# Category Archives: Interesting or Fun

## Tried and Tested: Solving with SMEG

(this follows on from my last post about the order of operations and the advantages of the acronym GEMS)

I had a revelation (or, blinding flash of the bleeding obvious…) in my third or fourth year of teaching: when solving equations*, we are inverting the order of operations.

If that is crushingly obvious to you, please enjoy a smirk at my expense, but be aware this might be an issue for your colleagues and they could benefit from your expertise!

If not, I’m glad it’s not just me. Let me explain:

To evaluate 10+3 x 5², our order would be:

Exponents: 10+3 x 25

Multiplication and Division: 10+75

If it were the other way around, as an equation (10+3 x x²=85) we would ‘know’ (more on this later…) to follow the following order:

Subtract 10: 3 x x²=75

Divide by 3: x²=25

(Eliminate) Exponents: x = 5, x = -5

In other words, we ‘know’ to reverse the order of operations. But we don’t say this to the children!

This implicit knowledge (or unknown known, for the Rumsfeld revivalists at the back) underpins the function machines approach to solving equations.

I have tried to conceal the origin of the image below– the teacher who shared this image from their work is generous to have done so! – and have included it to illustrate what I’m describing.

Using a function machine is a common approach for the early stages of teaching children to solve linear equations, rewriting

3x + 5 = 11

as

x -> x3 -> +5 -> 11

and then teaching the children how to ‘go backwards’ (i.e. use the inverse).

There are some problems with this.

• At worst, it collapses for harder questions (setting up a machine for unknown on both sides or questions of the type (x+3)/(3x-4)=7 or even 13 – x = 7 is a tortuous process and teachers have to furrow their brows to prepare it. Pity the novice learners!).
• At best, it is awkward when showing operations such as squaring.
• Eventually they will still have to learn to balance to solve (‘change side change signs’ and *shudder* ‘magic bridge’ are not defensible, in my opinion, and you should drop them*.)
• More fundamentally, this structure still doesn’t explain how to set it up successfully on their own. It doesn’t explain or teach anything, it is only a model for visualising. Structures are lovely, and can be illuminating once you know what is going on, but unless they can ‘see’ the order in which the equation was ‘built’ in, they will be trapped with random guessing when putting operations into the machine. Without explicit instruction about interpreting the order for setting up the equation, I think most children would make the reasonable inference that, if

3x + 5 = 11 becomes x -> x3 -> +5 -> 11

Then

5 + 3x = 11 becomes x -> +5 -> x3 -> 11

Because ‘5 happened first.’

Of course, you might have already spotted (or have already explicitly known) that the order of ‘building’ follows the order of operations: in 5 + 3x = 11 we can see that  happened first because multiplication takes precedence over addition. From this we can conclude that, to solve the equation we must do the inverse of the order of operations: invert the addition before inverting the multiplication.

My bet is that most teachers, particularly those in their first few years of teaching, haven’t realised this and certainly haven’t communicated it to the children.

Tried and Tested: SMEG

Our approach has been:

• Teaching the order of operations in Y7 using GEMS, and getting children to be confident and fluent with it for several months, including with negatives and fractions and forms such as 3(4+5×2-20)/2. We leave algebra until the end of Y7/start of Y8 (once their number work and mental work is secure, especially around fractions, factors and directed numbers).
• Spending a lot of time on expressions and collecting like terms and manipulation
• Teaching children to form expressions, using GEMS to set it up correctly (e.g. rigorously teaching and checking they can differentiate between “I think of a number, I add 4 and then multiply the result by 3” and “I think of a number, multiply it by 3 and then add 4” or “I think of a number, square it, add 5 and divide by 2” and “I think of a number, divide by 2, square it and then add 5” and so on)
• Introducing one-step equations for 2-4 lessons (!), depending on the group, with a big focus on balancing and no decision-making about ‘what to do first.’ The numbers used are deliberately NOT ones that facilitate mental calculation; the aim is to make it essential to learn to balance. We also teach checking methods at this stage and illustrate the idea of balancing with visuals (usually just images; an actual balance seems to be unfamiliar to this generation of children and hasn’t made any difference to their understanding, in my experience). We would also include examples where they have to simplify first, such as 2a + a -6a + 7a = 11 – 4 – 2, and so on.
• Teach technical terms, such as eliminate, unknown, isolate, balance and inverse. This allows them to compress more ideas into an explanation and reduces the chance of them making incorrect leaps in logic or imagining patterns that aren’t there.
• Introducing a range of 2-steps and explicitly teach “We use GEMS to calculate, we use SMEG*** to solve (i.e. undo calculations)”. We then show them how to use SMEG to undo the calculation.

e.g.

31 = 3x – 5

Would be explained as:

[Can we simplify either side? – this twist wouldn’t be in the first few examples and checks,  but introduced soon]

What are we trying to isolate? The unknown, x.

What do we need to eliminate to do this? The operations on x’s side, which are x3 and +5.

S is first in SMEG: is there a subtraction or addition to eliminate? Yes, the +5. Eliminate that from both sides.

We have eliminated everything in the ‘S’ group. Is there anything in the M group, a multiplication or a division? Yes, the x3. Eliminate that from both sides.

The unknown is isolated; we’re done.

(As I mentioned before, we try to avoid easy ‘in your head’ questions when teaching a process or a decision – although some will be shown after the first few examples so that they can see their intuition is correct – and try to avoid the most obvious layouts (e.g. not having most examples in the form ax + c = b) ).

This works even as the question becomes more complex:

(3-2a)/5 = -6

S: there is an addition (the +3 is added to -2a; we would have taught that idea already), but it is in a Group, so it needs to be dealt with at the end.

M: there is a division, we will eliminate that. Multiply both sides by 5.

3 – 2a = -30

The group is gone, we can start SMEG again.

S: There is an addition, not in a group. Eliminate that:

M: There a multiplication left, so eliminate that by dividing both sides by -2 and simplifying:

a = -33/-2 = 33/2

Unfortunately, we still have to teach that there is a special ‘first step’ with an unknown on both sides (that is, eliminate the smaller unknown). We explicitly teach them to distinguish between cases of one unknown and unknown on both sides, such as

3a +2 +a = 5

and

3a + 2 = a + 5

This might sound like a very intense level of detail! It’s been the product of several years of refining and trying to find out where the children come up against difficulties or their own reasoning and inferences can lead to errors or frustration. We’re pretty happy now that this approach is leading to success for close to 100% of children (most of our pupils would say maths is one of their favourite subjects, and equations are often given as an example of something they find easy to do).

(This photo is from 2015, when this pupil was in Y8 and much less confident with maths. It’s such a pleasure to see her now, as she is so much more assured – although still hilarious and expressive! – and has some really creative approaches. That said, I think a clear structure allowed her to achieve early success and self-belief, and her more methods are still underpinned by reliable processes such as balancing with SMEG).

*this is assuming the equation has a single unknown (i.e. no quadratics with  terms). I’m focusing here on the foundational phases of solving an equation.

** 1) It conceals why anything works. 2) You aren’t changing signs, you are inverting the operation! 3) it collapses as soon as you aren’t eliminating addition or subtraction.

*** So, obviously SMEG makes it sound gross. I usually mention the brand of fridges as an offhand comment so that they do unwanted wondering. I was tickled to learn that SMEG (fridges) stands for Smalterie Metallurgiche Emiliane Guastalla (“Emilian metallurgical enamel works of Guastalla“)

Filed under Interesting or Fun

## Tried and Tested: GEMS

One oddity about working  in a school that is well-known / notorious is that we are often asked about how we – that is, the maths department- approach or teach different things. I worry about saying anything, as we don’t have results yet (182 sleeps to 21st May 2019!) and our approaches haven’t been through the crucible of The Exam Process. I don’t want to give the impression of certainty as we are always changing, refining and even chucking whole strategies!

However, there is a pair of approaches we have been taking for several years that we do think are working well and I hope you will find useful too: GEMS and SMEG. I’ll write about GEMS below and its unattractively-named cousin, SMEG, in my next post.

GEMS is an alternative to BIDMAS/BODMAS, and is used more in the USA, although not widely. I can’t remember where I first saw a teacher blog about it; please let me know if it is you, so I can give credit where it is due.

GEMS stands for:

• Groups
• Exponents
• Multiplication (and its inverse, division)
• Subtraction (and its inverse, addition)

Why we use it:

• It forces teachers to make clear to the pupils that multiplication and division are equally important operations
• As above, but for for addition and subtraction
• *Even if* the pupils forget, in the case of addition and subtraction, having subtraction ‘first’ avoids errors such as 10 – 5 + 3 = 2. From what I can see, this type of error can only occur if they think they must add first.
• ‘Groups’ is much more accurate than brackets / parentheses, as it includes expressions within numerators, denominators, roots, etc.
• Exponents is no more difficult a word to teach than indices
• It has a nifty image that can go with it to give a sense of the hierarchy (and add general sparkly specialness 🙂 )

(images from Ms Lopez’s blog )

• No revision guide or revision video is using this acronym. You have to explain to them that it is the same kind of mnemonic as BIDMAS, and explain why you are teaching it instead.
• You have to be strict with yourself and with them about saying ‘multiplication…and its inverse division’, as it would be even worse if they had no idea where D and S have gone!
• In common with BIDMAS, it doesn’t solve the problem of ‘functions’ (e.g. if calculating 3 x sin40 + 5, GEMS doesn’t make clear that first they should find the sine of 40). We’ve been bit hazy with and let them sort of see them as an exponent without ever saying so…a sin of omission that makes things a bit easier, even if it feels a bit wrong!

If you want to make this change, please make it a whole-department change. It is a nightmare if one teacher does it and then pupils or classes move on to someone else. You may decide ‘only for Y7, and then add on a year each time,’ but ensure you have within-group consistency.

Bonus tip: introduce the order of operations after you have taught directed numbers. Children can quickly infer unintended patterns if all your examples have positive results. Ideally, early examples would be in forms such as this, to force them to preserve the order that is in question:

4 – 3 x 5

Many pupils, knowing they should ‘multiply first’, will conclude it also means ‘put the result of the multiplication first’ and write

15 – 4

for the next line. If you use only positive examples, such as 4 + 3 x 5, you won’t notice there is a growing habit of them writing 15 + 4, and so on, whereas their little leaps become obvious when negative numbers are involved.

Filed under Interesting or Fun

## Maths Muddle: alternate segment, same segment, alternate angles

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

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

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

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

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

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

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

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

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

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

Filed under Interesting or Fun

## Never Let Me Go

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

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

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

The tasks for the pupils to undertake following this example:

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

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

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

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

Here are some counter-arguments:

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

What we should be doing instead:

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

TL;DR

• Plan hard examples
• Practise explaining hard examples

Filed under Interesting or Fun

## Drill and Thrill

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

What is a drill?

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

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

Why Drill?

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

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

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

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

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

What can be drilled? And what should be?

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

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

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

Ones that didn’t work, or don’t suit

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

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

Rolling out: when and how

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

Joy Factor

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

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

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

Risks

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

Lastly…

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

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

Filed under Interesting or Fun

## Holier than thou

I try to avoid polemics, or being divisive. The topic below is one where, I suspect, opinion is already sharply divided and each side views the other with suspicion and some incredulity.

1) It really gets my goat when a teacher reserves an efficient strategy or rule-of-thumb for themselves, yet insists that pupils may only have access to a complex and more confusing version (i.e. The revered “correct” version). What happens in practice is that bright sparks quickly see the underlying pattern, check it for reliability, and then adopt the rule of thumb unimpeded by the teacher’s beliefs. The rest remain confused, experience low success rates in their work and thus low confidence and motivation.

2) If someone uses or shares a heuristic or strategy that isn’t technically accurate, but is efficient and reliable, it is bizarre to assume that they must be an ignorant teacher. Maybe they are. But maybe they’ve made a strategic calculation about how best to teach their pupils.

I have never met a maths teacher who doesn’t know that “the decimal point doesn’t move, the digits do.”

I get the impression most pupils have been told it too, especially in primary.

Low accuracy in this topic is one of the most common hallmarks of pupils who struggle in maths generally.

If a teacher teaches the strategy of “move the decimal point,” it is unlikely to be due to ignorance on their part. The cry “the point doesn’t move!” felt like a tired phrase before I’d even finished my PGCE year. It’s the “not all men” of the maths teaching world.

When multiplying or dividing by a power of 10, the crucial change is the relative position of the digits and the place value columns. Moving the digits, or moving the columns (i.e. the decimal point) will both get you there.

Overwhelmingly, numerate adults use the “rule” of “add a 0 to the end” when they are multiplying integers by 10. Overwhelmingly, numerate adults move the decimal point to multiply or divide by a power of 10. I don’t understand why we would block pupils’ access to this widely used approach, or than to be sanctimonious, to think “I’m not interested in them getting to the right answer, I’m interested in how they think, and in building deep understanding.”

Any decent maths teacher is interested in how pupils think and in building deep understanding. But is simplistic and arguably harmful to insist that learning only takes the form of “deepening understanding.” We learn to speak with (relative) grammatical accuracy long before we learn what it means to conjugate. I suspect most British people can’t parse their speech and writing. I also suspect that, if we’d forced them to learn to parce as the sole method of learning to speak and write, they would be terrible at it, and most would hate it.

We enjoy things we feel we are good at, and getting better at. Deep thought can only take place in the context of a rich landscape of examples, exceptions and intellectual self-confidence. Teaching a rule and building fluency creates the context for the surprisingly deep and difficult thought that underpins place value.

Filed under Interesting or Fun

## Under Pressure

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 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.”

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)

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]

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.