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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!

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

 

 

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Masses of Maths: what should pupils learn by rote?

Should maths be learned by rote?

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

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

calvin_education

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

jo boaler quotation

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

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

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

0.8 + 0.4 x 52 ÷ 0.01

A pupil has to think about all of the following:

  • The order of operations (that they should complete the multiplication and division first AND that, within that, that they should work from left to right)
  • What the notation []2 means
  • The value of 52
  • A strategy to multiply an integer by 0.4
  • A strategy to divide by 0.01
  • How to add 0.8 to the answer

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

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

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

…they will see this instead:

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

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

 

What does a knowledge grid have to do with it?

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

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

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

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

indices knowledge grid

This sets out what we expect pupils to know by heart if they are going to be able to tackle more complex or interesting problems involving indices (e.g. What is the final digit of 10100+999+598?). Knowing by heart that ab x ac = ab+c doesn’t replace knowing why this relationship is true. But, knowing it by heart – and practising explaining why it is true – frees pupils up to tackle problems like ‘evaluate 23 x 52 x 22 x 53′.

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

masses of maths 3

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

masses of maths 2.PNG

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

masses of maths 4

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

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

How is it used?

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

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

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

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

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

masses of maths 5.PNG

masses of maths 6.PNG

Pitfalls We Fell Into

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

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

Pitfalls We’re Still Trying to Avoid

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

masses of maths 7.PNG

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

 

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

 

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

 

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

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

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

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Long-Term Solutions (Or: Why Make a Textbook)

This is my sixth year of teaching and I think it’s the first time I have taught equations properly to a KS3 class. I was almost there last year, and thought I was doing it well, but I now know there are several topics where I completely let the pupils down. This post is about how I could have been better-prepared earlier in my career, and avoided leaving later teachers with a mess to clean up.

———

Naveen Rizvi’s piece yesterday in the TES caused a stir that surprised me. Many people had a negative reaction beyond what I would have expected (I won’t link to them) and was followed by some negativity – or at least concern and alarmed questions – when Bodil subsequently shared an example of two pages from the booklets we give to pupils.

As I see it, these are some of the main barriers preventing pupils from achieving their potential in maths that CAN’T be dealt with by better resourcing:

  1. Limited working memory (i.e. there is a limit to how many new concepts the pupil can form and connect in a single lesson
  2. Fear of maths; strong and paralysing anxiety around maths
  1. Poor mathematical foundations from primary age
  2. Poor literacy (insofar as it limits their access to everything in education, and their ability to practise independently)
  1. Unsupportive home environment that leaves the pupil unprepared for school in a practical or emotional sense
  2. Low attendance
  3. Fixed mindset around maths, often meeting its first major challenge at secondary
  4. Passive behaviour. This could charitably be called low motivation, or disengagement. It could less charitably be called laziness.
  5. Disruptive behaviour and avoidance techniques
  6. Their peers’ disruptive behaviour
  7. A class culture that doesn’t value effort and hard work
  8. A class culture that penalises mistakes and revealing or discussing errors
  9. A class culture that makes it uncool to want to see the links between ideas in maths
  1. A weak teacher who isn’t trying to improve (either wilfully, or due to disenchantment borne of circumstances)
  2. A weak teacher who is trying to improve but isn’t there fast enough (typically an NQT, a teacher transferred from another dept (usually PE or geography), or a teacher who has been neglected in terms of development)
Possible solutions:

Improved teacher pedagogy and understanding of how memories and connections are formed.

Improved teacher understanding of what fixed and growth mind-set actually is (not just a gimmick to console pupils when they underperform… my heart bleeds for Dweck).

Possible solutions:

Effective intervention and catch-up programmes in school (ideally supported at home).

Possible solutions:

School leadership foments a culture that challenges this (supported by classroom culture created by individual teachers), either through super-high expectations/tough love or alternative approach that challenges and changes issues that hold pupils back in school.

Possible solutions:

Head of Department leads maths-focused CPD

Caveats:

This is not easy. ITT doesn’t seem to cover this adequately, and it appears to be a relatively new part of most teachers’ pedagogy*, relatively complex to understand and highly complex to begin to incorporate into practice (particularly for the weakest pupils).

* This is, of course, excluding some very experienced and successful practitioners. In their case, it appears to be something they’ve come to understand intuitively and isn’t easily shared as it isn’t codified.

Caveats:

There are many programmes that appear to have high impact in closing the gap between pupils’ reading and chronological ages, or the gaps in their mathematical foundations. In particular, direct instruction programmes such as Connecting Maths Concepts (McGraw-Hill scripted direct instruction programme) and Lexia appear to be effective ‘off-the-shelf’ interventions (based on my own experience!).

Caveats:

Really brave leadership on school culture, especially in challenging circumstances, is too rare (in my limited experience). Many bloggers have written about the gap between their school’s behaviour policy and the ‘real behaviour policy’ (teachers are left to defend their own classrooms, with little or no back up). In the best cases I’ve seen, there is total clarity about the positive, learning-focused culture the headmaster/mistress seeks to embed, and the behaviour policy serves this and is always upheld.

Caveats:

This is incredibly time-consuming. Most HoDs simply don’t have the capacity to do this well. The number of conflicting interests they have makes this difficult: teaching as many of the critical/tricky classes as possible (as they are, hopefully, one of the strongest teachers), writing SOWs, managing staff shortage (it is maths, after all), retaining staff and keeping them happy, improving teaching quality. And, ideally, reading widely to prepare for new exam specs and maths education research…!

However, there are more issues than this that are – I think – relatively neglected outside of the rarefied atmosphere of online edu-chat and conferences.

Barriers created in lessons:

  1. A capable but exhausted teacher who can’t prepare adequately for lessons (their department is under-resourced and teach a full and varied timetable)
  2. Confusion about what they should be covering to prepare for the end of Y11 (it is unclear what the pupils covered in Y7-9, or in how much detail; there is uncertainty about what should *actually* be taught when they see ‘averages, 1 week’ on the SOW… Does it mean calculating the mean, median, mode and range only, or complex questions where some values are missing and then one value is changed?).
  3. Painfully optimistic allocations of timing to teach topics (expressions – 1 week; fractions – 2 weeks), due to insufficient clarity about what should actually be taught.
  4. A gap between what they cover in lessons (superficial) and the rigour of the exam (increasingly higher, hopefully). A recent example of this was the GCSE question: Solve for a: 2a + a + a = 18. This question is beyond trivial, but many teachers had not prepared their class for the possibility that simplifying and solving could be used in the same problem.
  5. Unclear explanations, or rule-based explanations, that makes it difficult for pupils to use their knowledge flexibly or to ask useful questions (e.g. “change side, change sign” to solve linear equations because it seems quicker and easier, or convoluted steps to solve simultaneous equations).
  6. Inadequately scaffolded and varied practice in lessons that doesn’t prepare them for the variety of forms maths can take in the real world (or in exams…) (We all suffer from textbooks that escalate the difficulty of questions too quickly, so that your weakest pupils get only 2-3 questions practising questions in the form a+3=10 before they’re moved onto the other three operations).
  7. The practice gap (i.e. getting much less practice than pupils in other schools). Most textbooks DON’T HAVE ENOUGH QUESTIONS. At all. Most of the newest books boast how many more questions they have. It is not enough. If a pupil has only just begun to grasp a procedure, they need to do it many times to build their confidence and then begin very careful and gradual variations.
  8. Pupils forgetting that they have learned something (“I swear down they never taught us that”). This comes from haphazard, or no, continuous revision or interleaving (weaving old topics into current topics).
  9. Pupils doing what seems obvious to solve a problem, rather than what is mathematically correct (e.g. writing that 3/4 + 1/2 = 4/6). As above, an absence of revision and interleaving.
  10. Pupils knowing they’ve learned something, but muddle it (e.g. calculating the mean when asked to comment on the median). Also as above…

I am increasingly convinced that a good textbook would begin to address these ten problems. A good textbook:

  1. Offers interesting talks and prompts for pupils to have high-quality discussions in pairs and with the class. These can range from puzzles to problems that provoke cognitive dissonance (e.g. which is closer to 1/2, 1/3 or 1?)
  2. Offers worthwhile questions that allow pupils to use multiple strategies to solve a problem or to calculate (e.g. 4.5 x 24)
  3. Plans for revisiting old topics, particularly those that are high impact (directed numbers, fractions, equations, manipulation, mental maths, calculation) or easily confused (e.g. minimally different topics such as perimeter and area)
  4. Has carefully and thoughtfully sequenced content in the big picture (e.g. equations preceding graphs) and in the fine detail (e.g. breaking down directed numbers into the many strands of understanding and procedure that pupils need to grasp).
  5. Has identified key examples that a teacher might want to use with a class, covering the most important problem-types for a concept or procedure.
  6. Offers clear and highly accurate explanations of WHY something works.
  7. Has distilled clear steps to scaffold pupils’ work as they begin to tackle a new procedure.
  8. Offers memory devices to help pupils retain and recall concepts or steps (Chants for the 7 times tables, or mnemonics such a KFC for dividing fractions (Keep the first, Flip the other, Change to times, it’s no bother).
  9. Offers LOTS of practise at each level of difficulty in a procedure.
  10. Has lots of interleaving available, but sectioned off, so that the teacher can judge the level of complexity students should experience.

None of this replaces planning lessons. You still want to share enthusiasm, build excitement, anticipate common errors and misconceptions, explain clearly, model explicitly and unambiguously, check for understanding, grow their confidence in the face of setbacks, celebrate success, maintain pace and focus in a safe and happy environment and – of course – go back and refine the plan and resource after you’ve taught it. This all takes planning, deep thought about your classes and huge love of maths. I don’t understand how the existence of such a resource would compromise the idea that teachers tailor their teaching to their classes.

Sadly, such a resource doesn’t appear to exist. That’s why we’re making a textbook. Please get in touch, have a look, and help up improve it!

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Generating Examples for Generalised Rules: #collabomaths

I was at the National Maths Conference on Friday in Sheffield and could easily spend a blogpost on summarising the many things I learned. Happily, many people have already done so (and took more photos). Instead, I’d like to focus on something I’m going to try doing differently as a result of the conference and invite you to join a very small, very geeky, Twitter party.

geek party 2

I was struck by what was shared in the Shanghai session, when those who visited showed examples of how teachers create progression in their examples for procedures. In particular, the design they employ appears to really build up concepts of underlying structures, by showing how varied they are. James Pearce gives an excellent summary here.

Specifically, Shanghai teachers seem to prepare their examples and explanations to help students see a broader range of applications for a rule. Here is an example for multiplying indices, in terms of the examples we might show to our students in the UK:

indices UK

The focus in Shanghai is on a broader range of applications, in order to make it easier for students to generalise the rule. Here is a rough example:

indices Shanghai

This would not have been instinctive to me, thinking about the cognitive load on my students and the risk that struggles with directed numbers or non-integers would cloud what was happening. However, few of the examples are inherently harder and it creates more opportunities for interleaving (in addition to illuminating the broader rule).

Here is another example, for difference of two squares:

difference of two squares

I particularly liked the final one, and how that would be so much better a preparation for the new GCSE spec! I’m wondering if, in my efforts to make sure that work is scaffolded and students’ working memories aren’t overwhelmed, I’ve presented too narrow a range of applications at the outset and thus made it harder for them to see how to apply it outside of that narrow structure.

With this in mind, Richard White and I thought we would use the approach we learned in Luke’s session to generate ideas, whereby there is a ‘splurge’ of initial ideas and we later sort them to decide the best range of examples to show to students.

Admittedly it was an odd way to spend the later part of a Saturday night, but we found it wonderfully, geekily enjoyable to focus on a narrow piece of the curriculum and think about how we could create more demanding examples that better exemplify a rule. Here is what we created in about half an hour:

surds 1surds 2surds 3surds 4surds 5surds 6surds triangle

It’s far from exhaustive, but is a much better basis for planning work on (simple cases of) multiplying surds and bringing rigour to a SOW (and supporting new or struggling teachers, as well as non-specialists). It gives a clearer goal in terms of “What should they be capable of by the end?” and “What examples will I share to get these ideas across?” Richard has since used the approach in NQT mentor meetings to help those teachers to think about planning in a more focused way (i.e. to move away from resources towards exposition). As a professional exercise, it was really enjoyable as it made for a happy marriage of focus and creativity.

We are planning our next topic and, due to living in different cities, are going to see if it’s possible to try generating examples via Twitter. We’re going to have our first attempt this Wednesday (30th September) from 4.30-5.30 using #collabomaths as the hashtag (better suggested will be accepted!). I am also trying to corral my maths teaching hero (the man who taught me almost all I know, in my first school, but who thinks using MS Word is the height of tech sophistication). We will probably go with expanding single brackets, but it’s TBC. If you would like to join the teeny party, you would be very welcome 🙂

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