Never Let Me Go

 

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

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

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

enlargements - modelled example

The tasks for the pupils to undertake following this example:

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

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

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

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

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

 

Here are some counter-arguments:

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

 

What we should be doing instead:

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

 

TL;DR

  • Plan hard examples
  • Practise explaining hard examples

 

 

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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!

drill and thrill 1

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.

drill and thrill 2

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… 

drill and thrill 3

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. 

drill and thrill 4

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)

drill and thrill 5

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

drill and thrill 6

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.

drill and thrill 7.png

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)

Design advice

  • 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
    • Add 1
    • halve
    • double
    • add 10
    • 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
    • pattern in answers
  • 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

 

 

 

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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.

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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 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.

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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!

—-

(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)

 

 

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Tell Me More, Tell Me More

There is a problem in how we teach:

We typically show pupils only the classic forms of a problem or a procedure.

It’s easily explained: time and fear. It takes ages to show lots of examples, and it’s also scary. They might be bored. They might not get it. When we want to be kind to ourselves, I think we say we don’t want to spoon-feed, but I suspect it is also due to thinking only about the specific questions we want them to be able to answer. It’s goal-focused thinking (typically good), but the goal is too narrow (that lesson’s questions). The upshot is that they can’t generalise…the most prized of mathematical skills. I’m tentatively concluding that this is a huge problem.

I suspect that, even if we fixed behaviour, bad leadership, too little contact time, and all the other things that can impede learning, this problem would still cause the long tail of underachievement we see across the country. I suspect it explains why pupils with anything less than an A* are bewildered by A-level maths (and even those pupils look pretty shocked), whereas pupils with As and Bs can cope in other courses.

We suffer from three problems, as maths teachers:

  • expert-induced blindness (we can’t see what we can’t see because we can’t remember finding it difficult to understand)
  • We can see the connections and logical conclusions in maths, and it is obvious how things generalise. We don’t even notice ourselves generalising.
  • Algebraic reasoning is really hard to teach. People who can reason algebraically feel like they are applying common sense. I suspect a more accurate description of is “a deep understanding of the order of operations (and willingness to test by substituting)”

It took me a while to realise why I have decent algebraic reasoning. I struggled a lot at A-level (which is why I like teaching maths, incidentally). Everything else before that seemed easy. As in, I filled time after finishing exercises with finding new ways to do them (e.g. if I’d solved equations by balancing, I’d go back and use trial and improvement, then try factorising to solve in a different way…). It seemed like filler at the time, but really I was amassing hours of practice and practising generalising. Our pupils don’t have that time (or inclination…I’d have sooner written two extra pages in silence than whispered to my neighbour). Their teachers have to close that gap by making it explicit.

One of the biggest disservices we do to our pupils is that we leave it up to them to reason and generalise at the point when examples and problems become least familiar. In particular, we disadvantage those who find maths most difficult. It’s scary to show them anything beyond the standard examples, as they become easily confused. It’s also frustrating and slow. But this is why they then can’t generalise: we didn’t show them anything non-standard or, if we did, it was in an exercise when they were floundering on their own with the least support.

how-maths-feels

I used to think this was the right way to prepare for a topic:

“What sort of questions do I want them to be able to answer?” and planning backwards from there.

This is too narrow. Instead, the question should be:

“What are all the possible forms this topic can take? What’s everything I understand, implicitly, when I look at those questions? Which things have I reasoned for myself, that I don’t even notice myself figuring out?”

I’ll try to make concrete what I’m describing, as I’m realising how far we still have to go in our department:

In the past few days, I’ve been preparing booklets (i.e. textbook chapters) on algebraic expressions (Year 8) and angles on lines and around points (Year 9). Even simple topics have alarming levels of depth to them if approached from the perspective of

“What if none of this was obvious?”

This is just the examples for coefficients. Imagine a weak pupil. Heck, imagine a normal pupil! Are any of these really obvious?

coefficients

Goodness knows that, until now, I probably only showed them two of these, three if they were lucky. And then I couldn’t understand their inflexible thinking. It will take a long time to go through all of these with my lovely fourth-quartile group, and it will take a lot of pausing, checking and mini-whiteboards. But if not now, when? If I don’t show them now, I am implicitly deciding they aren’t to know it. And, by implication, that top grades aren’t to be an option for them. Top grades might not be realistic, but it shouldn’t be because I didn’t teach them.

Here is one section – of many more – of the examples to show that angles can look different but be equal. And these aren’t even that different! But I had assumed most of this was obvious before now, or maybe only shown Image A and Image D. D and E as comparisons were thanks to a suggestion from a colleague.

equal-angles

Here are some of the question types I have been playing with just for angles on a line and around a point. And I have a horrible feeling it’s still only scratching the surface. There is so much implicit knowledge in here: matching angles, right angle notation, when and how to use algebra to solve, spotting straight angles in the midst of full turns, spotting two distinct sets of 180 on a single straight line… It is really scary to imagine teaching it. It is even scarier to think that for 6 years I have allowed pupils to walk into exam halls (or worse…the world) where they have no help or explanation the first time they come across it.

angles-on-lines-and-at-points

Given the sheer volume of implicit knowledge and modelling needed (especially for the second and third from last….they will melt the minds of all but the most confident. Terrifying!), we’re expecting that ‘angles on lines and around points’ will be at least three lessons.

So far, I think that’s my contribution when people say

“What do you mean by a mastery curriculum?” 

I think it might be simplest to summarise as

“We tell them everything we know, even when we didn’t realise we knew it.” 


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What Matters (Mathematically) the Most

 

Sometimes when maths teachers visit I feel frustrated that I can’t make suggestions to them of what they could change (or keep) to improve how their pupils do in maths, as the holistic approach of the school can make it feel as everything is inextricably tied together. However, there is one strategy that has made a big difference for us that can, crucially, be replicated in other settings:

  • Identify ‘high leverage’ topics
  • Teach them as early as possible in KS3
  • Give them lots of time when they are initially taught
  • Interleave them into every subsequent topic, whenever possible
  • Frequently revisit and retest them as stand-alone topics

 

  1. Identify high-leverage topics

These are the topics that:

  • Can come up anywhere (e.g. fractions in a perimeter question)
  • Can derail a lesson if they’re not in place (e.g. dividing by 10)
  • Tend to terrify all but the strongest pupils (e.g. calculating -629 + 433)
  • Are counter-intuitive (e.g. adding fractions)
  • Are prone to ‘System 1’ (i.e. unthinking) errors (e.g. squaring, order of operations)
  • Are the typical building-blocks we don’t have time for in KS5 and are the bane of A-level teachers’ lives (…..all of algebra?)
  • Are easy ways to gain (and lose) simple, predictable marks in exams (e.g. rounding)
  • Are easily confused: what Bruno Reddy describes as ‘minimally different topics’ (e.g. three measures of centrality (mode/median/mean), or perimeter and area)

The ones we’ve identified are:

  • Automaticity with simple calculations (adding and subtracting small values, halving and doubling small values, times tables (including division), adding 10 to any value)
  • Some aspects of place value (multiplying and dividing by powers of 10, in particular)
  • Efficient and reliable written methods for the four operations, including with decimals
  • Directed numbers (particularly the four operations)
  • Fractions (pretty much everything about them: simplifying, forming equivalents, comparing, ordering, the four operations…)
  • Simplifying expressions, especially when the variables look scary
  • Rearranging expressions, especially when there are negative coefficients
  • Solving equations (linear)
  • Rearranging equations (both linear and polynomials)
  • Substitution
  • Order of operations, particularly the importance of leaving addition and subtraction to the end of the calculation
  • Rounding (including to significant figures)
  • Area and perimeter (simple cases, focus on not muddling procedures)
  • The three averages (simple cases, as above)
  • Simple proportional reasoning (e.g. in an equation, being able to multiply both sides by 5, or doing opposites to both parts in a product (e.g. 12.5 x 16 = 25 x 8), etc. Obvious cases are finding the ‘best value’ product when differently-sized packets have different prices, but it also comes into percentages of amounts, etc).
  • ‘part + part = whole’ (e.g. in an L-shape, the two lengths on the right-hand side must have the same length as the height on the left-hand side), typically represented on blank number lines
  • Key vocabulary (integer, associative, inverse, eliminate, variable, etc)
  • Key number facts and relationships (the first 10 primes, 15 squares, 10 cubes, fraction-decimal conversions, the effect of dividing by 0.5 or 0.1, etc)

 

  1. Teach them as early as possible in KS3

That list is a lot! It takes up almost all of Y7 and Y8. Even then, some are only covered to the extent that they allow us to continuously drill, quiz and probe how well they differentiate ideas. For example, our Y8s frequently do questions with perimeter and area, but haven’t been taught anything beyond area of rectangles and perimeter of ‘any shape’. Similarly, our Y8s and Y9s are pretty good at applying the three averages as procedures, but have few insights (yet) into what they measure. That will come in Year 10 (I am convinced it’s hard to meaningfully understand statistics before then, so am hoping the gamble of separating procedures and understanding will pay off).

We put directed numbers as early as possible in Year 7, once their written methods with decimals are sound. This is followed by ‘all of fractions’ and then…Year 7 is almost over! Angles are studied to a limited extent, but as a vehicle for practising written methods (e.g. practising subtracting by finding the missing angle on a straight line).

Year 8 could probably be summarised as ‘algebra, all year.’

In Year 9 we go a bit wild and do percentages and then shape until the end of the year (angles, with understanding what an angle actually is, then Pythagoras, trigonometry, transformations, etc).

 

  1. Give them lots of time when they are initially taught

Teach every variation that you can think of

Take the simplest aspect of solving equations. 4a=12, or a+5=11 might spring to mind. That’s barely scratching the surface. These cases are relatively obvious to all but the weakest quartile. The examples below need to be shown to pupils.

2a + 2 = 12 (simplify first)

12 = a – 10 (unknown on the RHS)

7 = 3a (result is a fraction)

5.7 = a + 7.8 (decimals AND a negative solution)

3/4 + a = 9/10 (fractions requiring LCD)

10 – a = 20 (answer is a ‘surprise negative)

Failure to teach these explicitly disadvantages the vast majority of pupils. As teachers we frequently make the mistake of showing our pupils the classic examples of a case and thus never increasing the flexibility of their thinking. Inevitably, they freeze once the problem doesn’t take the form they’ve seen – it looks like a totally different problem!

 

Thinking of every variation is time-consuming, but also enjoyable. It forms the basis of fruitful and professional discussion with colleagues (and can be done via Twitter, I’ve discovered) and can be aided by trawling through very old textbooks and exams (enjoyable, if dusty, work!).

 

Plan for every misconception, and pre-empt problems by explicitly teaching about them

Many of us wait until the pupils say “but….why?” when we show them the procedure to add fractions. Inevitably, the explanation is messy and confusing, even for us. Sometimes we plan to ‘explain’ it by using diagrams. This might show it working, but still doesn’t give them the language to explain why one way makes sense and another way doesn’t. It should be planned for in advance, agreed with colleagues and scripted (to ensure clarity and economy of language). Some of my best explanations were in my PGCE year, when I used to rehearse the rationale for things; it should never have been dropped!

 

Include lessons that focus on addressing cases where pupils muddle concepts and are likely to make errors

In the past, I have finished each year thinking “I really should have done a lesson to get [class] to practise the difference between finding a percentage of an amount and reverse percentages” and then not done it. Because it is hard, and it is scary. The Y9 teachers are mentally preparing themselves for it: we’ve planned our percentages lessons in anticipation teaching what will undoubtedly be one of the toughest lessons (or 2 lessons…or 3 lessons…) of the year. It is really, really hard to help pupils see the difference between these two types of questions. This is why we have to force ourselves to do it; praying they’ll see the difference ‘on the day’ – what I’ve always done in the past – is to abdicate responsibility at the point we know they need us most.

There are many topics like this: order of operations questions with a negative result (e.g. 3 – 10 x 2) or questions that combine surface area and volume.

 

Do lots of drills, both on decision-making and on procedures

I was slowly losing my mind as I taught some of the weakest pupils to calculate with directed numbers. Showing them quick work-arounds for some questions (e.g. -4-5 = -9 can be quickly calculated because it is a total distance, or -3 + 7 can be quickly calculated by rewriting as +7 – 3 = 4) was causing as many problems as it solved as they were confusing when to apply them, even though they understood the ideas in principle. A teacher in the department asked me what drills I’d used to help them spot when to use them. It was a blinding flash of the bloody obvious: I hadn’t done any. I’d focused on calculation drills – completing those calculations – but not on decision-making drills – identifying the questions where those work-arounds are applicable. Drills aren’t mindless if they’re helping pupils to focus in on critical decisions, or improving pupils’ fluency and accuracy.

 

Do lots of extended and applied questions where they have to make decisions  

Once they know what they’re doing with the basics, make the questions TOUGH! It’s no good learning to multiply two negatives if they can’t extrapolate to four negatives. It’s easy to simplify the signs in -2 x -3; it can quickly become mindless and ineffective to practise. A better test of application is -2 x -3 x 4 x 0.5 ÷ -10. Once the pupil has to think about other things – such as how to multiply by 0.5, or divide an integer by 10 – we can see how well they really recall and apply the rules of negatives.

 

  1. Interleave them into every subsequent topic, wherever possible

Answering this question requires much more thought than “Expand 7(2a-3)”

“Form an expression for the perimeter of a regular heptagon with a side length of 2a-3.”

And

“Calculate the area and perimeter of a square with a side length of .”

Requires much more thought, recall and organisation than simply presenting the underlying calculations.

Simplifying expressions is relatively easy. Simplifying this expression is not:

7/8ab + 2/3a -1/2b + a/2 -ab

Interleaving in this manner, in every topic, communicates to pupils that everything they have been taught is relevant and important. They can’t decide “I’m bad at fractions, but that’s ok.” They’ll realise they have little choice but to improve and stay good. Similarly, their teacher will face the same reality: they can’t afford to give up on the topics that are most challenging to teach.

 

I have this checklist in my mind when I design questions for any lesson:

Could we include….

  • decimals?
  • fractions?
  • directed numbers?
  • the order of operations?
  • perimeter, area or angles?
  • averages?
  • indices?
  • more challenging language?
  • Variables instead of numbers?
  • Technical vocabulary (write ‘variable’ or ‘unknown’ instead of ‘letter’)
  • Technical syntax (“A number is picked such that…” “Demonstrate that, for all integers…”)
  • An opportunity to include some ‘scary’ generalist words (writing ‘nasturtium’ instead of ‘flower’ and ‘yacht’ instead of ‘boat’ is another way to bring valuable difficulty to routine practice and teaches them to be comfortable with not knowing every noun they see in questions)?

 

  1. Frequently revisit and retest them as stand-alone topics

A third of our weekly quizzes is given over to explicitly testing pupils on these high-leverage topics. This gives us valuable information about their retention and growing misconceptions, and forces us (and our pupils) to give over regular revision time to them.

 


 

All of this takes a lot of planning up-front, unsurprisingly. It also demands a pretty rigorous mindset when planning. Thankfully, it can be introduced gradually and has a snowball effect as pupils become accustomed to regularly revising tough topics. Additionally, this strategy can be applied in almost any setting, regardless of your school’s meta-approach to teaching, learning and behaviour, so can be operated without too much interference.

 

Think this sounds interesting? Come visit! We love having guests. It challenges our thinking and it boosts the pupils’ confidence to have people come in to see them.   

Think it sounds wonderful? Apply to join our team of enthusiastic maths nerds! We are advertising for a maths teacher, starting in September (or earlier, for the right candidate). Closing SOON: https://www.tes.com/jobs/vacancy/maths-teacher-brent-440947 


For those who spotted it, the title is ripped off from one of those poems that we fall in love with aged 15 and meet again 15 years later…

what-matters-most

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Filed under curriculum design, lesson design, pedagogy