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.


Filed under Interesting or Fun

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:


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

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


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



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Filed under Interesting or Fun

Tell Me More, Tell Me More

There is a problem in how we teach:

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

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

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

We suffer from three problems, as maths teachers:

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

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

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


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

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

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

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

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

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

“What if none of this was obvious?”

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


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

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


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


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

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

“What do you mean by a mastery curriculum?” 

I think it might be simplest to summarise as

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

Are you intrigued, enraged or perplexed by what we do? Come visit! 

Are you delighted or enthralled? Apply! We’re looking for a teacher for September 2017 (or sooner, for the right candidate).  The ad is on TES now, closing SOON: https://www.tes.com/jobs/vacancy/maths-teacher-brent-440947 

Get in touch on dquinn [at] mcsbrent.co.uk or [at]danicquinn on Twitter to chat or arrange a time to call in. 


Filed under curriculum design, Explaining conventions, lesson design, pedagogy

What Matters (Mathematically) the Most


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

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


  1. Identify high-leverage topics

These are the topics that:

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

The ones we’ve identified are:

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


  1. Teach them as early as possible in KS3

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

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

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

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


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

Teach every variation that you can think of

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

2a + 2 = 12 (simplify first)

12 = a – 10 (unknown on the RHS)

7 = 3a (result is a fraction)

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

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

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

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


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


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

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


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

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

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


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

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


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

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


  1. Interleave them into every subsequent topic, wherever possible

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

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


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

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

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

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

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


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

Could we include….

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


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

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



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


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

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

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



Filed under curriculum design, lesson design, pedagogy

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).


“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/


Filed under Interesting or Fun

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


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.


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!).


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.


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!


Filed under Interesting or Fun

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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