What Are You Thinking?

Compare the following:

 “First you add 4 and 20 and you get 24. Then you halve it and you get 12. Then you times by 8 and you get 96. Then put centimetres squared.”

 

“To get the area of a trapezium I’ll use . a and b are the parallel lengths, so I will use 4 and 20 as I can tell they are parallel because of the little arrows. I will use 8 for h as it is perpendicular to 4 and 20. The lengths are in centimetres, so the area will be in centimetre squares.”

 

If you are a maths teacher, you might have noticed they are both ‘correct’ answers to the same question: how do you get the area of this trapezium?

trapezium pic

One of the ways we have been trying to increase the level of rigour in maths is to shift the focus of our questions and the focus of pupils’ answers. In particular, we aim to increase the ‘transfer’ from specific examples in lesson to being able to see how an idea, concept or example applies to other cases, or how a strategy can be deployed in novel situations.

There are four main strategies:

  1. Explanations should mostly use general steps and reasoning, rather than specific values
  2. Clamp down on pronouns
  3. Teach the pupils what they need to know to be able to answer the question “How did you know that…?” and “How did you know to…?”
  4. Help them to build up expectations

 

  1. General Terms, not Specific Values

A lot of the explanations we give to pupils focus too much on the specific example and its values (e.g. 3cm, or y = 6) and not on the general underlying structure. Consider the difference between:

“To get the perimeter add 6 and 7 and 6 and 7 and don’t forget the units.”

And

“To get the perimeter I need to add the lengths around the outside. Some of them are blank: I will fill them in. I know that opposite lengths in a rectangle are equal; that is why these lengths are also 6 and 7. I will now add all the lengths. The lengths are measured in millimetres, so I will give the answer in millimetres.”

The second is longer, but focuses more on the general strategy of perimeter of any shape (find all the external lengths, calculate their sum, and use the information in the question to determine the units). Not only is the first explanation restricted to being an explanation and strategy for perimeter of rectangles, it is restricted to the specific rectangle in the question.

Here are some guiding principles:

  What? Why? Instead of this… Try this….
1 Restate the goal of the steps This means that the subsequent explanation is linked to a specific outcome, rather than feeling like an arbitrary collection of ‘steps.’ “To find the equation of the line, we will first…”

 

 

“So, first you need to…”
2 Make your explanation as generalised as possible, focusing on the names of parts This approach increases ‘transfer’, and allows them to learn a strategy or process as a general strategy, rather than as a single response to a single question. “subtract 53 from 360, so x is 307” “Use the fact that angles at a point sum to 360 to find x”
3 Used technically accurate terminology This will, in the medium-term, clarify their thinking and reduce the chance of them confusing ideas and concepts. It also allows you to find woolliness in their thinking or where they have ‘folksy’ understanding. “missing number” or “the letter”

 

“move around to get y on its own”

“the unknown [value]”

 

 

“rearrange to isolate y”

4 Be specific about what you expect to see Sometimes we are tempted to give simplified explanations and steps. This may seem less daunting for pupils to hear, but lowers their chances of eventual success. [to find gradient from a graph] “draw a triangle” “draw a right-angled triangle”
5 Make the implicit explicit, and only accept ‘complete’ explanations Sometimes we hope the most subtle ideas will become obvious, or gloss over aspects of a concept …and then are baffled when pupils lack flexibility. If there is underlying or implicit knowledge…share it with them! “because corresponding angles are equal” “because corresponding angles in parallel lines are equal”
6 Have the same standards for their answers and you have for your explanations If they can’t say back to you the accurate and ‘general’ form of a strategy or approach, they have not understood it or cannot remember it. This is crucial feedback for you! It also allows them to rehearse the general form, which is valuable for their retention, and puts pressure on them to attend to the most ‘high level’ version of what you say, not just the question they are looking at. “I need to do 50 divide by pi and then square root it” “To find the radius of this circle, I will use the formula  and substitute the values I know. I know that A is 50, and pi is a value, so there is only one unknown. I can isolate r to find the radius.”

 

Tips for implementation

  1. You need to change before the pupils can change! Explain to them what is changing in your modelling, and why, and what they should focus on.
  2. Ask smaller questions to check they can repeat back the steps. This is a mimicry stage, of course, but it is relatively easy and low-stakes, and allows them to rehearse using more mathematical ways of speaking. It also shifts their expectations of what they will hear from each other and what ‘sounds about right.’
  3. Use accessible ways to help them understand what to cut or keep in their answers and explanations, such as “Explain how to do it, without using any of the numbers in the question” or “Tell us how you’ll tackle it, making sure you use the words gradient and y-intercept and coefficient” or “I will be so impressed by anyone who can explain how to do ANY question where you have to find perimeter, and not just talk about this question.”
  4. Let them rehearse with their partners.
  5. If something they say is slightly off, tell them how to improve and get them to say it again (make sure you do this in a kind and supportive way!).
  6. Judge the room: sometimes it helps to use the easier and less rigorous approach if their confidence or buy-in is low and you need some ‘quick wins.’ If you do so, that is fine! What matters is that, in those situations, you have understood that the success you are aiming for is with their buy-in and confidence, and not necessarily with their learning.

 

Clamp down on pronouns

Watch out for when you – and more often the pupils – use pronouns to disguise uncertainty and wishy-washy thinking. This is usually ‘it’, ‘they’ and ‘them.’

For example:

Change “You multiply them” to “Multiply the base and the perpendicular height”

Change “You find its height” to “Find the slant height of the parallelogram”

Change “You add them and divide by how many there are” to “Add the values and divide by the number of values”

It has been fascinating how often a child has given a coherent-sounding explanation or answer but then faltered when I probed what they meant by ‘it’ or ‘they.’ In many of these cases, I think even the pupil didn’t realise the woolliness of their own thinking, possibly reflecting that I had used ‘it’ and ‘they’ so much that they (the pupil J ) didn’t notice it wasn’t making sense to them.

Teach the answers to the question “How did you know that…?”

A lot of the questions in maths classrooms have a 50/50 chance of being right:

  • “Are these lines parallel?”
  • “What is the value of x?” (if they need to choose between ‘subtract from 180’ or ‘subtract from 360’)
  • “Should I times or divide?”

…which means we are probably getting a lot of false positives, especially considering that our body language often conspires to give away the ‘correct’ answer.

“Are these angles corresponding [shakes head slightly] or alternate [happy inflection]?”

 

A more powerful question – and pupils must be built up to expect this – is to follow with:

  • “How did you know they were parallel?”
  • “How did you decide to use 6cm as the height?”
  • “Why did I subtract from 180, instead of from 360?”

If a pupil can’t answer that follow-up, then there is limited hope that they will transfer what they are learning in that question to a similar question, and even less hope they will transfer it to a superficially novel problem.

 

There is a big caveat:

If the ‘why’ and the ‘how do you know’ and ‘how to decide’ is not a core part of your teaching and explanation, it can’t possibly be part of the pupils’ responses!

A first step is to plan the answers to these questions when preparing your lesson:

  • “How did I know to add them all up first?”
  • “How did I know to rearrange?”
  • “Why did I set it equal to 540?”
  • “How did I know these angles would be equal? How could I tell they were corresponding? How did I remember that the F-shape gives us corresponding angles?”

 

Asking this has been a big shift in our practice, and it has made a huge change to the children’s confidence once they expected to be asked such questions…and to be able to answer them. Now that they expect us to ask those questions, they know to attend carefully to that part of the explanation. Even more gratifying is that they are asking questions such as “How did you know to multiply those two?” or “How did you know it would be a quadratic?” and are helpfully filling the gaps in my teaching!

 

Help them to build up expectations

This may be the most obvious, and I have seen many great teachers do this: make sure the pupils have some expectations around the answer before they get started! Similarly, teach them to use words in the question to begin visualising or imagining things.

For example, my Y10s have been practising the cosine rule and sine rule. One of the first things we do is build an expectation, such as

“x is opposite the biggest angle, so it should be longer than the other two lengths”

or

“one of the angles is 100, so my answer has to be less than 80”

or

“the length increased, so my angle should increase too”

or even

“I’m looking for an angle, so I will probably have to use inverse sine or inverse cosine”

 

Another example: today my Y11s were learning about how solve simultaneous quadratic equations. As we looked at the pair x2 + y2 = 25 and y = 2x+1 we first made little sketches of the two graphs to set up an expectation of where they might cross, and how many points of intersection we would expect. They then set off on the question expecting a solution pair where both values were negative, and one where both values were positive. We also discussed that we expected the solutions to be two pairs in the form (x, y).

 

A final example: when my classes are completing transformations, they have learned to put a little arrow pointing to the ‘rough’ area where they expect the shape to end up. This helps them when they become bogged down in completing a translation or rotation, and helps them judge how well they do at expecting a particular result.

 

A caveat: learning ‘what to expect’ and discussing this is usually an overwhelming and slightly awkward first step. I usually help the children feel familiar with the process and then move to expectations as, by then, they have a few examples in their mind to test their expectations.

 

 

Find this interesting? Find it challenging in a good way? Come visit, and consider joining our team! Visit mcsbrent.co.uk and check out vacancies for maths and economics

 

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Mean What You Say, Say What You Mean

There has been a lot of discussion in the past few months of what ‘warm-strict’ means, if anything. I am a big fan of it as an approach, and think it is a useful lens when reflecting on your teaching and relationships.

A conversation this week with our department hit something home to me that I knew, but had never made explicit: a correction or consequence isn’t given because a pupil has been ‘bad’ but because they didn’t live up to their ability to be ‘good’ (using ‘good’ and ‘bad’ in a loose sense!). When giving a consequence, or correcting behaviour, our words and emotions should show that it is straying from their good potential that is a disappointment. It doesn’t make sense to punish a baby for crying, or a small child for being messy when learning to eat, or anyone for not knowing things they haven’t been taught; they can’t live up to that yet. When correcting a child, they should hear “This consequence is not because you are bad, it is because we both know you can be so good.”

To that end, one of the things we are working on is making sure ‘corrections’ (i.e. admonishments) marry being stern with being constructive, positive and warm.

Stern

It is important to be some version of ‘stern’ when correcting a pupil, as they need to know you mean what you said and that you aren’t going to let it slide if they don’t try to improve. There are a few ways to show ‘sternness’, and they all involve being more dramatic than is perhaps natural, or we would ever do with an adult or emotionally mature child (THESE ARE RARE!)

  • Statements of expected norms: Direct and definite phrases such as “We wait our turn to speak” “Focus on the diagram” “Keep your hands in your section of the desk”
  • Tone: sharper, or deeper, or more urgent to create a contrast to your normal tone (which is usually calm, friendly, warm or excited).
  • Facial expression: an intense look, holding eye contact for slightly longer, a ‘firm’ expression that is slightly cross or shocked and different to your default relaxed expression (or happy but intense expression…!).
  • Body language: a sharp or definite hand movement (e.g. holding a hand up with the word “stop”, or chopping it down when saying “No! We NEVER interrupt other pupils in this class”), or moving closer to the class or leaning towards the pupil’s part of the room as you speak to increase the intensity of the interaction.

This might sound OTT to you. My experience suggests that most pupils require this level of drama to support corrections. It is their role, as children and teenagers, to test boundaries as they explore the world and social relationships. It is our role to make those boundaries clear and to defend them for the benefit of the child’s learning and safety (and that of the class). Doing this powerfully matters, so that they know it had a (minor) emotional impact on you when they did the wrong thing, and that you noticed and cared that it happened. This might mean they can perceive clearly that you are shocked if they kick someone’s chair while you are explaining an idea, or can tell you are unambiguously disappointed by a lack of good effort. Most of the time, this means hamming up your reaction. If teachers were actually shocked or profoundly disappointed by children turning around or kicking chairs or missing homework, they’d lose their minds!

If you are not sure, consider the toxic alternative: nagging. Nagging happens when pupils have learned to tune out what we say, and think all we do is moan and repeat things in a monotonous way. Managing to say something once, and with impact, is much better for your relationship with the child and the class (“I mean what I say”).

Constructive

The next element in a successful correction is making it constructive. With exceptions, this means describing the positive behaviour you want to see, not describing the unwanted behaviour. It also means finishing with a positive ‘pop’ in your voice…hence the frequent use of exclamation marks.

“You’re speaking too loudly to your partner”  becomes “Whispering voices with partners!”

“Don’t slouch” becomes “Strong spine!”

“That’s a demerit because you looked out the window” becomes “That’s a demerit, stay focused on your work to make sure you’re successful”

“Don’t talk to your partner” becomes “Listen to make sure you learn”

“This isn’t a good effort, I am so disappointed” becomes “I’m disappointed because we’ve seen you do such good work, I know you can do that”

“You aren’t trying” becomes “Let’s see your best effort”

This means the pupil isn’t getting attention for doing the wrong thing, but instead getting attention because you believe they can do the right thing. It also paints a picture of the successful version of the child and tells them how to get there. It also serves to remind the rest of the class what to do to meet your expectations.

There might be exceptions: if a child has visibly violated the norms of the classroom, or acted in a very immature way, you might want to show a strong ‘push back’ before pivoting to a description of what you expected to see instead.

“We NEVER snigger if someone makes a mistake. We show respect because they gave it a go and we know it is brilliant they are about to learn from it.”

“Do NOT grab sheets from others’ hands. Mature people wait sensibly for it to be passed along, because they are professional and patient.”

Either way, this has to be judged by the age and maturity of the group. I might use those phrases with more immature Y11s who need firm boundaries, but would be wary with a very ‘grown up’ Year 9 who will benefit more from a quiet and ‘grown up’ exchange. In general, shorter is better, and ideally down to a couple of words to avoid sounding nagging or pleading (even if the phrases are slightly odd!).

“Strong voice”

“Looking smart”

“Full focus”

“Top speed”

“Empty hands”

“Track the board”

 

Positive and warm

Some of the most energetic lessons I see, with the most good feeling, are ones where the teachers use small consequence (in our cases, this is usually a demerit, which knocks 1 point off their ‘merit balance’) to remind the pupils what they can live up to. They usually say them in a more urgent and energised voice, with intensity and speed to quickly move the lesson on without dwelling on the negative and to quickly give the pupil a chance for ‘redemption.’ They look positive and warm by the time they finish the sentence or phrase, perhaps having started with a cross or shocked look, to convey the message of “I am disappointed with what I see, I will be happy when I see your best, and I know you can do it.”

“That’s a demerit Daniel, I know you can focus on completing your work quickly. I’ll come to you in a minute to read out some top-quality answers.” (If a child is doing very little work, or being deliberately slow)

“Jana, that’s a demerit, you have to show all your working. I’ll take a peek at it when you have finished it with all the working to see how good your answers will be.” (If a child was warned they must show their method on their whiteboard)

“Ana, that’s a demerit. Help the class stay focused.” (If a child was trying to catch others’ eyes or distract them)

“That’s a demerit Jorge, let’s hear your strongest voice this time.” (If a child is repeatedly muttering or giving inaudible answers after you gave encouraging reminders to project)

 

In general, the newer an expectation, or the further a child is away from meeting it, the more narration is needed to ensure they know what you expect. As children know your expectations and simply stray from them by dint of being human (we all get distracted and need resetting!), the shorter the corrections can be, allowing you to say short and positive phrases in an assertive way.

 

Pre-empting

The best teachers I’ve seen notice the ‘creep’ of an unwanted behaviour (e.g. kids mumbling when answering questions, or being sluggish in a transition) and quickly narrate the positive version of what they expect to see, avoiding the need for corrections.

“I’m going to pick someone to read, I can’t wait to be impressed by their booming voice”

“When I say ‘tell your partner’ you will go WHOOSH! as you’ll turn so fast and get straight into telling them your idea”

 “Before we start, let’s plan how much we want to get done. Pop a little mark on the page where you think you can get to in 5 minutes. I wonder who is going to lead the way with how much good quality work they can do…” (Obviously this would only work for a lethargic class, not a class that would dash off haphazard rubbish in a bid to fill the page!)

“I love that Saman looks so strong when he works. He is totally focused on the job at hand, no wonder he is blazing through these.”

 

Lastly, these things are just as relevant in more private conversations with pupils, although you won’t need the same urgency and intensity as there is more time and no audience. When I am speaking with a pupil after school about their behaviour, the focus is on how they could be and on ‘what to do next time,’ rather than dwelling on what happened. Focusing on what pupils can control – their actions and choices in the future – makes the conversation about agency and improvement, rather than about blame or defensiveness. And it is always nicer to finish by saying you “can’t wait” to see this better version of them, and you believe in it.

 

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Tried and Tested: Solving with SMEG

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

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

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

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

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

Exponents: 10+3 x 25

Multiplication and Division: 10+75

Subtraction and Addition: 85

 

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

Subtract 10: 3 x x²=75

Divide by 3: x²=25

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

 

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

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

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

function machines cropped

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

3x + 5 = 11

as

x -> x3 -> +5 -> 11

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

There are some problems with this.

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

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

Then

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

Because ‘5 happened first.’

 

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

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

Tried and Tested: SMEG

Our approach has been:

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

e.g.

31 = 3x – 5

Would be explained as:

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

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

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

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

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

The unknown is isolated; we’re done.

 

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

 

This works even as the question becomes more complex:

(3-2a)/5 = -6

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

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

3 – 2a = -30

The group is gone, we can start SMEG again.

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

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

a = -33/-2 = 33/2

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

3a +2 +a = 5

and

3a + 2 = a + 5

 

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

8Z equations 2015

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

 

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

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

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

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Tried and Tested: GEMS

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

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

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

GEMS stands for:

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

 

Why we use it:

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

gems example

(images from Ms Lopez’s blog )

 

Disadvantages:

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

 

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

 

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

4 – 3 x 5

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

15 – 4

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

 

 

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Maths Muddle: alternate segment, same segment, alternate angles

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

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

angles 1.png

angles 2.png      angles 3.png

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

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

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

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

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

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

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

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

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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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