Must all maths teachers do this?

I had the privilege of being in Tom Kendall’s classroom recently, and saw something wonderful.

A child in Year 7 said something that showed he’d understood something essential to arithmetic development, or advanced conceptualisation of arithmetic, or an important threshold concept… I’ve never been sure how to categorise it, but everyone who’s mathematically competent recognises the ability to do this: take an expression like:

9 + 6 – 7 + 1 – 2

And see it simultaneously as ‘nine add six subtract seven add one subtract two’ and ‘positive nine add positive six add negative seven add positive one add negative two.’

David Chart introduced me to the idea of procepts – ideas in mathematics that can simultaneously be thought of procedures or concepts, leading to children who manifest as ‘better mathematicians’ i.e. often observed as being able to perform more calculation in the classroom, more quickly, more easily and more accurately, as actually undertaking an easier form of calculation in their heads due to this ‘proceputal understanding.’

I think I can articulate what this is now better than I’ve ever been able to before:

Case 1

This is how the vast majority of people, and all children at first, will see the expression:

(+9) + (+6) (+7) + (+1) (+2)

Case 2

But people who are mathematically trained also see it as:

(+9) + (+6) + (-7) + (+1) + (-2)

The result is the same in both cases, and the ability to switch between the two at will means you can make choices that simplify arithmetic calculations you have to perform in your head.

There’s a beautiful symmetry here; the symbols + and  unfortunately represent two concepts each: the operations of addition and subtraction and the position of a number relative to zero (many other ways of conceptualising it, but suffice it to say, ‘sign,’) so each case either:

(1) Holds constant the sign, and allows the operation to vary

or

(2) Holds constant the operation, and allows the sign to vary

Either you (1) see the expression as the addition and subtraction of positive numbers only, or you see it as the addition only of positive and negative numbers.

2-x-2

 

Method

I’ve always articulated this idea to pupils, but never had any sense of whether any of them were seeing arithmetic expressions this way, and never really known how to properly communicate the idea.

Perhaps it’s because it’s an idea that sits within an uncomfortable space which cannot be explicitly or directly taught/communicated, but what should we do about this?  I’m wondering whether Tom has the answer.

As Michael Fordham aptly points out here, people differ in their thoughts.  Curiosity is another good example of something that cannot be directly taught; while some believe that we can directly teach curiosity by having ‘lessons in curiosity’ and measuring pupils progress in ‘being curious,’ most believe that we must facilitate curiosity, but differ in how.  Some feel that the solution is to ask a big interesting question, and then leave children alone with books and computers to go answer it, sating their natural curiosity.  Others believe we need to emerge them in fantasy, a la the Mantle of the Expert approach.  Others believe that these approaches limit children to their current narrow experience, and so the solution is to do lots of good teaching of the stuff we can teach explicitly, believing we first need to know something about something before we can express curiosity – and the more we know about it the more curious we are likely to become.

This same problem strikes at the heart of mathematical problem solving, approaches to which I’ve felt were lacking since forever.  The typical approach is to say ‘we need to teach problem solving,’ then turn to Polya or his anaemic derivatives and create step by step approaches for children to follow.  For the longest time I’ve had little faith in this approach, partly because I see it everywhere yet see it making no difference, but mostly because the words of Daisy Christodoulou still ring in my head “We underestimate our own knowledge, and overestimate theirs;” consequently whenever I’ve approached a problem solving task I’ve viewed it not just in terms of ‘what steps am I undertaking,’ but ‘what mathematical knowledge am I bringing to bear,’ and ‘what previous problem solutions am I adapting here,’ as suggested by Willingham.  in other words, if we aren’t teaching the mathematical content well, and we’re not showing pupils solutions to a wide variety of problem types with overlapping features, then we aren’t going to develop their problem solving ability.

Further to this, I suspect that the ability to problem solve is rooted in the idea of ‘noticing’ expressed here.

As White (1967: 69) puts it:
We can ask someone how he [sic] `would’ discover or cure, but not how he
`would’ notice, although it is as legitimate to ask how he `did’ notice as it is to ask how he `did’ discover or cure. For the former `how’ question asks for the method, but the latter for the opportunity. Although appropriate schooling and practice can put us in a condition to notice what we used to miss, people cannot be taught nor can they learn how to notice, as they can be taught or can learn how to detect. Noticing, unlike solving, is not the exercise of a skill.

When we ask people how they solved a problem, they will at some point tell us ‘the things they tried,’ and what they eventually noticed following all this trying, but will not mention how they noticed it.

In this sense, problem solving might be an emergent property of other teachings, in the same way that Fordham suspects ‘curiosity’ is an emergent property of our existing knowledge, not a skill to be honed or developed in isolation.

Back to arithmetic

Tom’s lesson was about how to evaluate multi-term expressions, and I couldn’t help but wonder if his approach to this had increased the probability that any given child would notice these two ways of seeing the expression, as described above (whether or not the teacher also attempts to point to it verbally.)

So, we have:

9 + 6 – 7 + 1 – 2

And two ways of evaluating it.

Method 1 – Go from left to right.

9 + 6 – 7 + 1 – 2

15    – 7  + 1 – 2

8      + 1 – 2

                    9    – 2

7

This method focuses the mind on the early concept of ‘adding and subtracting positives.’ (I also like that there’s no = sign here either)

Method 2 – Sum the positives and negatives independently, then resolve

(Tom used column addition for all stages of this, but that’s a little harder to set out on a computer)

(+ves)   9 + 6 + 1 = 16

(-ves)         7 + 2  = 9

+ves subtract -ves:

169 = 7

 

That second method hints at Case 2 described above; the fact that you can treat all those subtractions as adding together negative numbers, and that you can treat addition of negatives as subtraction.

I don’t know yet if this is helping those children notice this important concept, or if the child who spoke up was just ‘one of the bright ones’ who learns more in spite of us than because of us.  I also haven’t yet had time to think up a way, or search to see if someone else has, of determining whether someone can see the expressions each way.  But just logically analysing the methodology… there’s certainly something here.  As I come to the end I wonder if maybe it needs a further activity to be introduced, just literally asking pupils for ‘the other way’ of writing:

(+9) + (+6) – (+7) + (+1) – (+2)

Or ‘the other way’ of writing:

(+9) + (+6) + (-7) + (+1) + (-2)

I also wonder, on reflection, if that would be an explicit/direct method of teaching the idea…

Either way, we’re not yet getting enough pupil to this realisation/conceptualisation.  This is something we must do.

 

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Money Value of Time – 3 – Mental Models for Education

In a former life I spent some time working as an Associate for a boutique consultancy that specialised in social enterprise and the third sector.

The MD asked me to codify the company’s burgeoning internal knowledge by producing a document that listed the different social enterprise business models that its consultants had encountered, along with their relative pros and cons for different social causes.

I sat with my line manager and mentor and explained that I would probably need an hour with all of the consultants to interview them and get that knowledge out of their heads and onto paper.  His response left me stunned…

“Okay.  You could, you could do that, we could make that happen.  But before we do anything you need to appreciate that each of these guys is billed out at £800 a day, so if you want to speak with fifteen of them for an hour each, what you’re really asking for is £1,500.  So, do you still want to go ahead with that, or shall we find an alternative solution?”

I had never heard anyone frame the value of time like that before, and it’s a lesson that stayed with me.

time-equals-money

Time is money

Flash forward to my years as a teacher – a world in which our social goals often leave us shy when talking money – and here are a couple of examples where I wondered whether the same robust analysis had been applied.

In one school we were moving from decant to a newly refurbished building.  The entire department spent two days decorating the corridors, making it a bright and welcoming place for pupils.  We then asked for plastic covers for the displays, which existed elsewhere in the building.  The request was denied, since it would cost a couple of hundred pounds.

Four months later, the predictable happened.  The general wear and tear of hundreds of pupils barrelling down the corridor had left things looking a little tatty, and we were expecting an inspection from the LA.  The maths department were ordered to spend an INSET day dressing up the corridor once again.  I ran the cost.

15 adults, on an average salary of around £100 per day (approx. £25k), for a whole day, that’s £1,500.

That’s an initial investment of £3,000, plus a further £1,500 spent repairing damage.

We could have spent £200 and not needed to repair, instead we spent £1,500, and I wondered if anyone even realised.

ice

Plastic display covers.  Cost: £200.  Cost of not having them: £1,500

At another school we were never expected to write out lesson plans, day to day.  Three times a year, though, there would be a monitoring visit from the academy chain.  The teachers to be observed by the monitors were mostly notified in advance, and as with any observation were expected to produce a written plan for that lesson.  But there was a general sense of ‘the monitors are ultimately free to go where they like, and so we must all be prepared,’ and with that general sense came a mandate for all of us to therefore have plans written up for all of our lessons on that day.

I was ready and prepared for the day in question, everything good to go by 6pm, except for the written plans.  I decided to time how long it took me to put the plans together for the next day’s four lessons.  Turns out it took 2 hours just to write up my thoughts into formal plans for four lessons.  I might imagine I’m just a bit slow at writing, but I wasn’t the only one still there at 8pm.  Again, I ran the calculation.

£30 for two hours’ work, for around 40 staff.  £1,200.

Most of that was waste, since as mentioned, the vast majority being observed were informed up front, so almost none of those 160 lesson plans were seen by anyone.

Further, these numbers are calculated before taking Opportunity Cost into consideration.  What else could those teachers have been doing with their time?  Would they have taught better the next day had they rested for a further two hours the night before?

And I wondered, was that cost, and that waste, ever considered?

waste-of-money

Do we waste money and not even realise it?

It’s not that either of these decisions were bad in and of themselves.  The first school was in a financial crisis, to the point of laying off half its workforce the following year, and so I’m sure spending £200 on almost anything seemed untenable.

I do wonder, though, if those stealth costs, and the stealth waste that follows them, had been taken into consideration when making the decisions; in most cases I would suspect not.  It’s been my experience that many (certainly not all) senior leaders tend to consider their staff as a ‘free’ resource, rather than thinking of them as a purchase of expert time which they are slowly spending across the year.  How differently might we consider the things we ask of teachers if we saw their value to the school in those terms: costly expert time.

Everyone I’ve shared my initial story with has met me with the same wide-eyed look of surprise I had when my mentor first pointed reality out to me.  And so I continue to wonder: what better decisions might we make in education if we all kept this financial reality in mind?  What wastage might we be more prepared to slough away if we see ourselves as haemorrhaging money.

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Opportunity Cost – 2 – Mental Models for Education

Time

I feel like joining London Business School to study for an MBA.  Let’s see what that’ll cost me.  £70,000… ouch!  But okay.  Actually hang on… I know I’m going to have to relocate to London to do this, so I’d better look into rental prices there.  It’s probably going to cost me £600 per month at least, that’s another £7,200, oh and I should probably add on bills and food etc. while I’m at it… so let’s call that £15,000 all in.

£85,000.

That’s the real cost for me to study an MBA.

Almost anyone seriously considering this would go through a very similar process.  That said, you don’t usually go for an MBA straight out of your undergrad, so actually most people seriously considering this will realise something else: I have to give up earning for a year.

Let’s say someone on a £40k salary wants to give this a go.  If they don’t go for the MBA, then they’ll earn £40,000 that year.  If they do, they give up the opportunity to earn that £40,000.

This is the Opportunity Cost of how they choose to invest their time.

Actual real cost of that MBA?

£125,000

 

Money

You have £20M to invest.  You find a way of investing this money that you’re confident enough will return £50M in five years, a huge profit.  Should you go for it?

A good answer is obviously ‘Yes.’  A better answer is ‘Are there any better options?’

See, if you choose to invest your £20M in this thing you found, you are giving up the opportunity to invest it in something else.  Your potential investment will return a £30M profit after five years.  If you could find another that would return £100M profit after five years the Opportunity Cost of not taking that option is £70M.

 

The first example was about investing time, the second about investing money.  Both are relevant to education, though time might be most important.

 

Education

I read this a while back.  It links to two other posts with the same idea.  In short, all three are defending ‘Pokémon Go’ lessons, and all three miss the point.  All three suggest there might be something magical or mystical that the teacher in question knows that we don’t that will somehow ‘light the fire’ of learning in their pupils.  In other words, all three suggest that motivation is all that matters, and all three can’t imagine that there might be a way to motivate pupils without appeal to popular culture.

Doug Lemov captured this failure to understand education beautifully for me recently, with the a line drawing a distinction between ‘Learning to read, versus reading to learn.‘  That’s important.  When we read in school we are not only doing it to learn how to read, but to learn from what we read.

If pupils are somehow using Pokémon Go to explore class conflict, as Debra Kidd suggests might be appropriate, then there is a question of opportunity cost.  Pupils *might* learn something about class conflict, but they will learn nothing more.  Is there alternative substance that would have allowed them to learn something about the real world or its history at the same time as learning about class conflict?  The same is true for any book.  That time, once used, is gone.  It behoves us to ensure that we spend it wisely on behalf of children.

The same is true of any content we choose to study.  The evisceration of knowledge in the minds of English teachers and educationists has led many to believe that one can study anything at all, a pamphlet if you like, and be learning as much as one could studying Austen.

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Effort:Impact Ratio – 1 – Mental Models for Education

6 years ago Joe Kirby decided to leave a lucrative career working for a social venture capital fund, take a massive pay cut, and join Teach First.

As we talked about how we were going spend the year nervously preparing for what was to come, Joe talked about the use of ratios in finance to build a rough picture of what is happening with an investment, good enough to make reasonable decisions from.

He proposed three possible ratios that might serve a similar function for us in teaching.  One, I can no longer recall.  Another, was equally the brain child of Doug Lemov, and found its way into Teach Like a Champion literally called ‘Ratio.’  This was the teacher:pupil work ratio, which is reasonably common in education.

For me, though, the most enduring and powerful of the three ratios Joe proposed was effort:impact.

The idea is simple: teachers should invest time in things that are low effort, but high impact.  The lower the effort, and higher the impact, the better.  The greater the effort, and lower the impact, the worse.

Although these things are difficult to quantify in the education world, you can develop a feel for ‘acceptable effort:impact ratio.’  If the effort just feels too high for the associated impact, don’t do it!

Would I have spent time fine-tuning the position of images in a PowerPoint so that they were all precisely placed, had I considered the actual impact of all that effort?  I felt that pupils in school deserved access to the kind of quality that we’re all used to receiving from commercial companies, like Google, Apple and Microsoft.  But just because I felt that something was important, did that make my decision a good one?

This is an important question, because while our feelings can, and most certainly should, inform our decisions, we often allow ourselves to be slaves to them.

Are comments in marking really worth it?  How about double or triple marking?!  Some people will say that it’s worth it because it has an impact; yes, but for how much effort?  Are there ways you can achieve the same impact for less effort?  If so, do that instead.

Effort-Impact.JPG

Illustrative only

Joe has since helped to establish a Free School, and, as for me, effort:impact has been the one ratio that truly endured above all others, now taking a central place in the school’s strategy.

The result is extraordinary.  The number of traditional school activities that Joe and his team have shown the confidence to reconsider would shatter the nervous system of most senior leaders.michaela-tweet

 

Initiativeitis – Scourge of School Leadership

Everyone wants to be seen to be making their mark on a school, whether driven by vanity alone, or a real sense of purpose.

A few years ago a friend, recently appointed to head of department, expressed her frustration that her team showed no enthusiasm for the changes she wanted to make.  From her perspective it was all about the kids, and would make a difference to their lives.  The first thing I asked her was if any of her new initiatives would result in her team having to spend more time working.  “…yes,” was the response I got.  My next question was whether any of their other work would be removed or reduced by the initiatives.  “…no.”  There’s the problem.  What at first might look like an assemblance of lazy no-gooders of Goveian nightmare were perhaps instead just a team of mortal humans already worked to their limit.

The unconscious desire to leave our mark can leave us pray to Action Bias, and what we don’t realise is that at the back of our minds it’s justifiable because the action is also good for us.  As the leader initiating the initiative, if it works, we take the credit, so there’s a bit of extra incentive there for us which doesn’t carry through to the whole team.  Whether we realise this consciously or not, it’s always there, it’s unavoidable, and if we don’t realise it consciously, we still know it subconsciously.

One lens, one filter, one mental model to rule them all, Effort:Impact ratio.

That one simple idea has meant not only that the Michaela leadership say no to any new initiative that won’t return enough impact for the effort of its team of teachers, but it led them to jettison what was, until recently, a cornucopia of received wisdom and sacred cows.

falling-cow

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Overview – 0 – Mental Models for Education

Decision making is tough.  We are riddled with cognitive biases, from action bias to the halo effect and sunken cost fallacy.  They seem contrived only to make us terrible decision makers.

Teachers, HoDs, Heads, all need to make decisions.

Shall I tick ‘n flick those books?  Or write comments?  Or do neither and go to bed?

Should the department invest most of Year 7 studying number, or dedicate equal time to geometry, algebra and stats?  Shall we study Skellig, or Oliver Twist?

Should I implement that new initiative, or not?  Should I spend money on one experienced hire, three trainee teachers, or four teaching assistants?

Mental models: you could call them strategies, or lenses, or filters, or heuristics – whatever you call them, they are simple tools designed to aid decision making.  As I moved through the last five years in education there were times I desperately wished some mental models I’d encountered in my former life were widely known and understood in education.  Once known, the absurdity of some of the decisions we make is revealed.  If not the absurdity, then the very real costs, or the risks, that otherwise remain hidden, and go unnoticed.

So I’m going to chuck a few out there, and see what people make of them.  Some you will have heard of, others will probably be new.

To start with, I’m going to try to cover the following, over the next few weeks:

  1. Effort:Impact Ratio
  2. Opportunity Cost
  3. Money Value of Time
  4. Cost-Benefit Analysis
  5. The 80:20 Principle
  6. Objective Oriented Mindset
  7. MECE (pronounced Mee-See)
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Why SLANT?

At a past Michaela debate I heard Peter Hyman describe a desire for education ‘…with a smile, not a SLANT.’  The implication was not only, bizarrely, that the two are mutually exclusive (Bright Face is another technique taken from Lemov’s Teach Like a Champion) but that ‘SLANT’ is somehow not desirable.

bright-face

The acronym stands for Sit up straight, Listen, Ask and answer questions, Nod your head, and Track the speaker.

It’s largely a ‘child’s guide to active listening.’

…super hard to wrap my head around what’s egregious about this.

 

In my experience, some of these, like ‘nod your head,’ tend to fall away quite quickly, and SLANT becomes more of a shorthand for how you expect pupils to sit: straight backed, eyes on the teacher, and with hands interlocked, or arms folded.

This last part’s the interesting bit.

If you struggle to picture it, you can see Colleen Driggs use a hand gesture to remind a child to return to a ‘hands interlocked’ position in this video (the others are already there.)  You have to be pretty quick to see it, mind.  The whole thing takes only a second or two, and Colleen doesn’t break her flow for it at all.

I’ve been in a few trainee teacher classrooms recently where I saw something strange.  When they counted down to silence, they got it pretty quickly.  When they were speaking, the pupils were largely silent.  Maybe they did, but I couldn’t remember my classes going that well in the first couple of months.  Despite this, I noticed something else… I couldn’t for the life of me focus on what the teacher was saying!  Despite the relative quiet, I was constantly distracted by a cacophony of ruler waving pen tapping hand stomping fidgeting that was endemic in the classroom.

What’s a teacher to do?  At most, I expect you could ask that the trainee focus on a firm expectation of ’empty hands.’  That has two problems.  The first is that it doesn’t really deal with the hand stomping or finger tapping.  The second is that… well you know what it’s like, we will all sometimes mindlessly pick things up and play with them when we’re trying to focus for a period of time.  Not just *low level* disruption; this is, for the most part, likely, completely *unintentional* disruption on the part of pupils, yet disruption it is.  I couldn’t focus and I was trying really hard!

How could children focus in this environment?  Truth is most couldn’t.  Yes there was quiet, but there wasn’t a high degree of attention being given in most the classrooms I saw, and I couldn’t blame them – distractions abound.  As lessons drew on, the relatively high level of respectful quiet that the trainees had initially commanded began to wane, I suspect, in direct relation to how much pupils felt they weren’t really learning.

And so I was met with a newfound love and appreciation of SLANT;  at least, that part of it that asks not only that pupils ‘sit up straight,’ but that gives a clear expectation of what to do with their hands during teacher instruction so that they don’t accidentally disturb those around them, and so all have the opportunity to think deeply about the content.

In schools that make use SLANT or similar, I’ve seen this expectation around ‘folded hands’ fade with older pupils.  Despite this, I also saw those pupils exhibit more of the typical ‘mature adult’ mistake – far fewer of the class were wont to fiddle and fidget in general, and those who did needed only a quick, gentle reminder, and they immediately emptied their hands, a little embarrassed, pretty much any of us do when we realise we’ve accidentally started clicking a pen or tapping something that might distract those around us.

So, long live SLANT!

(or some version of it.)

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When is a mathematical process not a process? This changes everything.

When it’s a concept*.

The best example I have so far to explain this distinction, and its importance, is expanding single brackets compared with double brackets.

Transformations

The type of concept at work here, in Engelmann’s taxonomy, is a transformation.

There is an input, the transformation concept is applied to it, resulting in some output.  For this to be conceptual rather than procedural it should be possible to directly infer the transformation that is taking place by exposure to carefully chosen examples of input followed by output.

transformation

The transformation is the hidden concept we want pupils to infer from observation the changes from inputs to outputs

Single

Concept: Distribution

I used to think of expanding a bracket as a process.  There are many ways of communicating it, but one might be:

algebra-2

However the lines are drawn, however the working is set out, and whether you go for this or a grid method, or a variation of that which links it to rectangle areas, it will still largely be communicated as a step by step process.

Here’s why I don’t think it is a process.  Consider the following as an alternative method of instruction, the one I would probably opt for now:

algebra-3

Ideally these would be written on a whiteboard, with pupils able to witness which numbers you rub out and change, and which stay constant

By chance, I actually did something very close to this with my Year 7 bottom set class in my second year of teaching.  The sequence, language and examples were less carefully chosen, and I didn’t use the identity symbol, but the principle was the same: rather than treating this as a process, I used a few examples from which they were able infer what was happening.  This was a group of poorly behaved children who barely knew their times tables, but still they could see that I was multiplying the first number by the other two, 100% success.

Depending on the group, either the initial instruction sequence (the examples) or the later sequence of teacher-led pupil activity can be expanded to include all manner of important variations on that concept, including:

  • Have three or more terms in the bracket
  • Swap the order of the unknown and the constant
  • Include negative terms
  • Include more than one unknown
  • Interleave with index laws
  • Include a term outside the bracket which isn’t multiplied by its contents
  • etc.

So much to cover, so little teacher exposition needed.  The concept is ‘multiplication is distributive over addition,’ and it can be inferred through carefully chosen examples (this declarative specification of the concept can be provided at some later date.)

As a caveat, I think verbally pointing out that ‘Ten times three is thirty, and ten times five is fifty,’ *is* an option, or ‘Ten times three ex,’ if you prefer, but I would suggest waiting until the second expression has been written, and pupils have had a few seconds to read it; explain verbally what happened *after* the fact, rather than while you’re writing it up on the board.  We tend to like doing this because it feels like it helps us manage behaviour.  Having tried both, I felt that silence on my part actually helped draw attention to my writing, while speaking at the same time possibly added to the cognitive load of the explanation; after all, it required reading and listening at the same time, which we all know aren’t possible! (split-attention effect)  So this part of the advice isn’t about silencing the teacher – as so much ‘Ofsted Outstanding’ training is wont to do – so much as it is about judicious use of language – Economy of Language, as Doug Lemov might call it.

Compare this now with expanding a pair of brackets.

Double

Process

First, my above suggestion will not work for a pair of brackets.  The reason is obvious:

algebra-4

It’s barely possible to infer directly how the input – the expression on the left – resulted in the output, the expression on the right.  At a stretch, a small minority might spot the relationship between the terms in the bracket and the first and final term in the right hand expression, but few, if any, will realise where the middle term came from.  If they did, how long would it take them to form that realisation?  It wouldn’t be instant at all.  Indeed, forgetting about the middle term is the classic mistake we see.

So my earlier method won’t work here.  Expanding a pair of brackets is not a concept.

I’m now going to set this out a little differently to how I most often did this, and how I’ve seen most teachers teach this, using neither FOIL, any of the many crossed line set ups, nor grid multiplication.  Instead I’m going to set out clearly the three lines of processing that are actually taking place.

algebra-5

Each of these three steps represents the application of a transformation.  The distributive property is applied twice, albeit in very different contexts, and simplification by collection of like terms is applied once.

This changes everything.

Here’s why this changes everything.

Traditionally I would have simply taught the process for expanding a pair of brackets, step by step.  I would have asked that pupils replicate my steps.  Now I would do something very different.

Each of those three lines I would treat as a separate concept, to be communicated separately.  Each one might take a lesson, more than one lesson, or perhaps two of them would appear in the same lesson.  They might be separated by only one day, or perhaps a hundred days; however this is structured, they are treated as separate ideas and fully explored independently of one another.

One lesson might consist of applying the distributive property to the pair of brackets only once (getting to Step 1, though it wouldn’t be called that yet.)  That first line becomes ‘the answer,’ no more.

Notice that, since each of these lines *does* express a transformation concept, they *can* each be communicated in the same way I outlined for single brackets.  It takes a little more time to see how you get from a pair of brackets to step 1, but not that much more, and importantly it *can* be inferred from carefully chosen examples.  It could be done with three and four term expressions in the brackets, rather than binomials, or with three or four pairs of brackets rather than just two.

The same is true in going from step 1 to step 2, and 2 to 3.

Eventually, once all three concepts are fully explored and embedded they would be ‘chained’ together in one lesson, or if needed, more than one.  Going from a bracket pair through to step 2, or from a given Step 1 to Step 3, or all the way from start to end.

By this point, pupils should be fully able to answer the traditional, slightly boring and pointless ‘expand this pair of brackets’ question on an exam paper, but they will also have a much deeper understanding of the concepts masked by that step by step process, and hopefully be better able to deal with anything an exam paper could throw at them.

 

What do you think?  What are the limits of this?  What won’t it work for?

 

 

* We could get into a discussion about whether or not a better categorisation of this is actually ‘procept,’ discussed here by Gray and Tall.

For the sake of simplicity, I’m not going to do that for now.

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