi@hotmail.co.uk.

Creating Problem Types – Circle Theorems Part 1

Last summer, I made as many different problem types for the topic of Circle Theorems. I looked through different textbooks and online resources (MEP, TES, past papers). I did this because when I last taught circle theorems at my previous school there weren’t enough questions for my pupils to get sufficient deliberate practice. This was a two fold issue. Firstly, I would find a practice set of questions which would not provide enough questions for a pupil to practise one particular problem type. Secondly, the sequencing of questions in terms of difficulty would escalate too quickly or not at all. Here I will outline the different problems types I created (using activeinspire) and then explain the thinking behind them. I have been very selective with the problems I have included here; I have made more questions where certain problems types are more complicated which I shall discuss at the end. I shall more in the following posts.

I made two different categories of problems for each circle theorem. The first type would explicitly test a pupil’s understanding of the theorem to see if they could identify the circle theorem being tested.

The second type would be testing two things. Firstly, such a problem type would be testing their ability to determine the circle theorem being applied in the question. The second aspect of the problem type would be testing related geometry knowledge interleaved which can be calculated as the secondary or primary procedure in the problem e.g. finding the exterior angle of the Isosceles triangle.

One common theme in these questions is that procedural knowledge applied is executed in a predetermined linear sequence. Hiebert and Lefevre wrote that “the only relational requirement for a procedure to run is that prescription nmust know that it comes after prescription n-1.” Multi-step problems such as the ones that you will see show that procedures are hierarchically arranged so that the order of the sub procedures is relevant. Here are the different problem types for each circle theorem where I explain how many items of knowledge is being tested in each question, and what each item of knowledge is.image-0Figure 1: A triangle made by radii form an Isosceles triangleimage-1

Figure 2: The angle in a semi-circle is a right angleimage-2

Figure 3: The opposite angles in a cyclic quadrilateral add up to 180 degrees (the angles are supplementary)image-3Figure 4: Angles subtended by an arc in the same segment of a circle are equal

image-4
Figure 5: The angle subtended at the centre of a circle is twice the angle subtended at the circumferenceimage-5Figure 6: Tangents to circles – From any point outside a circle just two tangents to the circle can be drawn and they are of equal length (Two tangent theorem)image-6aFigure 7: Alternate Segment Theorem – The angle between a tangent and a chord through the point of contact is equal to the angle subtended by the chord in the alternate segment.

To conclude, there are many different problems types for the topic of circle theorems and the complexity of the problem can be addressed in many ways such as:

  • arithmetic complexity
  • Orientation of the problem
  • Multiple representations of the same problem type
  • multiple subprocedures to determine multiple missing angles
  • Interleaving the application of multiple circle theorems.
  • Interleaving the use of basic angles facts
    1. as a necessary step in the procedure to find other angles
    2. as an independent step in the procedure where finding one angle is not necessary to find another angle.

I would be keen to hear any thoughts. Please don’t hesitate to email me onnaveenfrizvi@hotmail.co.uk.

Making the Most of Every Minute

Since starting at Michaela in September, I’ve learnt that every second counts. I’m still shocked at the speed of routines, and how they enable us to squeeze every last second of learning in every lesson.

There are lots of things that slow lessons down which Michaela has almost managed to eradicate. For example, chaotic corridors at lesson transition time, settling pupils into the classroom, lengthy periods wasted waiting for silence, and time-draining activities that yield less learning than alternative methods. In science, the added pressure of getting through experiments requiring lots of equipment can further remove the focus away from learning.

To demonstrate exactly what this looks like in practice, here is an outline of a typical Michaela science lesson, broken down minute-by-minute. This would work with any class, but this lesson in particular was with a year 7 set 3 class – their second lesson on a new chemistry unit. The title of the lesson was ‘Evidence for atoms’. Times are approximate.

10:30 Pupils are waiting patiently outside the classroom in silence whilst another class is dismissed.

10:31 Pupils say a hearty “Good morning” as they come in to the class, take their pencil cases out and are then SLANT-ing.

10:32 “Good morning year sevens. Drill questions page 3. Ready…go!” Pupils sit, hand out exercise books and immediately start their drill questions on page 3 (see more on speedy entrances here). Drill questions are one worded answers to recap over the whole unit.

10:34 I say “3,2,1 SLANT” to get their attention. I tell them to get their green pens (for self-checking) ready. “Ready…set…go!” I read out the one-word answers to each drill question and ask for class feedback.

“Hands up if you have question one incorrect?…two incorrect? Etc.” The majority of the class puts their hand up for question 7, so I provide a brief explanation and recap on Aristotle and the four elements.

10:36 “When I say go and not before, recap questions at the back of the book. Ready…go!” Pupils complete five recap questions in silence. Recap questions are sentence answers linked to the previous lesson.

10:39 “3,2,1 SLANT. Green pens ready. Ready…set…hands up! Question 1…” Pupils give full sentences ending with a Michaela full stop (Sir/Miss). I take a hand up from Arnold*: “Aristotle thought that the world was made up of four elements which were air, fire, water and earth, Miss”

“Excellent, that’s right! Make sure you’ve all got that down in green pen.”

10:42 “3,2,1 SLANT. When I say go and not before, tracking line one. Quickest row gets to read first. Ready…go!” Pupils put two hands on their ruler holding a blue pen in one hand. Each pupil reads one line each of the text and annotates their text book accordingly.

10.47 We pause after line 9 for questions. “3,2,1 SLANT! During which period were Alchemists practising?” More than 50% pupils put their hand up. I non-verbally ask for a choral response: “1,2,3….” “Renaissance!” After a few more questions we continue reading about Robert Boyle (and his theory) with frequent pauses for questioning.

10:58 “3,2,1 SLANT! We have comprehension questions to complete. What is the title?”

Tim: “Evidence for atoms”

“That’s great! What must you remember? 1,2,3..” “Capital letters at beginning of sentences!” “Ready…go!” Pupils start their comprehension questions.

11:09 “3,2,1 SLANT. Green pens ready. Ready…set…hands up! Question 1…” Sarah: “The two main aims of Alchemists were to find the elixir of life and to turn cheap metals into gold, Miss” “Great answer, merit for Sarah. Now check your spellings for Alchemists. A-L-C-H-E-M-I-S-T-S” Pupils check by ticking every letter.

11:13 We continue to read about John Dalton, again pupils annotating their textbooks and pausing for questioning.

11:18 Pupils complete second set of comprehension questions.

11:23 “3,2,1 SLANT. Green pens ready. Ready…set…hands up! Question 1…”Pupils give whole class answer in a similar fashion as the first comprehension.

11:27 “Put one book in another (i.e. textbook folded within exercise books) 5…4…3…2…1 SLANT! Books passed down 20…19…3…2…1 SLANT! Behind your chairs 10…9…3…2…1 SLANT!”

11:28 I read out a list of all the merits and demerits I have given during the lesson. In the final few minutes, I ask a few extra choral response questions. “Dalton said that atoms cannot be broken down. Another word we use is…1,2,3,…” “Indivisible!”

11:30 Pupils are dismissed, filing out into the corridor. It’s always lovely to hear a “Thank you, Miss” when they leave.

Another Michaela lesson complete. The simplicity of each lesson and the consistency of routines enables maximum learning to be achieved.

Click here to see a minute by minute history lesson by Mike Taylor.

*names have been changed for anonymity

Under Pressure

Posted by Dani Quinn

I’ve had two interesting conversations this year with some of our weakest pupils.

Fadekah is in Year 8. For all of Year 7 we despaired if she would even be able to pick and microwave her own meals, or complete routine tasks to earn a wage. She was a seriously spaced-out kid, if very sweet. She never did her homework, and would wear a dopey expression of “I’m cute and helpless and can’t do anything” when her form tutor chastised her for this. Sometimes she giggled if she was being told off. Everything seemed to pass her by. She struggled with the simplest of abstract concepts and didn’t know any times tables. She didn’t know the number before 1000 and seemed unable to remember it no matter how many times I told her. In lessons she did little work, grinning in a far-away manner if given a consequence for not working or not listening. I didn’t see how she could get a G, let alone a C, in Y11. I didn’t see how she could have a good future.

At the start of this year, I had her class again. On the first day she was the star of the lesson. That night she did her homework. And the next night, and the next. She came after school frequently to ask questions about what was learned and took copious notes recording explanations and tips I gave in lessons. Her test results are now typical of the class, despite finding the material difficult to grasp and often feeling confused by the work (Year 8 is mostly algebra). She never needs to be corrected in lessons for not listening or not trying; she is frequently pointed out as a role model. Her questions are insightful and thoughtful. Her homework is always early, she often does extra.

I asked at the end of September what had happened; why had she changed?

“I decided I wanted to do well. So I decided I would do my homework and do work in class.”

That was it. She had nothing to add to it. She just decided, and then she did it.

A colleague had a similar conversation with a similarly transformed pupil. His answer was simple “I decided I should try working instead of daydreaming and the work seems really easy now.”

Another girl in the same class, Jana, had appalling results in maths, and every other subject. She struggled to answer the most basic questions (How do you get home? What’s 4+10?). I assumed she must have a very low processing speed and a very limited working memory. Even an instruction like “pick up your whiteboard pens” seemed to be received on delay. I decided in November that being helpful and understanding wasn’t the right approach; she was getting less than 10% in year group exams where the average always exceeded 70%. I tried being tough. In lessons, if I asked a simple question which she couldn’t answer, and then told her the answer and asked again and she still couldn’t answer (i.e. hadn’t listened), she’d get a demerit. In many lessons she would get two demerits this way, meaning a detention. I was worrying that I was punishing a child who maybe had a fundamental problem.

At the start of this half term, she was different. She was answering everything. She was slightly slow, but her working was always clear and always led to good quality solutions. Her errors made sense and were typical of a Year 8 (i.e. she was doing as well as everyone else, making mistakes that reflected thought). Her hand was always up, provided there was thinking time. She asked good questions. She got lots of merits. I asked her what had changed.

“I realised that if I listen then I get it.”

I couldn’t tell if that delighted me or made me furious. But she has maintained the change, and has stopped looking worried and lost in lessons. She seems to enjoy maths and feel proud of what she produces. I suspect she won’t turn back.

These experiences underline for me how much of pupils’ underachievement, even where they seem like cognitive or social outliers, has a simple explanation. They are not listening properly, they’re not really thinking, and they’re hoping they can fly under the radar with minimal cognitive effort. They are not disrupting, but they are not learning. Their precious time at school is being squandered.

Few normal (i.e. non ‘bright’) pupils get good results, or have good life chances, if they stay stuck in this rut. Teachers need to motivate and inspire these pupils, but we also need to keep them under constant pressure to listen carefully, think deeply and feel accountable for their work, both on the page and in their brains. These are the strategies that we have come up with in the maths department (with lots of input from Olivia Dyer, the head of science).

Strategies for pushing more accountability onto the pupils

  1. After an explanation or example, posing questions that put the onus on the pupils to seek more help or clarification:
    • “Who needs me to explain that more?”
    • “Who would like to see another example?”
    • “Who needs me to say it in a different way?”
    • “Who needs me to ask them a question?”

2. No Opt Out (described in Teach Like a Champion). This comes into play when a pupil doesn’t know an answer to a question. Lemov describes well the why and the how. In summary:

a. Pupil A doesn’t know the answer

b. Tell them you will come back to them (eventually this can be dropped)

c. The answer/explanation is supplied

d. Go back to them

e. If correct: well done, you went from not knowing and answer to being able to say it (I know this is a shallow description of ‘knowing!’. It is the first of many steps…). If incorrect: give consequence for not listening / opting out

Levelling up: Narrate why it is important to listen carefully and be ready for the teacher to return to them. Encourage pupils to remind you to come back to them (by putting hands up politely), thanking them for reminding you and taking responsibility for being held accountable. Praise it as behaviour that shows they really want to learn.

  1. If a pupil looks a little spaced out, or often is a poor listener, saying “I am about to ask three/five questions. You’ll be picked for one of them.”
  2. Everybody answers: before you accept answers to a question, every pupil writes their answer down. This gives more thinking time to the slower thinkers. It also holds them accountable, as it is visible if a pupil is writing or not. This is common in maths with the use of whiteboards (provided there is a good routine in place for pupils to write the work in a secretive fashion and show it simultaneously, so that pupils can’t copy each other).
  3. Describe – and enforce – the body language you expect to see when you ask a question. These are the ones I typically expect and insist upon:

a. Looking at the question on the board, with an expression that shows ‘thinking’ (no vacant expressions). This is usually a focused or intense face. Some pupils faces really screw up their expression when they’re thinking, some look quite calm. This depends on you knowing your pupils, but the absence of focused thought is generally quite obvious.

b. Looking at the question on the page, with a thoughtful expression (as above).

c. Doing working on the sheet / whiteboard. In maths this is typically jottings for a calculation, or other things to relieve the burden on working memory.

d. Hand up, waiting to answer.

With some classes, I’ve said “If you stare vacantly at me once I’ve asked the question, instead of looking at the diagram, I will know you are wasting thinking time. That means we’ll have to wait for you, and is stealing time from the people who started thinking straight away.” I’ve moved to giving a demerit if they persist in it after the warning. That might seem harsh, but the explanation of why I do it means the pupils seem to find it very fair (it’s always palpable when pupils think something is unjust!) and the quality and pace of responses has jumped up. I wished I’d moved to this sooner.

6. Give more thinking time for questions. We all think we do it. We all know we don’t! A colleague pointed out that, for our many EAL pupils, they must hear the question, translate to their home language, think about it, decide an answer, translate back to English and THEN put their hand up. It also puts positive pressure on both teacher and pupils:

a. If more and more hands are gradually creeping up, the coasting pupils think “Yikes! Better think of an answer” as their non-participation is becoming obvious. If you really want to keep them on their toes, you can ask the 1-2 without hands up to tell you what the question was. If they know, but can’t answer, that’s fine. If they don’t know…make clear this means they are throwing away a chance to learn and to test themselves.

b. If the number of hands going up stops, you know the problem is probably you: you need to tell them again, and make it clearer. You also might need to improve the question, so it is also clearer.

7. Pause before asking for hands up. Give the thinking time, then say ‘hands up.’ This means many more hands go up at once (giving the message “It is normal to participate in this classroom” “It is normal to be eager to answer”) and slower pupils aren’t dispirited by their neighbour who has an answer before the teacher has finished speaking.

8. Show almost all of the question, but leave out the final element. This means no one can put hands up until you are ready, but they can begin thinking. For example, you could give this simplification: 4a3 x 5b? and leave the question mark blank for 5 seconds, allowing them to plan their answer for the rest of the question. This gives the slower thinkers time to catch up, and creates a slight element of drama when the number under the question mark is revealed.

9. For recaps when pupils seem unsure, give word starts:

“What is the name for a triangle with two equal lengths and two equal angles?”

[few hands]

“It begins with i…..”

[many hands] [take an answer]

Ask the question again

To be  clear, the strategy above isn’t helping them connect ‘isosceles triangle’ to the definition. It is probably only helping them to remember the name of a triangle that begins with i. But, it can be a good way for pupils to see they know more than they realise, and to build up their confidence. It also helps you see if the problem is remembering a word at all, or connecting it to a definition.

10. Reverse the question. If you’ve asked a question, like the isosceles one above, you can reverse it straight away: “Tell me two special features of an isosceles triangle.” Assuming you made sure that everyone listened to the first answer, it is now not acceptable to not know the answer. This makes clear to pupils that they need to really listen to your questions, not just jump to answers.

11. Interleave questions. If pupils are struggling to match together a word or procedure and its definition or process, or to explain a concept, you need to ask it several times. However, repeatedly asking the same question means pupils quickly start to parrot back sounds, rather than strengthen the connection between words and ideas. Interleaving the important question with other low-stakes facts that they know forces them to listen more carefully and to do more recall (rather than repetition). For example, if the key question is how to find the sum of angles in a polygon, you might mix it in with easier questions like “What does n stand for in the formula?” and “Which polygon has an angle sum of 180?” and “What is the formula for the area of a triangle?” This forces more thinking and practise of contrasting the new answer (the formula) with other faces that seem similar.

Make them accountable for helping you to check their understanding

The main challenge with pupils who are struggling is that they can be adept at disguising it. Many options for ‘whole-class AFL’ are technology heavy, or fiddly in one way or another. We like the following:

  1. Heads down, fingers up: if the groundwork is done, this can be a very quick way to check understanding. It works best for questions with two options (yes/no or true/false) but can be also for ‘answer 1, 2, 3, 4 or 5.’

a. Pose a question (typically focused on misconceptions)

b. Give time to think and decide secretly on an answer

c. “Heads down!” Pupils put their heads down in the crook of their arms (to avoid a ‘thunk’ and bruised forehead!) and one hand resting on top of their head

d. The teacher calls each of the options and pupils raise their hand up a small amount (so the movement is imperceptible to their neighbours). It is important the teacher gives the same amount of time for each possible option, so as not to give away the answer. Counting to 4 in your head can help.

e. “Heads up!” …give them a few seconds to readjust to the light… Having their heads in the crook of their arms means they don’t get as zoned out as having it straight on the desk, which is also helpful!

2. Routines for whiteboards that keep answers secret from each other (described above). You must narrate why it is not only important not to look at others’ boards, but also why keeping one’s own board secret it essential. Narrate how it might seem kind to let someone see your answer, but it is in fact unkind as it stops them from getting the help they need.

3/ When answers are given on whiteboards, praise good-quality written explanation. For example, I will pick out and praise the clearest workings, showing them to the rest of the class and praising how it let me understand what they were thinking. A colleague encourages his pupils by intoning, in a very funny way “…let me see your brains.”

Levelling up: I have recently moved to giving pupils demerits if they show me the wrong answer with no working. This has made a huge difference in two ways: it means that children who are quick thinkers are forced to slow down, so the others aren’t intimidated or disheartened when they need more time. It also means I don’t waste time trying to guess where they went wrong. Full working allows me to quickly identify the point of error and give better feedback. Because this was narrated and ‘trialled’ for a lesson, the pupils who had demerits for this weren’t upset when they got a consequence and, more importantly, have changed their ways.

4. If you are faced with the problem of a big split between how many get it and how many don’t, and you feel bad for the ones ‘waiting around’ for the rest, you can try:

a. Writing up the exercise they will do once you judge they understand it

b. Posing a question to check competency/understanding, telling them to wipe their board quickly and start the exercise if you tell them they’re correct.

c. As you see each correct answer, saying simply ‘correct/well done/correct’ and letting them get on with it.

d. Get a show of hands of who has not started the exercise, then tell those pupils they are going to see more examples and be asked more oral questions. I find that, once I start on the re-teaching, many pupils then say “Oh! I get it now” and then they join in the written exercise, quickly narrowing down how many I am trying to help.

Laying the ground for purposeful written work

Strategies that I’ve tried and seen others use to good effect are:

  1. For short-form questions (i.e. those requiring only 1-2 steps), go through it first as an oral drill, cold calling pupils. Then, use it as a written exercise. There are several benefits: every pupil has had a chance to ask for clarification on questions where they don’t understand why that was the answer, or to note down hints to help them start it on their own; pupils can begin work quickly and in earnest, knowing that it is something they can do with more confidence; you get twice as much ‘bang for your buck’ with an exercise. This works best for things that are highly procedural, but I think it also works well for questions where the ‘way in’ must be found. If a good chunk of the exercise has been done orally, the written attempt will still require them to recall and decide how to begin.
  2. Drill on step 1: If the exercise is focused on decision-making (e.g. an exercise mixing all fraction operations, where the main challenge is that pupils muddle which procedure goes with different questions), it can be done as an oral drill just for step 1. For example, “For question a, what will you need to do? Find the LCD. Question b? Find the reciprocal and multiply.” This can be a lower-stakes version of the exercise to allow you to check how ready they are before embarking on the more extended task of completing the calculations.
  3. Before starting exercises, particularly more extended ones, or quite visual ones (e.g. an angle chase), give the pupils 30-60 seconds to scan for any that they think they don’t know how to start. Then, give hints and tips for those (depending on the pupils, you might model a very similar one on the board for them to look at when they get to it). This prevents you from running around from pupil to pupil as they encounter the problem, and gives them confidence when they get to it…and no excuse for just sitting there waiting instead of attempting it!
  4. If several pupils are struggling with the same thing, or asking the same question, or making the same mistake: STOP THE WORK! Make them all listen to the additional instruction, explanation or example. This prevents you from creating lots of low-level noise as you help others, and gives help straight away to them all.

Culture of Thinking: do I understand this?

The ideal situation is that pupils themselves are thinking deeply about what is being taught. This usually can be observed when they ask question in the form:

  • Did ____ happen because _____?
  • Is ____ like this because _______ is like that?
  • If that is the case, does that mean that ____ is the case?
  • Is this similar to the way that _____?
  • I thought that because _______ we couldn’t ________?
  • What happens if you try it with 0 / 1 / 2 / a negative number / a non-integer / a power?
  • I think there is a pattern in this. Is it __________?
  • Will the answer always be positive/negative/an integer/a multiple of __?
  • I have an idea to help remember it: ___________.

Praise such contributions! Narrate that this is the sort of thinking that makes someone good at your subject, and makes it stick as they are forming connections with other ideas. Their memories of the ideas will be richer and more powerful. You can also narrate how this is beneficial to the other pupils, and to you as a teacher, and express gratitude.

Culture of Thinking: What do I need if I want to succeed?

A good place to get to is if the pupils themselves identify what they need, and flag it up. This is usually seen with questions like “Could we try one first on whiteboards?” or “Could you show another example, please?” or “Could we do another question together before we begin writing?” This means they are really thinking about if they understand something (or, can complete a procedure) and aren’t relying on teacher validation. Things that can help to bring this about:

  • Narrating why you show examples
  • Narrating what you want them to think about when you explain things, or show examples
  • Narrating what they should annotate and why
  • Narrating why you are asking questions
  • Narrating what should be happening in their minds when they think about something

As above, narrate how this is beneficial to the other pupils. You can even say “Who is glad that ____ asked that? Next time you can be the person who everyone else is thanking, by being alert and giving me helpful advice.”

Miscellaneous suggestions

  1. Choral response is nice to deploy to help practice new and difficult pronunciations (combustibility, hypotenuse, consecutive, and so on). It is utterly pointless otherwise, unless it is being used to make pupils think. Choral response is great for an oral drill for questions like,
    1. a1 = ?
    2. a0 = ?
    3. 1a = ?
    4. 0a = ?

…but is pointless if they are simply repeating sounds. It needs to help them put ideas together, or be a low-stakes way to practise recall of facts or saying tricky words.

2. Use as many memory aids and links as you can. They more ways that pupils can recall something and know that they are remembering correctly, the better. There is no use in a pupil correctly recalling the process to find the median if they doubt they have it correct. That is nearly as bad as not remembering at all, as it will feel futile to proceed. Even the weirdest memory aids can be valuable: my Y9s suggested remembering median with two prompts: (1) think of it as medIaN, because it is IN the middle, and (b) it sounds like medium, and medium is the middle size. These are not sophisticated, but it allows them two have two ways to recall the process, and two ways to feel they are on the right path.

3. Set a goal for the lesson. Our deputy head described this as being what a learning objective was meant to be (as opposed to exercise in the time-wasting that can be seen – and enforced – in many classrooms today). I sometimes start the lesson by silently modelling an example of the kind of question I hope they’ll be able to do by the end, then putting a very similar question right by it. This will be on the left of the whiteboard. Then I use the remainder of the whiteboard during the lesson. Often I can be only 15 minutes into the lesson before (some) pupils’ hands shoot up, thinking they know how to answer the ‘goal question.’ This puts positive pressure on the others, as it gives the message “We’ve been taught enough to be able to do this! You need to keep up!” and lets pupils feel smart, and feel intellectually rewarded, for paying careful attention.

4. Have a set of stock phrases to denote things that REALLY matter and make them feel motivated to push themselves mentally. Olivia, our head of science, uses phrases such as

“I’ll bet my bottom dollar this will be on your GCSEs”

“This is the sort of question that only pupils who get an A* can do”

“Pupils who master this always find A level much easier”

I hope these strategies are useful to you. We are trying everything we can to get 100% of our pupils to do well in their GCSEs (and generally, be smart and confident people), so would love to hear about other approaches. These strategies are, of course, in the context of a school culture that celebrates curiosity, a love of learning and the belief that hard work is the path to success. This post focused on some behaviourist strategies, which we believe are the most efficient and effective approach, but in the bigger picture we focus on goals for the future and the instrinsic motivation of being an educated and confident person.

If you find it exciting to think about strategies to motivate and challenge children who often fall behind, consider joining us. Our ad is on the TES, or you can visit our website. You can also email me on dquinn [at] mcsbrent.co.uk if you want to know more.

Vertical vocabulary

One of the great things about working at Michaela is the symbiosis that happens between colleagues. Inevitably for me, this exchange of ideas is most keenly felt between the Humanities and English Departments and, specifically, between the essays that our pupils write in English and history. As Katie has written here, she and the English Department have developed the ‘show sentence’ in order to combat the problem of what happens after the classic sentence starter ‘This demonstrates that…’
A similar problem afflicts history teachers. We can drill pupils all we like on significant dates, people and places, but that does not, on its own, give our pupils the vocabulary to express what these dates, events and people mean. Unless history is to become ‘just one damn thing after another’ we need our pupils to be able to know what I call ‘vertical’ as well as ‘horizontal’ vocabulary. If the names of dates, people and places help our pupils to describe the ‘horizontal’ narrative of history, our ‘vertical’ vocabulary expresses the themes that link seemingly disparate periods of history together. This is the vocabulary that we need to express change and continuity, cause and consequence, conflict, power, economy and ideas. Crucially, this vertical vocabulary is always domain specific: the causes of the First World War and the causes of the Enlightenment will require different explanatory toolkits. An how are our pupils expected to master this vocabulary if we are not the ones to give it to them?

The big mistake that I made when I first started teaching was to presume that this ‘vertical’ vocabulary would just be discovered over time. I think there are many of the ‘enquiry’ bent who still think that this is something that just appears via osmosis. But I now see that this is as much as a nonsense as presuming that pupils will discover any other form of knowledge. Words like ‘tension’, ‘consolidated, ‘exacerbated’ which express historical ‘analysis’ are as much a form of knowledge as ‘King John’, ‘1215’ and ‘Runneymede’. Pupils must be explicitly taught such vertical vocabulary (and even occasionally make mistakes with it as they use it in different contexts) so that an initially inflexible concept bends and flexes over time.

Over the last two years, we’ve toyed around with various different ways to teach our pupils what a good paragraph looks like, each time desperately trying to get away from the straightjacket of PEEL and towards an understanding of paragraph formation that actually concentrates on what it is that the pupils find most difficult. As the English department have found, ‘this demonstrates…’ is not the part of the paragraph that pupils find the most challenging; it’s what comes next. The challenge for history teachers is not just making sure that our pupils remember the ‘horizontal’ vocabulary of dates, people and places, but to remember the ‘vertical’ vocabulary, too. This is exceptionally difficult, not least because the vertical vocabulary pupils need to express themselves with sophistication changes between periods of history and therefore from unit to unit.

What we’ve tried to do is break down paragraph construction into its component parts:

Point: What is your answer to the question?

Information: What dates, people and places have you remembered (we use this term deliberately) that support your point?

Explanation: Why are these dates, people and places significant? As my colleague, Mike, puts it – ‘So what?’

So if we were preparing our pupils for a question on the causes of the English Civil War, we would use an example show them the sort of structure and vocabulary that we’d expect to see in a good paragraph. We broke this essay down for our weakest pupils so they had to write an essay which included paragraphs on religion, power and finance as well as a conclusion. Have we had to over-simplify it as a consequence? Yes, of course. That is the price we have to pay to give them the foundational knowledge that they will critique as they learn more. I haven’t here got into the debate about long and short-term causes, although colleagues teaching more able pupils certainly would.

Here’s how we structured one of the model paragraphs on the religious causes of the Civil War for our weakest group in Year 8 (bottom quartile). The vocabulary that I’ve emboldened has been written into the booklets we use in class so that, by the time I get to the lesson, they should already be familiar with it. We need to be as confident that our pupils know the vertical vocabulary before an assessment as we are that they know the horizontal.

Question: Why did the English Civil War break out in 1642?

The English Civil War broke out in 1642 because of religious tension/enmity/division/animosity/hostility between the Puritan Parliament and the Arminian Crown.

In 1625, Charles I married the Catholic French Princess Henrietta Maria. This was important because the Parliament was predominantly Puritan and it increased suspicion that Charles was a secret papist.

In 1633, Charles appointed William Laud, an Arminian, Archbishop of Canterbury. This created further division because Laud attacked Puritans and banned their books.

Finally, in 1639, this tension escalated into violence after Charles attempted to impose his English prayer book on Scotland. It was this final act that led to a Scottish rebellion which forced Charles to recall Parliament and sparked off the civil war.

These events show/illustrate/demonstrate/reveal/emphasise that war broke out in 1642 because of increasing religious tension between the Arminian Crown and the Puritan Parliament

How is this different to how I used to teach paragraph construction?

  1. We’re not presuming that pupils will just discover words to express ‘tension’ and the changing relationship between Parliament and the Crown in the period – we’re explicitly giving them to the pupils. We create drills so that the vocabulary they need to express this changing relationship is automatized.
  2. We’re deliberately structured the evidence which supports the point in chronological order to help them remember the examples and to encourage them to think about how this relationship changes leading up to 1642.
  3. Each time they use an example, it’s followed by one sentence of explanation. When we’re constructing this paragraph on the whiteboard we’re saying ‘Yes, but why is it significant that Charles appointed Laud as Archbishop? How does that support your point? SO WHAT?!’

This essay was completed by a pupil in our weakest group in Year 8 at the end of last year (from memory). The pencil mark in the corner is the mark 15/25. What I was most pleased to see was the inclusion of the ‘vertical’ vocabulary that we’d be teaching in the lessons preceding the assessment. We’ve still got a long way to go on this, but I think it shows we’ve made a start.

img_20161021_131443021

You Turn Me Right Round

This is just about some ideas for angles we’ve been using in the department.

Polygons: angles as turn

We’ve been trying to demonstrate to the pupils what it actually means to say the sum of the angles in a triangle is 180 degrees. On a straight line it is easy to demonstrate the half-turn, and remain faithful to the idea that angles are a measure of turn.

I find that the activity of tearing the corners from a triangle and arranging them on a line doesn’t seem to stick. I suspect it also reinforces the idea of a line more than the idea of a half-turn.

Using a board pen or, in my case, a small toy bird, we’ve tried this instead. Because it is small-scale, I’ve used an anthropomorphised key…

It starts facing forward. At each vertex, it turns. Upon returning to its original position, it is facing the opposite direction. It has completed a half turn🙂

It then extends nicely to quadrilaterals (it is facing the same way), then pentagons (a turn and a half – facing backwards) and so on. It allows the pupils to see that not only does the number of triangles increase (the standard and much-loved way of showing progression in polygons), but also that each time they increase by a half-turn.

Vertically opposite angles

We were finding it tricky to help pupils spot vertically opposite angle when there were more than two intersecting lines. One pupil* suggested that they position their rulers to rotate around the point of intersection, turning to hide the one being focused on. The one that is revealed has the same turn, so must be equal. They are the vertically opposite angles. Here are two examples of her suggestion:

It is obviously easier if a finger is placed at the point of intersection, as it is easier for the ruler to rotate. The limits of one-handed camera phone filming!

Lastly, another pupil had a good suggestion to help with spotting vertically opposite angles. If each separate line segment is highlighted (and there are no non-straight lines!), then the ones ‘trapped’ between the same colour-pair will be vertically opposite to each other:

vertically-opposite-highlighting

This works better than just giving highlighters, as the additional rule of ‘trapped between the same colours’ gives them a little more to hang onto!

If you have tips to make it easier to spot alternate angles than ‘a Z shape’…please tell me!

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(I’ve partly made a fuss of this way of modelling because the pupil is a solid fourth quartile kid. It’s been really exciting to hear her come up with her own ideas for demonstrating what she understands. I plan on showing her this video tomorrow)