MECE – 7 – Mental Models for Education

Pronounced ‘Mee-See,’ this is another fave from the world of consulting.

When mapping out a curriculum, assessment, taxonomy, decision or driver tree, consider whether its components as you’ve outlined them are Mutually Exclusive, as well as Collectively Exhaustive.

In other words, make sure your categories don’t overlap, and check they cover everything.

 

I use MindsetKnowledge and Craft as a taxonomy for teacher training.  Each of these breaks down further.

Mindset: Beliefs and behaviours (thanks to Matt Hood for that breakdown suggestion)

Knowledge: Theory, Subject, Pedagogy, Context

Context breaks down further still.

Context: Historical, Contemporary

At each stage, ideally, those boxes wouldn’t overlap, yet together they would incorporate everything a teacher needs to teach effectively.

Teacher Training.JPG

 

At each stage, ideally, there would be three boxes, and never more than five; remember, this is a tool to aid thinking and help make decisions.  The more complex it becomes, the less it will aid thought.

When thinking about observing lessons, you could try to observe planning, instruction, behaviour, assessment (thanks to Joe Kirby for this example.  To be clear, I’m not speaking about judging lessons through observation, here, merely thinking about things that one can possibly observe.)  Can you really observe planning, though?  It might be inferred, but it cannot be observed.  Alternatively it could be paired back to just Instruction and Behaviour.

What I like about the potential of this as a MECE example is that it could be considered Teacher Behaviour and Pupil Behaviour, since behaviour is ultimately the only thing that can be observed directly.

Teacher Behaviour:  Instruction, Assessment, Management

Pupil Behaviour: Compliance, Self-Direction

Teacher behaviour is therefore broken down into Instruction – things the teacher observably does to communicate ideas to the pupils – Assessment – things the teacher observably does to assess whether or not those ideas were communicated successfully – Management – things the teacher observably does to communicate directions to pupils, and maintain order.  Are there any things a teacher will do that don’t fit in these three boxes?  If so, the three are not Collectively Exhaustive.

Pupil behaviour is interesting.  Can you think of any pupil behaviours that are not compliance with teacher direction, or acted upon of their own direction?  Unless I’ve missed something, it’s a perfect dichotomy, and as an observer you are looking for the extent to which pupils comply with directions from the teacher, and the extent to which pupil actions outside of teacher-direction are good or poor choices.

Classroom Obs.JPG

When thinking about teaching, cognitive science currently considers all knowledge declarative or procedural.  No overlap between them, and if the models are correct, it shouldn’t be possible to think of an example of knowledge that is neither of the two.

I’m not saying that my examples above are perfect, and they might not even be frameworks that you would want to use, but I hope they are at least examples of how this kind of thinking can aid the structuring of complex systems, and in turn aid decision making.

I would recommend reading a few short resources here, here and here for further, and possibly better, examples of MECE from its traditional place in the world of business and management.

If anyone can think of better examples from the world of education, please share!

 

 

 

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Objective Oriented – 6 – Mental Models for Education

I’ve written about this before, so will only touch on it again briefly.

There are two modes of thought when doing something:

  1. Process oriented
  2. Objective oriented

The first asks ‘What must I do?’ then tries to do it.  The second asks ‘What must I achieve?’ and then tries to figure out how best to achieve it.

Most people are naturally process oriented, all the time, and it has two dangers.  The first is that it can make us feel we have ‘achieved’ simply be executing the process, whether results were realised or not.

In government, for example, having ‘distributed leaflets’ or ‘run x number of local information sessions’ might be given as measures of success to justify a programme’s spending, yet, the results (sometimes called outcomes) of those activities is never actually mentioned.  Did they change people’s behaviour in the way you hoped?

The second is that it can make it difficult for us to see ways of achieving better results.  We tend to focus on tweaking our existing process, rather than imagining an alternative way of reaching our objectives.

I give a clear example of this from the world of hand dryers in my original post, linked above.

In teaching, being process oriented would mean you see your job as turning up, doing some stuff that we’ll call teaching, and so long as you do that, you’ve discharged your responsibilities as a teacher.

Being objective oriented would mean you’re always asking yourself ‘But did the children learn what I intended them do?  How do I know?  If not, what am I going to do about that?’

In short, an objective oriented mindset leads naturally to the process of reflection.

What I find ironic in education rhetoric is that by asking teachers to be ‘reflective practitioners,’ we are focusing them on a process, rather than the objective…

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The 80:20 Principle – 5 – Mental Models for Education

In 1896 the Italian economist Vilfredo Pareto noticed that roughly 80% of the land in Italy was owned by 20% of its population.

The Pareto Principle, or more often, the 80:20 principle, suggests that 80% of your sales usually comes from only 20% of your products, services or clients.  It’s the reason a First Class ticket to New York costs £7,000 when an economy class ticket costs only £400.

 

20-80-rule-photo-700x400

 

This one’s brilliant for time management and decision making in general.  It can be combined with Effort:Impact to help us to realise how, actually, all that time we spend trying to make super nice PowerPoints probably falls into the 20% impact for 80% effort category.

The most important use of the 80:20 principle I’ve found in education is curriculum design.  Will Emeny produced this incredible network map of the GCSE maths curriculum, and in doing so he revealed the 20% that underpin the 80% of the curriculum.

 

whole-network-with-labels-low-res

  • Multiply and divide whole numbers
  • Add and subtract whole numbers
  • Multiply and divide decimal numbers
  • Add and subtract decimal numbers
  • Understand place value
  • Multiply and divide negative numbers
  • Add and subtract negative numbers
  • Order of operations
  • Round to decimal places
  • Round to significant figures
  • Powers of 10
  • Fraction of an amount
  • Connect between fractions, ratios, decimals and percentages
  • Plot and identify coordinates

 

If your pupils can’t do those things, they’re not doing much of anything else.  Therefore, focus your Year 7 curriculum here, and make damned sure they all succeed!

Michel Thomas is a master of this.  He pointed out that the 100 highest frequency words in the French language made up 50% of the everything people said in everyday conversation, so with only 100 words you’re half way there!  He made sure to put them front and centre of his language curriculum.

He did the same again by finding rules that opened up the most words possible for English speakers, and introduces them immediately, rules such as all English words ending in ‘-ible’ and ‘-able’ being the same in French, just with a different pronunciation.  Doing so gives you an immediate vocabulary of thousands of French words, so that only 90 minutes in he’s asking you how to say in French ‘What is your opinion of the economic and political situation in France at the moment?’

 

In physics, chemistry, biology, what are the 20% topics?

What about geography and history?

English?  (Stop saying it’s about nothing more than skills!)

What about music and art?

 

Figure it out, redesign your curriculum, leverage the power of the 20%.  I put that down as one of the most vital ingredients in the success of Bruno Reddy’s maths curriculum at KSA.

 

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Cost-Benefit Analysis – 4 – Mental Models for Education

When considering the benefits of your new initiative, think also of the costs.

In business this can be a simple calculation: what are the projected costs?  What is the expected revenue or saving?  Does the revenue / saving exceed the cost?  If so, profit.

It’s rarely ever so simple in education, but the principle is the same.

Towards the end of Elizabeth Green’s book Building a Better Teacher, she tells a story of a teacher who deals with a recalcitrant child by tolerating her slightly abusive tone, and engaging in her game until she ultimately guided the pupil to the point she had wanted all along: asking a question about a film clip they had seen, and with a more responsive tone in her voice.  Green describes the exercise as being like ‘a Tai Chi deflection.’  Her telling of the story is a positive one – this is a teacher to be admired.

Green explicates the benefits, however, without listing the costs.  How much of the other pupils’ time was being wasted while this child played her games?  If only a minute, that’s half an hour of learning time in total (assuming a class of 30).  That number scales up rapidly if they spent two minutes, or three, in this back and forth conflict.  What lessons was this child learning throughout?  That her poor choices and negative actions carry no negative consequences in turn?  That this is positive behaviour, in which she should often engage?

Green outlined the legitimate positive outcomes of the teacher’s decisions, but didn’t pause to list any of the potential costs.  If she had, we could pose the equally legitimate question ‘did the benefits exceed the costs?’

We’re now at a point where we can see that the costs can take on many forms:

  • Money
  • Time
  • Money, in the form of time
  • Opportunity

Cost-Benefit Analysis is like an advanced version of Effort:Impact ratio.  Where the ratio is a very simple heuristic for good decision making, the analysis can be a more formal exercise.

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Financial cost-benefit analysis

 

We can’t necessarily set forth all our costs and benefits quite so quantitatively, but the principle is the same.  All benefits carry associated costs, and we need to be on the look out for them, in whatever form they take, before making our final decisions.

 

 

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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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