You Turn Me Right Round

This is just about some ideas for angles we’ve been using in the department.

Polygons: angles as turn

We’ve been trying to demonstrate to the pupils what it actually means to say the sum of the angles in a triangle is 180 degrees. On a straight line it is easy to demonstrate the half-turn, and remain faithful to the idea that angles are a measure of turn.

I find that the activity of tearing the corners from a triangle and arranging them on a line doesn’t seem to stick. I suspect it also reinforces the idea of a line more than the idea of a half-turn.

Using a board pen or, in my case, a small toy bird, we’ve tried this instead. Because it is small-scale, I’ve used an anthropomorphised key…

It starts facing forward. At each vertex, it turns. Upon returning to its original position, it is facing the opposite direction. It has completed a half turn 🙂

It then extends nicely to quadrilaterals (it is facing the same way), then pentagons (a turn and a half – facing backwards) and so on. It allows the pupils to see that not only does the number of triangles increase (the standard and much-loved way of showing progression in polygons), but also that each time they increase by a half-turn.

Vertically opposite angles

We were finding it tricky to help pupils spot vertically opposite angle when there were more than two intersecting lines. One pupil* suggested that they position their rulers to rotate around the point of intersection, turning to hide the one being focused on. The one that is revealed has the same turn, so must be equal. They are the vertically opposite angles. Here are two examples of her suggestion:

It is obviously easier if a finger is placed at the point of intersection, as it is easier for the ruler to rotate. The limits of one-handed camera phone filming!

Lastly, another pupil had a good suggestion to help with spotting vertically opposite angles. If each separate line segment is highlighted (and there are no non-straight lines!), then the ones ‘trapped’ between the same colour-pair will be vertically opposite to each other:

vertically-opposite-highlighting

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

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

—-

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

 

 

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Tell Me More, Tell Me More

There is a problem in how we teach:

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

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

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

We suffer from three problems, as maths teachers:

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

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

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

how-maths-feels

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

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

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

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

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

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

“What if none of this was obvious?”

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

coefficients

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

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

equal-angles

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

angles-on-lines-and-at-points

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

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

“What do you mean by a mastery curriculum?” 

I think it might be simplest to summarise as

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


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What Matters (Mathematically) the Most

 

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

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

 

  1. Identify high-leverage topics

These are the topics that:

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

The ones we’ve identified are:

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

 

  1. Teach them as early as possible in KS3

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

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

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

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

 

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

Teach every variation that you can think of

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

2a + 2 = 12 (simplify first)

12 = a – 10 (unknown on the RHS)

7 = 3a (result is a fraction)

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

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

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

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

 

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

 

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

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

 

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

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

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

 

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

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

 

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

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

 

  1. Interleave them into every subsequent topic, wherever possible

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

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

And

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

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

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

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

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

 

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

Could we include….

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

 

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

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

 


 

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

 

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

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


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

what-matters-most

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Masses of Maths: what should pupils learn by rote?

Should maths be learned by rote?

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

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

calvin_education

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

jo boaler quotation

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

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

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

0.8 + 0.4 x 52 ÷ 0.01

A pupil has to think about all of the following:

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

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

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

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

…they will see this instead:

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

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

 

What does a knowledge grid have to do with it?

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

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

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

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

indices knowledge grid

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

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

masses of maths 3

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

masses of maths 2.PNG

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

masses of maths 4

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

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

How is it used?

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

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

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

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

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

masses of maths 5.PNG

masses of maths 6.PNG

Pitfalls We Fell Into

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

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

Pitfalls We’re Still Trying to Avoid

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

masses of maths 7.PNG

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

 

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

 

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

 

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

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

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

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

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

———

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

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

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

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

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

Possible solutions:

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

Possible solutions:

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

Possible solutions:

Head of Department leads maths-focused CPD

Caveats:

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

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

Caveats:

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

Caveats:

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

Caveats:

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

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

Barriers created in lessons:

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

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

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

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

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

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Generating Examples for Generalised Rules: #collabomaths

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

geek party 2

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

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

indices UK

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

indices Shanghai

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

Here is another example, for difference of two squares:

difference of two squares

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

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

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

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

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

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

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In Praise of Being Boring

“Which would you hate more: to overhear someone say you were stupid or say you were boring?”

“Oh definitely boring! That’s got to be the worst thing to hear.”

“You don’t mind someone thinking you’re stupid?”

“If someone thought I was stupid I’d assume they lack the intelligence to realise how clever I am.”

I won’t embarrass the speaker by naming them, but suffice to say that everyone present agreed that being called boring is the most crushing of criticisms.

Our most recent maths survey (we do one each term) yielded mostly positive results: over 90% saying they’d take maths if it were optional; the modal response to ‘Do you like maths as a subject?’ was ‘I love it’; only 1 of 259 respondents disagreed that ‘there is usually a positive atmosphere in lessons’; only 1 disagreed with the statement ‘my teacher tries her best to be fair’; the modal description of work in lessons was ‘challenging, but I can do it.’

The results don’t vary with significance when filtering for different teachers or sets, which reflects the hard work of my colleagues in the department as we try to offer the students a consistent experience of maths (we all believe in allowing group 4 to access the same curriculum and expectations as group 1, even if it can feel like a profound challenge at times).

Obviously, I should be delighted with this. I am. I feel so blessed to have colleagues that let their students feel so positive about maths.

But…

In the free-text section for ‘What 3 words do you associate with maths and maths lessons?’, 12 students said ‘boring’. My immediate reaction was to feel wracked with guilt and feel we’d let them down. Maths is amazing and beautiful, we work hard to convey our enthusiasm for it and help all students access it. We must have failed these 12 students! Even though I know it’s a huge overreaction, it niggles at part of my brain.

In a recent review of the school (as part of the Bradford Partnership), the observers commented that students were very serious-minded in maths lessons and expressed some dismay that we never rarely have occasion for the students to work in groups. Retorts spring to mind: “Of course they were serious when there was a stranger in the room!” “You only saw 10 minutes!” “There’s a lot of evidence in favour of a serious atmosphere in maths learning!” and so on, ad infinitum. I’m trying very hard not to reject this feedback just because it doesn’t fit with my current conclusions about how best to teach maths.

Are we missing an obvious solution?

I was intrigued to read a blogpost by Matthew Smith, arguing – amongst broader points – against what he perceives to be needlessly boring lessons:

“When planning a lesson on adding fractions, a bland, one dimensional lesson might involve a few examples by the teacher and a worksheet with a range of questions. A highly effective lesson that encourages all learners to make progress and see the beauty of maths might start with the following question; “Can all unit fractions be expressed as the sum of two other unit fractions?” This leads to an investigative lesson, no one knows the answer straight away and it is accessible to all students. In short, mixed ability teaching can and does work.” [his emphases]

I have selected Matthew’s post not because of anything specific about him or his school – I haven’t neither met him nor spoken with him; his posts suggest he is a committed teacher who works hard for his students – but because I think that the passage is very representative of a dominant school of thought around maths teaching.

I think it’s completely incorrect. The question – can all unit fractions be expressed as the sum of other unit fractions? – is definitely very interesting and even as I type I feel a little bubble of excitement thinking about working on it later. It’s easy to convey that excitement to students and to work together to answer that question. There is definitely a place for these questions in maths lessons – and not just for higher groups or the keenest students – for all students’ lessons.

But…

How on earth can a student ‘investigate’ that question without knowing how to add and subtract fractions? It is not an easily-learned process. It’s fiddly to actually do – selecting an appropriate denominator, forming equivalent fractions, resisting an understandable instinct to add the denominators together – and it takes a significant understanding of proportion to make the leap from grasping the initial ‘why’ (e.g. when exploring diagrams) to seeing its link to the how (having a common denominator). Even a student who can confidently explain why we need common denominators to add will occasionally relapse into ‘add everything.’ It’s not that they don’t get it, it’s that they don’t remember it at all times. It’s not obvious. It takes a huge chunk of working memory, even for A level students. Insofar as I can claim to understand psychology, it strikes me as exemplary of how System 1 can trump System 2 unless we are in a state of serious-minded vigilance as we work.

All my experience so far suggests that, particularly for students lacking mathematical confidence or mathematical knowledge (the latter label encompasses almost all students), it is profoundly unhelpful to offer too rich a context as a vehicle for learning, especially for elements of maths that require consistent, fluent and (necessarily) fiddly processes. This doesn’t mean you don’t share the opportunities to then use and enjoy that knowledge, it just means that they don’t have to happen in tandem.

My (limited) experience so far suggests this:

  • Convey that the topic or skill is interesting (not necessarily useful. It usually isn’t and I’m cool with that). If it can be introduced with a ‘huh???’ moment, then that is great. It’s not always possible and it’s better to just dive in than to create a pseudo-hook.
  • Show them how to do it. Scaffold it carefully, being sensitive to your group. For the love of God please don’t ask them to discover it. It’s great to ask them to notice and think about patterns, and to explain what is happening, but it is a terrible disservice not to also explain it very clearly and very carefully. And to check that they can also explain it very clearly and very carefully.
  • Never stop telling them about how awesome you think this bit of maths is, how proud they’ll feel when they can do it fluently and connect it with other things.
  • Let them practice. LOTS.
  • Use formative assessment throughout to unpick and discuss misconceptions.
  • Let them keep practising. LOTS. On their own. In purposeful silence in a supportive, happy atmosphere.
  • Keep using real-time assessment to match the pitch and pace of the work to avoid students being stranded or coasting (constrained by what is possible without being torn in 30 directions).
  • As more of the process is chunked into long-term memory, and confidence in the idea ‘I can actually do this’ grows, build the links between the how and the why. For example, if learning to divide by a decimal, this is the time to cement the link between the idea of ‘3÷0.5 is 6 because I am seeing how many half-cakes are in 3 cakes’ and what is happening when 3.4÷0.2 is rewritten as a fraction, multiplied by 10/10, and simplified.
  • Keep returning to it throughout the year – and their time in school – so that they don’t lose what they learned. Interleave it with increasingly complex and rich contexts (not necessarily ‘real-life’. Learning to appreciate the beauty of the abstract is a great gift to see them through the tedium of adult life). Let them experience the delight that their hard work in learning how to divide by a decimal, or add a fraction, or form and solve an equation, is allowing them to work on an intriguing puzzle or fascinating context.

Maybe it’s fine that it’s boring sometimes.

When I practise A-Level topics, or practise French, I do sometimes find it boring. Not dislikeable, just sometimes tedious. But it’s an acceptable – even welcome – boredom; it’s like the way that exercise is sometimes boring. There are more fun forms of exercise, and there are more fun forms of maths, but I can’t really do either without making sure the basics are solid, and examined in isolation without the trappings and context of more exciting or intriguing problems. It’s a satisfying boredom that is part of the journey to a worthwhile goal.

In the same free-text question over half of students described maths as ‘fun’, and 49% made ‘interesting’ their word of choice. Their later comments suggested that the teachers and the atmosphere are the drivers behind this, as few made any mention of the activities in lessons. These are reasonably representative comments, to give an idea (currently all the teachers are female – hence the default to ‘she’) –

“she brings a huge buzz to the class and tries to make it as enjoyable as possible” (a Y9 who is pretty clear that he doesn’t actually like maths itself)

“They all talk about how much they like maths and when they explain something to you, you understand perfectly” (I should employ this kid to write my blogs, as he’s clearly able to say the same thing but in fewer words).

“My teacher(s) do like maths because they have a such pleasure and passion in teaching it and also they go over something more than once for those who do not understand which proves FAIRNESS.”

“they all are very passionate about it, for example refer it as cool constantly”

“For me maths is difficult however on the other hand I feel happier every time I do it because I then learn and understand something new which is very important to me.”

 “I love maths because I know that DTA does it for our benefits and the teachers make us feel comfortable with maths.”

In summary:

  1. Love maths. Show it.
  2. Teach the processes explicitly.
  3. Let them practise it lots, and in a context- and confusion- free way.
  4. Interleave and revisit. Build up the confusion and context. NOW is the time to decide if there is time and merit in more ambitious activities (I am partial to a Barbie bungee, but not as a vehicle for teaching).

If you think this sounds awful, I am interested to hear from you. If you think this sounds awesome, please apply to work at our school. Bradford is great and so is working at our school.

tt rockstars

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