Sunday, 18 October 2020

Don't pack sausages

I spent nearly all of my teaching career seeking out and working in schools in economically deprived communities – what used to be euphemistically referred to as ‘schools facing challenging circumstances’.  These are my favourite types of schools and favourite types of communities.

A couple of decades ago, I taught in a school in a small, working-class town where the main employer was the local food processing factory.  Rumour had it that the most soul-destroying job in the giant metal building was to continually pack sausages for eight hours every day.

Recently, I visited the school and spent a couple of happy hours afterwards in the pub with a very dear, old friend who is now the headteacher there.

We recalled how we used to ask our pupils in our classes about their lives and what their parents did for a living.  Many pupils would tell us that their mother or father packed sausages at the factory.  We would ask them, ‘what about when you are older, will you pack sausages too?’  And their reply was always, ‘hell, no!’

At parents’ evenings, we would chat with their young parents about how their son or daughter was progressing at school and their factory worker parents would beam with pride and joy to hear tales of hard work and drive.  And, without exception, those parents who knew the mundane days of packing thousands of sausages into boxes would urge their child, ‘you must do well at school.  Always try your hardest.  You must not end up like me.’

These parents understood what has been understood by good parents since time immemorial; their ‘job’ as a parent was to ensure that their child lived an adult life with better conditions than their own.  The direction of travel was upwards.  They found meaning in their life by creating the narrative for their child of progress.  Be better than me, they urged.  That’s what I want for you.

And so many of our pupils accepted the challenge.  There were many who became the first in their family to head off to university and break out of their own class and into the professional classes.  Many of these pupils faced the common unease of becoming alien in their own community, but their parents delighted in their leaving and continuing the journey of progress.  Many of our pupils stayed in the town, but rather than ending up in the sausage factory, they became receptionists or clerks or estate agents or shopkeepers or entrepreneurs or a whole host of other lives away from the meat packing plant.  To some, these jobs might appear to be modest leaps up the social mobility ladder, but to those pupils and to their parents, they represented something profound, life changing and the source of great pride.  The parent was able to feel accomplished – they had ensured their child would not face a future of sunless days acting like a machine.

My friend and I remembered together how driven those pupils and their parents were and how determined they were to weave into existence a better life.  Social mobility was at the heart of why we sought out and chose to work in such schools.  We had zero interest in patronising and zero interest in creating conditions where the pupils in our schools should have to be dependent.  We wanted them to push upwards, to live lives of meaning, purpose and with autonomy.  That’s what their parents wanted too.

As we spoke some more about this reason for teaching, my friend told me the saddest thing I have heard in years.

Today, when a teacher asks the pupil if their parent works in the sausage factory and whether they will too when they are older, rather than the defiant ambitious response of, “hell, no!”, teachers are now met with the response, “you have no right to criticise my dad” or “there’s nothing wrong with working at the sausage factory.”

Worse still, at parents’ evenings, the parent who once was so determined that their child should not live a life of servitude, now protests, “it’s done me no harm” and “if it’s good enough for me, it’s good enough for her.”

This is heart-breaking.

How did this happen?  And why did this start to become the prevailing attitude around five or six years ago?

My friend, a good headteacher still driven by a desire to make social mobility a real possibility in these days when social mobility in the UK has long since ceased, tells me that he started to notice a very strange shift in attitude some years back when pupils started to celebrate being thick.  This had always been a taboo for our pupils – to be considered thick by anyone, but especially their peers, was one of the most awful insults they could receive.  Yet, suddenly and seemingly from nowhere, pupils in the school started to actively seek the status of being completely clueless.

I asked my friend if he had any idea where this had come from.  He told me, at first, they couldn’t work it out, but then kids started to refer to each other as characters from the television show, The Only Way is Essex.  One character, in particular, Joey, was held up as a hero by the pupils.  Of course, it was not just this one show, but rather a full-blown attack was underway to keep the working classes in their place.

Very wealthy, very mobile, very autonomous people have created a narrative that competence, knowledge, merit, hard work and wit are irrelevant qualities for success, but instead what children should be aiming for is shiny teeth and a complete lack of nous. 

In the UK in the 1980s, the two most popular responses amongst working class teenagers when asked what job they would like to have as an adult were ‘teacher’ and ‘doctor’.  Today, it is ‘footballer’ and ‘celebrity’.

Today the teacher who tells the pupil that it is not ok to work in the sausage factory is chastised for not promoting the pupil’s self-esteem.  This argument is made by those who will never have to work in the sausage factory – they claim that saying such things is tantamount to insulting the pupil’s parents and their background.

This is a deliberate lie.  It is a verbal trick to stop teachers telling their pupils the truth.  It is a sanctimonious sleight of hand to keep the poor in their place.  It is a deliberate conflation; pretending that to set ambitions for a pupil to be able to choose to do any job they desire, is the same as to demean a particular job or those who do it. This is mendacious and seeks to silence those who wish to open a world opportunity to working class pupils.

There is nothing about working in the meat processing factory that helps to promote an individual’s self-esteem.  There is nothing about living an adult life drained of autonomy that boosts one’s happiness.  There is nothing about knowing one’s parents were able to live in better conditions than oneself that makes one feel super.  There is nothing in knowing that one has made no progress that makes life chipper.  These are lies told to the working class by people who want them to accept their lot.

Teachers should tell their pupils the truth.

And the truth is this: It is not fucking ok for the well off to tell working class kids that they need to accept their lot and work in the sausage factory.

To those pupils at my old school today, I say, recapture that defiant ambition of those who came before you. Rise up.  Be more.  Don’t let anyone keep you trapped.

Monday, 14 September 2020

What is an education for?

The state broadcaster in the UK, the BBC, has lately been campaigning to have additional public funding appropriated for the further expansion of its child targeted television channel under the guise of education output.  The BBC has long since created television shows for children, many of which have been given the education badge and many of which have endured for decades.  They are good television programmes. Because that is what television producers are in the business of producing – good television.  This is not the same thing as a good education.


All of the BBC’s ‘education’ output is, of course, educational.  In the same way that all television programmes are educational or all books or all conversations or all trips to the theatre or all walks along the beach.  Everything is, to some degree, educational.  But that is not the same thing as a good education.


Television producers have one aim in mind; enticing people to watch television.  By this measurement, the BBC’s education output for children is a great success.  There is nothing wrong with good television shows.  The only thing wrong here is the BBC’s pretence that their output is in some way complementary and supportive of school learning and, therefore, deserving of public funding.


Around the world, broadcasters have been producing television shows for decades under the banner of ‘education’.  They are incredibly successful.  And understandably so.  Children’s television gives comfort to parents who cannot or will not teach their child to read or write or become familiar with number.  This comfort is easily acquired when they see their little one fixed on the antics of a puppet singing a song about counting.  It is easy to assume that such television must be equally as good, if not better, at teaching a child how to count or read or spot pattern than the parent could.  So, the television acts to assuage the guilt that a parent feels about not being able to give their child all of their time.


The UK government’s reactions to Covid-19, in common with many other governments, included the forced closure of schools, meaning that over 8 million children suddenly found themselves without a teacher for an extended period of time.  Schools responded with their own provision, largely via web conferencing, emails and sending print materials home.  Some parents responded by taking on the role of teacher or purchasing the services of a private tutor to continue the curriculum their child had been following.


The BBC responded by claiming that they were able to step in and teach the nation.  It would need more funding, more content and more airtime, of course.


This has been the plea of television broadcasters since the 1960s; that they can be the school.


But they cannot.


Public funding has been justified because of the assumption that television learning bolsters and improves school learning.  We have long known this not to be the case.  Television producers are rightly only occupied by making good television.  Good television, as we have known for a very long time, is defined by the desire of individuals to watch.  As all broadcasters are very aware, there are core reasons for ratings.


Good television entertains.  It amuses.


The claim made by the BBC producers, and their justification for public funding, has been that their education output encourages children to love school.  This is true, it does.  But only if school is like television.  Learning is not just about the content – the way in which content is delivered also teaches children something.  It teaches them an orientation towards learning. Television viewing teaches children an orientation towards learning that says all learning must be entertaining, all learning must amuse.  The television orientation to learning is one where every single lesson – every single television show – must be completely free of prerequisite, there must be no barrier to entry and nothing needs to be known beforehand.  The television orientation to learning does not consider learning as an edifice, which requires solid foundations and careful construction, but rather as a mishmash of disjointed pockets of content.  The television orientation to learning tells the pupil that learning is optional, that there is no need to listen if they do not wish to, that learning is individual, that the ‘teacher’ is at their control – a flick of switch and they are gone.  It teaches the pupil that nothing need be retained, that the end point is simply to be amused; that fun is the primary aim.  The television orientation to learning requires no adherence to public decorum or good behaviour.


Television does not instil in pupils a love of the classroom, it instils in them a love of television.  It creates in society an orientation so counter-productive to classroom learning as to be actively hostile.  Becoming acculturated is difficult.  It requires an entirely different set of dispositions to those that television has promoted.


In very recent history, television has been joined by new broadcast media.  The internet and, in particular, social media, also create an orientation towards learning.  They go further than television.  The social media orientation to learning is one that teaches the pupil that feelings are more important than thoughts and ideas, that facts are less important (indeed of no importance whatsoever) when compared to popularity.  It teaches the pupil that ideas are never hard to communicate and if something cannot be said in pithy terms it is worthless because one should never, under any circumstances, be required to think.  The social media orientation to learning teaches the pupil that anyone holding views that require them to think can simply be muted and blocked.  The social media orientation to learning is one in which the primary aim is the absence of discomfort.


The internet abounds with orientations to learning that are hostile to classroom learning.  It does not instil a love of school or of learning.  Take, for example, the many millions of educational videos available on YouTube.  They are good videos precisely if many people view them.  That is what they are for.  This is not the same as what an education is for.  The orientation teaches the pupil to skip chunks of information they do not immediately find interesting.  It teaches them that learning comes in small chunks and must always, always be instantly understandable.


Television producers produce television shows that make good television viewers.  Social media ‘influencers’ produce content that make good social media followers. YouTubers produce images and sound that gets the most likes.


Our addiction to, and obedience to, these media lead to an entirely new phenomenon in schooling; classrooms in which teachers abandon what the classroom is good for and try instead to create an orientation to learning that serves television, the internet and social media.


There are rules to the orientation these media demand:


·      No prerequisites

·      No moments of confusion

·      No requirement to pay attention

·      No view of good public decorum

·      No discomfort

·      No need to retain

·      No moments lacking fun

·      No exposition

·      No fact trumps a feeling

·      No debate or reasoning

·      No platform for opinion

·      No connection between episodes


Should we be dedicating public funding to promoting this orientation to learning?  I think not.  Anyone who thinks, thinks not.


Technology has always played a role in education – teachers, contrary to popular belief, have long embraced technologies that they believe will improve classroom learning.  I am not suggesting that there is no place for television or the internet in education; but I am suggesting that its place is as a resource, like any other, which teachers themselves use critically and where appropriate in their teaching.  Television producers choose to make educational content not based on thinking carefully about what content is good for education but based on thinking carefully about what content is eminently televisable.  As I write, the BBC includes in its content a mathematics programme about espionage, a show about downhill mountain biking, an investigation by a ‘graffiti detective’ in the Netherlands, and a programme about dolphins in the North Sea.  There is nothing at all wrong with any of this content.  But this content exists because it is what television is good for and not what an education is good for.


We must keep our focus on what an education is good for.  This requires teachers to continue their age-old tradition of critically analysing technologies and deploying them as appropriate rather than allowing them to disrupt and, eventually, destroy classroom learning.  We are witnessing the same errors made with television being made again now with the internet and social media.  Just like television producers, the producers of learning platforms, educational apps, online schools and educational social media content are understandably in the business of producing good websites and social media, the measure of which is popularity rather than creating an orientation towards good classroom learning.  And, just like television, there are one or two notable exceptions to this – those who put good classroom learning and what an education is for at the heart of the television programmes, websites or social media they create.  Alas alas, since the measure of these is good education rather than popularity, they often struggle financially and require a philanthropical angle.  It is well known in the world of education technology, for instance, that there is absolutely no correlation between the educational quality of a product and its commercial success.  This is because the end users are looking for products that amuse rather than products that promote good classroom learning.  As someone who invests in companies creating education technologies, I make it my rule to give my money only to those who hold as their core belief an orientation to learning that promotes and strengthens traditional classroom learning, where knowledge is interconnected, coherent and building into something meaningful and acquiring such knowledge requires the pupil to give serious attention to exposition.  These technologies are rare and, because they do not consider fun as an end in itself, they tend to be largely ignored by the market – but, in my book, losing money on such ventures is far better than making money from technologies that place popularity at the heart of their strategy.




How did we get to a point where the rules I list above could form the most widely accepted orientation to learning on the planet today?


I suggest that the major causal factor is the deliberate destruction, and now complete lack, of meaningful grand narratives.


An education must be for something.  We send millions of pupils to school, spend vast sums of money and cause enormous disruption to society in order to make an education happen – so surely it must be for something.


That something is very hard to find today.  In recent years, a concerted effort has been put into ensuring the opposite – that schools should be for nothing.  Individual schools should not be on the side of anything at all.  That what we should seek is a consensus approach to our education system.  But consensus is just the same as believing nothing at all.  I do not want consensus schools, I want conviction schools.


Conviction requires a grand narrative.  An enduring and cohesive set of ideas and attitudes, which permeate all areas of the curriculum and give purpose and meaning to what it is to be an educated person.  Without a grand narrative, the curriculum is no education at all; it is a meaningless collection of subjects lacking a moral, social or intellectual centre to their content.


The deliberate acts of vandalism, which removed all of humanity’s grand narratives from schooling and society, have often come from a place of good intent.  But the consequence is nihilism.


One argument I often hear for the lack of a grand narrative in our publicly funded schools is that there could not possibly ever be agreement on which grand narrative or narratives should be present in schools since the taxpayer is so varied in view and belief.  These are the people who seek consensus – a doomed venture.  What I propose is not popular at all, because I propose this; allow individual schools to choose grand narratives.


This is not popular because people will protest, but what if they do not choose to fit my beliefs!


I want to set out the grand narrative I would like to see permeate a school.  Before I do so, I will here explore some other possible grand narratives.


It is the purpose of such grand narratives, through their figures or images to direct one’s mind to ideas and a story.  This is not just any kind of story, but one that tells of origins and envisions a future. A story that constructs ideals, prescribes rules of conduct, provides a source of authority, and gives a sense of continuity and purpose.  A grand narrative has sufficient credibility, complexity and symbolic power so that it is possible to organise one's own life and one’s own learning around it.  Without such a transcendent narrative, life has no meaning. Without meaning, learning has no purpose.


Nietzsche once remarked, “he who has a why to live, can bear with almost any how.”  This is as true of learning as it is of living.  Without a why, without a purpose, schooling will be consigned to history and we will lose one of the greatest achievements of humanity.

Let us begin with the four grand narratives that formed the basis of humanity’s meaning and purpose until the very recent past; the glory of God, nationalism, patriotism and inductive science.


The glory of God

The oldest of the grand narratives and the driving force behind the very existence of schooling and education.


It is the grand narrative that has given countless people a reason for living and a reason for learning.  In the Western world, beginning in the 13th century and for 500 years afterwards, celebrating the glory of God was sufficient justification for the founding of institutions of learning, from grammar schools, where children were taught to read the Bible to great universities where individuals were trained to be ministers of God.  Even today, there are some schools in the West and many in the Islamic world whose central purpose is to serve and celebrate the glory of God.  In such schools, there is rarely a school crisis. There may be some disputes over what subjects best promote piety, obedience and faith, there may be pupils who are sceptical, even teachers who are non-believers, but at the core of such schools, there is a transcendent, spiritual idea that gives purpose and clarity to learning.


I believe the disappearance of this grand narrative is a great loss.  Many argue that schools cannot celebrate a God because we would not be able to decide which God to celebrate.  Again, the consensus argument comes forth and makes everything anodyne.  Again, I say, let schools choose.



There have been many examples throughout history of learning being built around allegiance to the nation state.  It is a narrative far too inspiring of obedience and has the common trick of morphing from a love of the nation into a love of the government, which then finds its way into schools as it did in the Soviet Union and China.



Unlike the nation state narrative, which is really a story of administration, patriotism is an entirely positive grand narrative rooted, as it is, not in a love of government but in a love of the place, the earth, the very land on which we make our home and on which we feel at home.  It is about a love of home and it drives a very deep desire to conserve and protect the land to such a degree that many people have given their lives to do so.  They do this not in support of some administration, but out of a love for the fields and walls and trees and rivers and sky and buildings and lanes and hedgerows and wildlife and sounds that make them feel they belong to a place.  Patriotism, with its foundation in the very soil beneath our feet, is an alliance of all those who also feel at home in some place.  It is patriotism, not nationalism, that results in a true desire to learn about the agriculture and culture of home and to protect the aesthetics of its environment.  It is the grand narrative, more than any other, that declares good taste, beauty and good public decorum.


Inductive Science

Bacon, Descartes, Kant, Kepler, Leibniz, Locke, Newton, Rousseau, Smith, Voltaire and others brought about the great Age of Reason and a long-awaited enlightenment after many years in the dark.  Not only does the Enlightenment demonstrate the incredible power of the human mind, but it reminds us, starkly, that humanity’s progress is not assured and can be undone and must therefore be defended or, as in the case of the Enlightenment, be brought back to life.


The first storytellers in the grand narrative of inductive science did not think of their story as a replacement for the great Judeo-Christian narrative, but as an extension of it. In fact, the point has been made more than once that the great age of science was based on a belief in a God who was himself a scientist and technician, and who would therefore approve of a civilisation committed to such an enterprise. “For all we know,” Eric Hoffer wrote, “one of the reasons that other civilisations, with all their ingenuity and skill, did not develop a machine age is that they lacked a God whom they could readily turn into an all-powerful engineer. For has not the mighty Jehovah performed from the beginning of time the feats that our machine age is even now aspiring to achieve?”



God was slain by Darwin, nationalism slain by Marx, inductive science slain by Freud and even Einstein. With each of the four grand narratives dismantled, a void opened up into which came new stories.  I consider a few of those new stories now.



Communism, Fascism and Nazism

The great religions of the 20th Century were ushered in promising Heaven, but all ended in Hell.  These grand narratives were possible in a world that had turned its back on God.  They are empty.  They are each an abyss.


Yet, they are still grand narratives.  You will protest, I am sure, that there is no place for these narratives in our schooling.  I still say, let schools choose.  This is often met with understandable horror, but I believe that people are wiser than is often assumed of them.


Open a school, declare its narrative as Fascism and see what happens.  This battle is won.  There is absolutely no support for these ideas in the Western world today.  There are, for certain, a tiny number of individuals who espouse the ideas of Fascism and Nazism, but they are lunatics and have zero popular support.  No such school could open, because it would be empty.  The 20th Century taught, in no uncertain terms, the lessons of Fascism and Nazism.  We know that evil lies there and nobody wants it.


Whether or not the 20th Century also taught humanity the lessons of Communism is much more difficult to discern.  On any intelligent appraisal, it is a grand narrative with equal horror and evil.  A narrative that should rightly go the same way as Fascism, yet still there is widespread ignorance about its impact.  Would a school opened under the declared grand narrative of Communism gather support?  I do not know.  I hope it would not, but I am not convinced.  Should we, therefore, have legislation that outlaws particular grand narratives?  I find this a most difficult question, but I err on the side of avoiding the banning of thoughts and ideas.




Traditionally an aspect of patriotism, which seeks to protect the environment in all regards, the new environmentalism is fixated on one issue alone; climate change.  It is a narrative with all the characteristics of a religion and is surely set to become one.  The reduction of environmentalism to climate change activism is justified by its proponents by stating that climate change is of such catastrophic importance that all other environmental considerations pale in significance.  The result is that our rivers are polluted, hedgerows strewn with litter, towns and cities uglified, farming practices bastardised and bucolic scenes lost.  There is no doubt that climate change is happening and that it does indeed trump these other concerns in terms of potential for long term harm, but the problem is so vast, so complex, so unknowable that the reality is almost nothing is done other than talking and signalling that something should be done.


My view is that the environmentalism of the patriotic was more useful and more likely to solve the issues we face today.  It was an environmentalism of local people taking real action to protect their local environment.  By looking after these ostensibly smaller issues but in a real sense, I venture that humanity might also make better progress with the larger issues.  Hitting ‘like’ on a social media post, signing a petition or forcing an entire school to watch a propaganda movie is nothing compared to actually changing something, actually making a difference.  For this grand narrative to have any substance at all, its proponents need to give up their cars, stop taking the train, stop buying iPhones from China or meat from thousands of miles away whilst a local farmer goes out of business down the road.  But nobody ever does.  The new environmentalism is a grand narrative built on empty gestures, superstition and inertia.  What good would come of a schooling system based on a narrative of doing nothing?


The new environmentalism replaces the patriot’s positive action to protect with a religion of repudiation and separation.  This is such a missed opportunity.  We could choose, instead, to make clear the interdependence of human beings and their need for solidarity. If any part of our single habitat is poisoned, then all suffer—which is to say that the extinction of the rain forest is not a Brazilian problem; the pollution of the oceans is not a Miami problem; the depletion of the ozone layer is not an Australian problem. It follows from this, of course, that genocide is not a Bosnian problem, hunger not a Somalian problem, political oppression not a Chinese problem.  Considering the Earth as a single habitat, making clear that when one harms any part of that habitat, one is harming one’s own home and that there is only this home, makes the idea of racism irrelevant and ridiculous, and binds us all together.


Some will ask, but why should I care what happens on someone else’s ship – after all,  I’m a-ok journeying through life on my ship?  The response is, of course, this is a pretty stupid question. Have you not noticed that you are all on the same ship? That you must rely on each other to survive, and that you have not taken sufficient care of your home?


Emotional health

There have been several attempts to build schooling around the grand narrative of emotional health.  Most notably a Rogerian or Maslovian model of schooling, which values above all else the development of one’s emotional life through the quest for one’s ‘real self’. Such a story leaves a curriculum irrelevant, since only ‘self-knowledge’ – one’s feelings – is considered worthwhile. Carl Rogers once wrote that anything that can be taught is probably either trivial or harmful, thus making any discussion of schools unnecessary. The proponents of this grand narrative seek to destroy rather than build.  They know that all their talk of not teaching content in schools is mendacious – they want it only for the poor, so that the poor will remain in their place.  Meanwhile, they ensure their own children are taught the very knowledge that allowed them to rise to such lofty positions as being able to declare that knowledge is useless. The ladder is pulled up, leaving the poor unknowing and incapacitated, whilst lifelines are thrown to their own.  Followers of the cult of Rogers and Maslow want our schools to stress only to importance of ‘self’. I suggest our society is already so saturated with the glorification of self that it would be redundant to have the schools stress it too.


Economic utility

Perhaps the most common response when teachers are asked to give a reason for schooling is that it provides the foundation for their pupils to go on to live financially secure lives.  They may not couch it in these terms, but instead talk about having good careers or independence.  Of course, we all need to be able to sustain our lives – we need to eat.  But the argument that an education is for this purpose does not stand up to interrogation.  If it was the case that an education was for economic utility, would we not stop the teaching of poetry, music or art?  We don’t, because it isn’t.  The economic utility narrative does not even stand the simplest of tests – the wealthiest nations are not those with the highest standards in education.  The economic utility story tells pupils that if they pay attention, do their work, study at home, and pass their tests then they will be rewarded with a good job with which to fill the rest of their lives.  It is an optimistic story, for sure – it plays to the protestant work ethic and is a narrative that Adam Smith would have endorsed.  It places autonomy in reach of everyone should only they work hard.  It is a meritocratic story, one where there is a direct correlation between effort and reward.  A good job can be yours.


But there is a great lie at the centre of the economic utility story; the lie that there exists a good job for everyone.  There does not.  There are awful jobs and they are going to be filled by countless numbers of individuals.  The economic utility narrative is really one of preparing pupils for entry into the economic life of a community.  The community may prosper, but that does not mean the individual will.


The economic utility narrative must necessarily be deceptive.  The story cannot include the reality that there are jobs in the millions that require no education at all.  Jobs that need nothing from schooling.


And if the aim is to enable pupils to go on to become heart surgeons, engineers, lawyers or accountants, should we not dedicate their time in school to learning those skills and just dump all of what we do in schools that does not lead to a tick list of marketable skills?  No. No. No.


Specialised competence can come only through a more generalised competence, which is to say that economic utility is a by-product of a good education. Any education that is mainly for economic utility is far too limited to be useful and so diminishes the world that it mocks one’s humanity.



The handmaiden of economic utility.  Or perhaps it is the other way around.  Whereas the grand narrative of economic utility says that humans are what they do to make money, the story of consumerism says that humans are what they buy with that money.  Whoever has the most toys when they die, wins.



I know of no other contemporary grand narrative so clearly a religion as the story of technology. People believe in technology.  That they believe technology works, that they rely on it, that it makes promises, that they are bereft when denied access to it, that they are delighted when they are in its presence, that for most people it works in mysterious ways, that they condemn people who speak against it, that they stand in awe of it, and that, in the born-again mode, they will alter their lifestyles, their schedules, their habits, and their relationships to accommodate it. If this be not a form of religious belief, what is?


I have earlier discussed the orientation to learning the grand narrative of technology inspires, so will not labour the point further here.



Of the modern grand narratives, the story of equality is the most widely adopted by teachers and schools.  It is a story that comes from a good place, but one that has been hijacked by those who have ill intent.  The good place is a place where each and every one of us is treated fairly and with dignity.  A place where one’s immutable characteristics are irrelevant and individuals are judged by their words, deeds, ideas, accomplishments, decency and character.  It is a place where social mobility is truly achievable, since it is not relevant if one was born poor and merit is given based on competence.  The grand narrative of equality began as the narrative of equality of opportunity.  A noble and right story to try to weave into existence.


The grand narrative of equality has lately returned to its darkest side of communism and Marxism.  Its proponents have no interest in equality of opportunity for all.  Rather, they seek equality of outcome for all bar themselves, who will, of course, as always, have more.


They seek to enact revenge on any and all who do not subscribe to their dogma.  And all must subscribe, because their dogma is righteous, and they are virtuous.  How quickly we have become blind to the fact that history is littered with armies of the self-proclaimed virtuous exterminating their fellow humans in vast numbers.  It would seem that soon, the new virtuous will fire new shots and we will be, once again, on the road to Hell.  I suspect, in my lifetime, I am for the Gulag.


In this vengeful equality story, nothing can ever be forgiven and nothing ever forgotten except, of course, for every single thing that they deem to be forgotten – ignorance, particularly of history, is the main mechanism of this narrative.  It is repugnant.  It is evil.


And even if this original story of goodness had not been hijacked by the vengeful, the desire for all to be equal, rather than for all to be treated equally fairly, results in a great loss.  It results in the death of diversity.  Diversity, particularly diversity of thought, is an entirely optimistic narrative, one that ensures a defence against the loss of all creative energy.  One that protects humanity from uselessness.


The second law of thermodynamics outlines the bleak story that everything in the universe tends toward uselessness.  With entropy comes universal sameness, and when matter reaches a state in which there is no differentiation, there is no employable energy.


But there are negentropic forces in the universe, energies that retard sameness and keep things useful. Every time we clean our homes, or look after our environment or build a dyke or mow the lawn or use information to solve a problem or plan a meeting, we are combating entropy, using intelligence and energy to overcome the inevitable decay of organisation.


Diversity is the key to overcoming entropy.  A new word in the language, a new melody written, a new style of art.  It is how culture remains alive.


Sameness is the enemy of vitality and creativity.


The attempt to make all people the same is the attempt ring the death knell of culture and civilisation.  People are not the same and thank goodness for that.  It is the diversity in people that makes life worth living.  And this diversity is easily and positively embedded in societies that take a cultural pluralism narrative as opposed to an equality narrative.


Diversity, through cultural pluralism, does not diminish excellence as is often the fear of those who campaign against it.  Rather, it is sameness – the lack of differences – that prevents robust standard of excellence from developing.  This is as true in the genetic code as it is in science, literature, art, language and music.  Every culture within a cultural pluralistic society does, of course, have its own view of what constitutes good music or art or… or all things.  These views of good meeting each other, across cultures, expands our view of excellence such that excellence becomes something comprehensible to all.  I do not want pupils to study the music of Tchaikovsky because he was a homosexual, but because his music attained that highest standards of excellence.  Do we learn about Einstein because he was Jewish? Marie Curie because she was Polish? Aristotle because he was Greek? Confucius because he was Chinese? Cervantes because he was disabled? Do we listen to the music of Grieg because he was a short Norwegian, or Beethoven because he was a deaf German? In the story of diversity, we do not learn of these people to advance a political agenda or to raise the level of pupils’ self-esteem. We learn about these people for two reasons: because they demonstrate how the vitality and creativity of humanity depend on diversity, and because they have set the standards to which civilised people adhere.  This is how diversity makes intelligent humans of us all.


The equality narrative in its contemporary form is attempting to make the only thing about Tchaikovsky of interest be his sexuality.  I find this so terribly sad.  When I was a young man at university, my friends and I would often respond to any meathead who protested at gays who sought equally fair treatment in law with the quip, ‘Oh, you are so right.  We have no desire to be equal.  We are perfectly happy being better.’  This long running joke that gay men once used to put imbeciles in their place has now come back to haunt us as the vengeful equality champions seek to punish and belittle heterosexuals and force them to repent and take their place as an inferior or be dismissed from society.  This is not what any of us who went through actual hardship to advance equally fair treatment want.  We are not filled with hate or vengeance and we are rather tired of people being offended on our behalf.




These new grand narratives are found from time to time in schools, but I suggest they are largely hollow and do not meet the test of being a story of sufficient credibility, complexity and symbolic power so that it is possible to organise one's own life and one’s own learning around it. 



The ascent of humanity

And so, to the grand narrative that I support and promote.  The English poet and cultural critic, Matthew Arnold, argued for a culture that made available to all ‘the best which has been thought and said in the world’.  There has been a recent resurgence of this view, driven by educators worldwide who have found freedom of association on social media and have been able to gather in sufficient numbers as to make their voice heard.  These educators argue for the grand narrative of knowledge.  That we can use schooling as the mechanism to acculturate children, immersing them in our cultural legacy, ensuring they have an equal access to the greatest minds of our civilisation and that our pupils become intoxicated by the splendour of knowledge and become addicted to learning.  Pupils in the schooling system that is built on the grand narrative of knowledge, leave school with fine taste and decorum, with an appreciation of beauty in art, music, literature and architecture.  They know the key foundational knowledge that has defined the culture that they live in and know what science and mathematics have proposed.

It is encouraging to see so many proponents of the grand narrative of knowledge shouting from the rooftops about their cause.  It is a fine cause with a noble aim to ensure that all children, regardless of background, are considered capable and deserving of an equal right to the knowledge that humanity has accumulated.


But it is not the grand narrative of knowledge that I espouse.  For me, the narrative lacks an essential component.


The narrative of knowledge seeks to fill pupils with the knowledge that has already been acquired.  It is paternalistic in its aims.


Rather, I espouse, as Jacob Bronowski espoused in his book The Ascent of Man, a more optimistic grand narrative than the knowledge narrative.


The knowledge narrative is paternalistic because it leaves the pupil paralysed and dependant, unable to play their part in the ascent of humanity.  It sets out the greatest thoughts and ideas that have already been thought and said but forgets to instil in pupils the mechanisms by which those thoughts and ideas came about.  This is the grand narrative I wish to see in schools; the narrative that it is humanity’s destiny to continue to become more and more enlightened and that this happens if we all play our part, all contribute our verse.


Unlike the knowledge narrative, the narrative of the ascent of humanity knows that subjects are not completed.  Rather, every subject represents a battleground.  Every idea that is currently accepted in every subject marks the point at which one idea was defeated by a new, more powerful idea.  Subjects have a history, they are moving and continuous, which means that those schools seeking to give pupils the mechanisms for bringing about the best which will be thought and said in the future, must present subjects as histories.


There are many things learned through classroom learning far beyond the content of subjects.  Through the way in which we structure our classrooms and the words that we say as teachers, we teach an orientation towards learning.  In the grand narrative of the ascent of humanity, that orientation is one of genuine interest, one where knowledge should be acquired but also debated, one where exposition is vital in advancing the construction of the edifice of learnedness, one where questioning and positing are expectations, one that reaches into the past and into the future, binding the dead to the unborn.


In the grand narrative of the ascent of humanity, to become educated means to become aware of the origins and growth of knowledge and knowledge systems; to be familiar with the intellectual and creative processes by which the best which has been thought and said has been produced; to learn how to participate in what Robert Maynard Hutchins once called ‘The Great Conversation’.


Education here is not child-centred, not training-centred, not skill-centred, not even problem-centred. It is idea-centred and coherence-centred. It is also otherworldly, inasmuch as it does not assume that what one learns in school must be directly and urgently related to a problem of today. In other words, it is an education that stresses history, the scientific mode of thinking, the disciplined use of language, a wide-ranging knowledge of the arts and religion, and the continuity of human enterprise.

Above all, education in the grand narrative of the ascent of humanity is an education the leaves pupils knowing that knowing does not end, that no person can ever know all there is to know, but that our purpose is to continue expanding humanity's knowledge.

As Walt Whitman so perfectly put it, 'that the powerful play goes on, and you may contribute a verse'.




I am often challenged to defend my view that schools should be free to choose grand narratives.  Good people, who have thought carefully about these problems, assert that, if we are to have grand narratives in education, it must be for the state to decide what they are.  But this simply does not work.  The direction of travel for politics is more consensus and less democracy.  Trying to find a consensus is one of the main reasons we have ended up here.  So, am I not afraid that schools would not choose the ascent of humanity as their overarching purpose?  I guess that would be an understandable concern, but I have great faith in individuals.  I have faith that, if schools espouse their grand narratives, parents will choose good over evil, enlightenment over darkness, autonomy over subjugation.


Schools should and could be places of conviction.  They could glue together subjects to form a true course of study, where the curriculum is coherent and has purpose.  They could tell their communities what they stand for – and by this, I do not mean adding three apparently randomly selected words to a strapline on their website and headed letter paper.  That does not cut it at all.  A school driven by a grand narrative has purpose and meaning behind all that it does – every person embodies the narrative, its values, morals and principles. I have enormous faith in the individual’s right to choose, in the right to free association and I believe that those grand narratives of substance and goodness would prevail.



An education must be for something if it is to have meaning.  To give up on meaning, give up on a defending a purpose for schooling is to give up on schooling itself and it will simply cease to exist, replaced instead by amusement and the rules of the new orientation.  It is striking to observe the change in educational discourse in recent decades.  There are two great debates to engage in.  There is the debate about the mechanical; the methods of how to teach so and so.  Then there is the debate about the metaphysical; the reasons for teaching so and so.  The former is now front and centre, the latter almost forgotten.  Without a return to fighting for education to be for something, it will become pointless and justifiably replaced by amusement.


Our schools can be places of conviction but only if we defend their right to be.

Monday, 27 July 2020

Models, Metaphors, Examples and Instruction

Many people have asked me over the years why I so often use the phrase ‘models, metaphors, examples and instruction’ when talking about mastery approaches to teaching.  This short blog is a high-level response to that question.



Within every mathematical idea, problem or situation lies technical detail.  This technical detail is key to allowing anyone learning mathematics to be able to take part in The Great Conversation of mathematics, such that they are not simply a passive recipient of the subject, but rather are in a position to contribute meaningfully and identify subterfuge.  Study of this technical detail is akin to the study of grammar of the English language.


"Grammar is important because it is the language that makes it possible for us to talk about language. Grammar names the types of words and word groups that make up sentences not only in English but in any language. As human beings, we can put sentences together even as children—we can all do grammar. But to be able to talk about how sentences are built, about the types of words and word groups that make up sentences—that is knowing about grammar. And knowing about grammar offers a window into the human mind and into our amazingly complex mental capacity."


"People associate grammar with errors and correctness. But knowing about grammar also helps us understand what makes sentences and paragraphs clear and interesting and precise. Grammar can be part of literature discussions when we and our students closely read the sentences in poetry and stories. And knowing about grammar means finding out that all languages and all dialects follow grammatical patterns."


Haussamen, Brock, et al. "Some Questions and Answers About Grammar," 2002


Technical detail in mathematics makes it possible for us to talk about mathematics.


Here is a mathematical problem:


It is not enough for the teacher of mathematics to consider this problem to be self-contained within a single lesson titled ‘Pythagoras’ Theorem’.  The teacher is able to consider the problem from this perspective because they have an often-unconscious appreciation of the technical detail that lies beneath, which enables them to talk about this problem as though there is only one technique at play.  Yet there are many.  Here is some of the technical detail that makes up this seemingly routine Pythagoras question (this is not intended as an exhaustive list, but hopefully conveys the point):




The equals symbol, ‘=’, denotes that, for any two given expressions, applying an isomorphic function to each will maintain equality.  Teachers should continually reinforce in pupils’ minds that the equals symbol gives them permission to ‘break and fix’ equality in any way they choose.  Fixing merely requires applying the same function that has been applied to one expression to the other.



Two expressions have equality if they have the same value or represent the same mathematical object.


Equivalence (relation)

A binary relation that is reflexive, symmetric and transitive.  These three conditions must be met:

1.     a = a (reflexive property),

2.     if a = b then b = a (symmetric property)

3.     if a = b and b = c then a = c (transitive property)




An equation is a mathematical statement asserting that two expressions have equality.



The area is the extent of a 2D shape on a plane.  This can be given a numerical quantity to communicate its extent in relation to standard extents.  At school level mathematics, the archetypal area unit is a square.  All areas can be defined in relation to the archetypal area unit.  This archetypal unit can be chosen to best fit the scenario (for example, a square of 1mm by 1mm, a square of 1cm by 1cm, a square of 1m by 1m, and so on to fit the circumstance)



Addition is of the two key operators required to create a mathematical field.  Addition has key properties as expressed in the Field Axioms.



Multiplication is of the two key operators required to create a mathematical field.  Multiplication has key properties as expressed in the Field Axioms.



The characteristic of equality being maintained when the order of an operand is changed.  Addition and multiplication are commutative.  For example, 2+5 and 5+2 have the same value, 5x6 and 6x5 have the same value.  When teaching both addition and multiplication, teachers should continually stress that the operations are commutative and use varied examples to expose the commutativity of these operators repeatedly.


Inverse function (anti-function)

An inverse functions returns the output of a function to its original form.  When teaching, teachers can use the analogy with pupils that an inverse function ‘reverses’ or ‘undoes’ a function.  Teachers should use varied example, on many occasions throughout the curriculum, to demonstrate the impact of an inverse function.  For example:


·      adding 5 can be undone by subtracting 5

·      multiplying by 8 can be reversed by dividing by 8




If the function f applied to an input x gives a result of y, then applying its inverse function g to y gives the result x, and vice versa, i.e., f(x) = y if and only if g(y) = x



A mathematical operation with a power, n, acting on a base, b.  This can be written as bn.  In the case where n is a positive integer, the action corresponds to repeated multiplication.  For example, bn = b x b x b… x b, where the multiplication has been performed n times.  For example, 45 = 4 x 4 x 4 x 4 x 4 = 1024.


nth root

In the equation, rn = x, the nth root of x, n, has the value r.  When teaching nth root, teachers should first ensure pupils fully understand exponentiation.  The nth root can then be discussed in terms of familiar exponent scenarios.  For example, given that pupils are comfortable with the fact that 23 = 8, teachers can then discuss finding the number which gives 8 when raised to the power 3.


Plane angle

The figure formed at the common point shared by two rays.



Turn is the archetypal unit of plane angle.  At school level mathematics, the most common values used to represent this unit are 360 degrees or 2 radians.  Teachers should continually reinforce that one full cycle is a unit of turn.



Expressions can be stated to be equal when they have the same value or represent the same mathematical object.  When expressions are not equal, their relative values can be compared by stating whether the value of one expression is greater than or less then the value of another expression.  This gives a range of comparisons which can be made


·      Equal to

·      Greater than

·      Less then

·      Greater than or equal to

·      Less than or equal to




To manipulate an equation such that one expression is a single variable.  This is sometimes referred to as ‘transposition’ or ‘making x the subject of the formula’, where ‘x’ is the variable to be isolated.  The ability to isolate is key to being able to solve equation problems or evaluate formulae.  Teachers should rehearse this skill regularly with pupils.  Often, a pupil who appears unable to work with equations or formulae, is actually struggling to isolate.



A two-dimensional surface extending to infinity.



A calculation from zero or more input values to an output value.  At school level mathematics, pupils work with unary and binary operations.  Unary operations, such as trigonometric functions, involve only one value.  Binary operations, such as addition, multiplication and exponentiation, involve two values.



A process for relating every possible value in a set to exactly one value in another set.



A defined collection of objects.



Field Axioms

Just nine properties provide the basis for all of arithmetic and algebra. The axioms of a field – or, as often stated, the field axioms – give the mathematician a rules-based approach to working with number and variables. Knowing and being able to work with the field axioms means that a pupil can attack any arithmetical or algebraic problem they might face in school-level mathematics. These properties always hold true; they are the same properties at the beginning of primary mathematics as they are at the beginning of calculus. They hold for whole numbers, fractions, negative numbers, rational numbers, letters and expressions.


Establishing their truth and utility is therefore the key aim of the teacher of early-stage mathematics.


The nine field axioms are split into properties for addition, properties for multiplication and a property connecting addition and multiplication. It is important to note that none are presented as subtraction or division.


The properties of addition:






Associative property of addition

(a + b) + c = a + (b + c)


Example: (2 + 3) + 4 = 2 + (3 + 4)


Commutative property of addition

a + b = b + a


Example: 2 + 3 = 3 + 2


Additive identity property of 0

a + 0 = 0 + a = a


Example: 3 + 0 = 0 + 3 = 3


Existence of additive inverses

For every a there exists –a so that
a + (–a) = (–a) + a = 0.


Example: 2 +(–2) = (–2) + 2 = 0


The properties of multiplication:






Associative property of multiplication

(a × b)  × c = a × (b × c)


Example: (2 × 3) × 4 = 2 × (3 × 4)


Commutative property of multiplication

a × b = b × a


Example: 2 × 3 = 3 × 2


Multiplicative identity property of 1

a × 1 = 1 × a = a


Example: 3 × 1 = 1 × 3 = 3


Existence of multiplicative inverses

For every a ≠ 0 there exists 1/a so that a × 1/a = 1/a × a = 1


Example: 2 × 1/2 = 1/2 x 2 = 1


Connecting addition and multiplication:






Distributive property of multiplication over addition

a × (b + c) = (a × b) + (a × c)


a(b + c) = ab + ac




The minimum number of coordinates required to specify a point in a mathematical space.  Pupils’ early encounters with dimension include specifying a point on the number line, which requires only one coordinate and so is said to be one dimensional.


Coordinate system

A system for uniquely identifying the location of a point or points.  Pupils’ early encounters with coordinate systems include specifying a point on the number line.  Teachers should build on pupils’ familiarity with and understanding of the number line when introducing Cartesian coordinate systems.



Any definite mathematical object.  At school level mathematics, pupils most common encounter with mathematical objects will be as numbers.  Numbers are values.



The relative size of a mathematical object.  Magnitude can be described using comparison.



For any two given entities with numberness, ratio is a relationship stated numerically of how many multiples of the first entity compare to the stated multiples of the second entity in a given context.  For example, a class containing 13 girls and 8 boys could be described as the relationship of girls to boys as 13:8.


Numberness (numerosity)

A set of physical objects is said to have numberness if the objects can be counted.



A mathematical object used to count, label, measure, compare, order or discuss items that have numberness.



A symbolic representation of numberness.



Individual symbols in a given base system used to create numerals.


Radix (base)

The number of unique digits used to represent numbers.  At school level mathematics, pupils work mainly Decimal (a system with radix of ten), using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 to create numerals.  The teaching of multi-base arithmetic is critical is pupils are to true grasp the meaning of place value and, therefore, how algebra is used to communicate working in an unknown base.  When pupils are secure in working across different radix systems, they are able to make strong connections across arithmetic and algebra, giving them the power to work in the generalised form with meaning.


Place Value

The relationship, in a given radix (base) system, between numbers, numerals, digits and numberness.



The existence of equal parts


Simple fraction (common fraction or vulgar fraction)

A rational number, expressed as a/b, where a and b are integers and b is not equal to zero.



A magnitude or multitude.  Quantities can be mathematical objects or physical objects.



This technical detail matters enormously, not just when attacking the problem personally, but fundamentally when teaching pupils about these types of problems.  It is not enough for the teacher to have expert mathematical knowledge, rather the teacher needs to have expert knowledge in how to enable a pupil to gain meaning and understanding of the novel idea, which means the teacher must be alert to all of the technical detail involved in any given problem or topic.


James Fey considered this preparing of technically detailed and true mathematics for the teaching process as one of ‘elementarization’.  As Fey puts it, “the translation of mathematical concepts, principles, techniques, and reasoning methods from the forms in which they are discovered and then verified by formal reasoning to forms that can be learned readily by a broad audience of students.”


This is a fact that is too often absent from mathematics teacher training and professional development; there needs to be some ‘translation’, some process, some experience that enables a novice to grip an idea and that this ‘translation’ necessarily requires considering the mathematics in a different way to its technical truth.  We require a medium in order to communicate the mathematics.


I think of this as didactics of the mathematics.  The range of actions teachers take in order to transfer an idea known to them into an idea known to the pupil with technical accuracy.


I suggest an element of this didactics of the mathematics as the hierarchy







I choose these words to label some phases of action.  They could easily be different words, but I have mulled it over for years and think these words work well.


I introduced this idea above as a hierarchy, but actually, I don’t even believe this myself!  This is much more like Hemel Hempsted’s Magic Roundabout, with mini-cycles inside a cycle.


Beginning at the base layer.  Instruction is the way in which the teacher communicates an idea in front of a class or pupil.  It is the words they use, the hand gestures, the sentences, the body language. It is the images they draw and where they point when talking about them, it is the time they take and the pauses they throw in.  Instruction is one key element of a teacher’s pedagogy.  But pedagogy without didactics is mere performance.  Instruction is how teachers introduce and speak about the examples they will use.


Examples are the questions teachers pose and the batting to-and-fro with pupils as they undertake their own attempts on examples and articulate to the teacher their thought processes.  They are the in-the-moment tweaks and changes that teachers make in response to these articulations and they are the pathway to automaticity and fluency.  Examples are the way in which teachers convey the metaphor they are hanging the new meaning around.


Metaphors are the connective stories that teachers use to convey novel information from a viewpoint of a pupil’s familiarity.  They are the everyday, they are the common sense, they are the narrative.  Metaphors make it possible for the novice to comprehend surprising, disruptive and difficult new knowledge.  Metaphors are communicated through models.


Models are the words, images, apparatus, tools and dynamic representations of mathematical ideas that enable teachers to discuss the chosen metaphor.  They are the ways in which pupils manipulate, rearrange and converge towards new insight.


We choose an appropriate model for the metaphor that we have chosen.  We choose our metaphors because we know our pupils.  The choice of models and metaphors determine the choice of examples that we will use with our pupils.  And the choice of model, metaphor and example at any given moment informs our approach to instruction.


So perhaps the ‘hierarchy’ looks something like this:


Let’s consider an example.  Suppose, for instance, the prima facie idea at hand is ‘subtraction of negative numbers’.  Just like with the problem above, which appeared on the face of it to be about Pythagoras’ Theorem, any problem in the idea ‘subtraction of negative numbers’ will also have technical detail at its heart.  Here, though, I want to look at an example of the ‘hierarchy’.


How do I start the thinking process?  How do I get to a point where I can perform an impactful translation from what I already know to be technically true, through a medium, to convince a pupil of what is technically true?


To begin with, I think about my pupils and what I know them to be already familiar with.  This gives me the beginnings of my story telling.  The beginnings of choosing my metaphor.


For this particular idea, there are three metaphors that I think will work with my pupils:


1.     Addition

2.     Difference

3.     Displacement


I choose these because I know that my pupils are familiar with all three, which means I can bridge from the familiar to the unknown by telling a connected story, in a logical and precise way, through the use of models, examples and instruction.


Suppose I choose addition as my metaphor.  This requires the introduction of the idea of additive inverse.  I know I am able to do this, because I know my pupils have been mathematically conditioned to be comfortable with the idea of doing and undoing. I can now choose a model to convey this metaphor.  If dealing simply with numbers, I will choose to use two-colour counters, if dealing with the general case, I will choose to use algebra tiles.  This model is chosen carefully to enable the pupil to see the logic in the technical detail of additive inverse – they can see and appreciate the existence of zero pairs.


Suppose I choose difference as my metaphor.  I know I am able to do this, because I know my pupils have been mathematically conditioned to be comfortable with comparing.  There are several models I could choose, but my go-to choice would be either Cuisenaire rods or a simple pictorial bar model.


Suppose I choose displacement as my metaphor.  I know I am able to do this, because I know that my pupils have been mathematically conditioned to be comfortable with the ideas of movement, comparing and doing and undoing.  The model I would choose here would be that of vectors.


So, by knowing my pupils, I can make choices about metaphors, which in turn determines the models I have available to me, which in turn authors my examples and determines the words, gestures, pauses and detail of my instruction.


The hierarchy also serves us well when we wish to engage with ‘corrective’ teaching.  That is to say, when a pupil has not yet gripped an idea and we undertake some new actions to remedy this.  I do not for one minute buy the common protest, “oh that kid doesn’t get it” or “this pupil isn’t very good a mathematics.”  Either we believe that teaching can have a profound impact or we do not. I, for one, am firmly in the former camp.  All pupils can learn well given the right conditions and their sincere effort.


Suppose we are in the game of tennis that is batting questions back and forth with a class as examples and problems.  And we are listening to them articulate their thinking when we spot a moment of confusion or error.


The hierarchy acts as a check list for action taking.  Rather than the typical practice that is often seen in mathematics classrooms, where a teacher reacts to a pupil ‘not being able to understand’ algebra tiles, say, by immediately changing the model, we can use the hierarchy to more carefully consider corrective teaching.


So, if a pupil has not gripped an idea, let’s start at the base layer.  Was there some aspect of my instruction that was unclear or muddled?  Did I demonstrate the use of the model accurately and avoid misconception?  Was my emphasis appropriately timed to draw pupils’ attention to the most important aspects?


We can then choose to change our instruction.


If confident that the instruction is not the issue, the next layer can then be considered.  Have the examples and problems been chosen well?  Do they reveal underlying relationships? Do they rely on knowledge the pupils have in order to complete calculation?  Are the variables suitably selected?  And so on.


We can then choose to change our examples.


If confident that the instruction was clear and accurate and that the examples are known to be effective, then to the next layer…


Did I choose a suitable metaphor? Did I really take into account my pupils’ true starting points and foundational knowledge?  Is the metaphor known to be impactful for communicating the mathematical idea at hand?


I can then choose to change the metaphor, which, of course, necessitates the changing of the model.


In sum, I so often use the phrase ‘models, metaphors, examples and instruction’ because I think it acts as a helpful framework in creating the actions and the medium through which teachers can create pedagogy with didactics classrooms rather than classrooms that focus purely on pedagogy with no regard to technical detail and the need to translate one’s own mathematical knowledge into a form that can be received by pupils.  Pedagogy without didactics is mere performance, pedagogy with didactics ensures all pupils can learn well by building accurate, technical detail.


It is not enough to merely expose pupils to what Matthew Arnold famously referred to as ‘the best that has been thought or said’.  That is no education.  What I want for pupils is for them too to have the mechanisms and technical knowledge that make them able to be the adults who think and say the best.  This means they must study the grammar of mathematics – its technical detail – so that they, themselves, can speak about the mathematics.