Category Archives: pedagogy

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

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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.”

And

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

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

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

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

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

 

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

Could we include….

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

 

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

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

 


 

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

 

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

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


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

what-matters-most

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Filed under curriculum design, lesson design, pedagogy