by Toby Payne-Cook
So, maths is everywhere at the moment: in the press and all over “edutwitter”, triggered by Rishi Sunak’s – Prime Minister of the United Kingdom – recent speech and his personal mission to make maths compulsory in schools until 18.
I have resisted the urge to respond and tweet on this matter myself as such tweets are inevitably laden with personal, unconscious bias and are often misinterpreted or reacted to in the wrong way, so I thought I’d cobble together my trademark meandering thoughts, observations and opinion on multifaceted matters of mathematics instead.
First of all, I was amused by the Sunak announcement, with my enthusiasm for the philosophy of Alain de Botton and the school of life ringing in my ears: “…Yet what is distinctive is just how selective we are about the topics we deem it possible to educate ourselves in. Our energies are overwhelmingly directed towards material, scientific and technical subjects – and away from psychological and emotional ones. Much anxiety surrounds the question of how good the next generation will be at maths; very little around their abilities at marriage or kindness. We devote hours to learning about tectonic plates and cloud formations, and relatively few fathoming shame and rage.”
Sunak is clearly more worried than most about how good the next generation will be at maths. Why is he worried about this? Well, presumably it’s an economics thing. Employers, industry and commerce – the people who grow the economy – are dissatisfied with the numeracy and mathematical skills of the average school leaver, apprentice and / or graduate. Maybe it’s a global competition and ego thing too, with our global ranking in PISA maths somewhat lower than we’d like it to be. And perhaps it’s a statistics of the modern age thing: to be meaningfully engaged with modern technological and data rich culture we all need a far deeper grasp of numbers, data and their associated fudging in the fake news and propaganda politicians and the media are so inclined to bombard us with.
In many ways, Sunak is right about maths, our collective shoddiness at it and it being – arguably – the most important subject we can educate ourselves in. But the most reactionary, and least imaginative amongst us, look at the current system and think he’s mad: the ridiculous Gove GCSE reforms; the clunky, overly full curricula; the pupil disengagement with the subject from quite early on in school life; problems with recruitment, retention and quality of maths teachers at a national level and the fact that an awful lot of people struggle to attain a meaningful ‘pass’ at GCSE at 16. For me, those calling for fundamental 14-18 reform including the probable removal of GCSEs and those suggesting a fundamental review and re-design of the 5-16 (or 5-18, or 5-14 in partnership with the former) maths curriculum are on to something. If Sunak’s big idea is just another bolt on to an education system built on sand, to make us a bit more like Singapore, he’s deluded in the extreme.
The problem with people like Sunak and Gove tinkering deeply with our education system is that they are highly educated people, with astonishingly quick minds. The sort of people who would have been top of the class through school and university, the sort of people who maths comes easily to. I know people like Sunak. I have a nephew in law, in his early forties, with an engineering 1st from Cambridge, who then became an investment banker at Goldman Sachs, then worked for a hedge fund and now invests in crypto currency and other such unfathomable worlds. His mind is astonishing and he has used it to make, and continue making a lot of money.
Having worked in research and development in three major multinational science based companies I have also worked with some exceptional scientific and creative thinkers with quick mathematical and analytical minds. These are the sort of people who would attain grade 9 at GCSE with minimal effort, and A or A* at A level – without excessive tutoring, cramming or revision. They are exceptionally bright. They may be socially inept, lack common sense and be a little strange too (not always) and in the ‘real world’ are likely to need great communicators and collaborators to work alongside them to bring their ideas, knowledge and algorithms to life.
Our education system, and the concept of ‘an education’ was – and continues to be – designed for them, and us reading this too. Sunak and Gove and most politicians (not all), economists and political or economic commentators are likely in the school measured top 1-5% of all learners. Anyone interested in reading and thinking about this stuff – including you and me and all the prevalent edutweeters and commentators are likely in the top 20 – 30% of all learners too – using school, psychometric or IQ measures.
I once worked with a maths teacher, a former commodity broker in the city, who said that – for a competent mathematician – it doesn’t take 12 years (from Reception to Year 11) to learn GCSE maths. I have taught some exceptionally gifted young people (9-13 years old). With CATs scores of 135+ in both verbal and non-verbal reasoning. They could master the GCSE curricula in maths and the sciences by 14, perhaps younger in some cases. Their brains need feeding and they lap learning with considerable aplomb – albeit – potentially – annoying quite a few teachers along the way! Then I’ve taught a lot of children with CATs scores in the range 110 – 130 and they will have no trouble in excelling within our prescriptive and narrow academic subject based education system at GCSE and A level – if they chose to comply with a limiting system. Those with CAT scores in the 90-110 range may, if pushed and encouraged at school and home, embrace the school system and earn a seat at the table in the competition of modern economic life, if encouraged to do so. Those with CAT scores below 90, and certainly below 80 are going to feel like a constant failure throughout our education system. The flawed idea that gaining some grade 2s or 3s and possibly two or three 4s at GCSE is going to feel like a success – for them – is utterly ridiculous.
The vast majority of the population will have CATs scores in the 90-110 range. Some of them will have mastered enough maths to be mathematically competent and economically useful to society by 16. Others will have not ‘fulfilled their potential at 16’ – as will be true of a great many children across the whole 70 – 140+ CATs (cognitive ability tests) range. Because of the nature of adolescence, and lack of any adult perspective on life and work aged 12 -16, a great many children disengage with academic learning and consequently underachieve in the arbitrary and overly measured public exams we inflict upon them aged 16. But yet we continue to pummel them with heavy and abstract maths and science curricula (and overly prescriptive English and humanities curricula too). The Nuffield foundation did some research about a decade ago, showing that children found learning and attaining in GCSE maths far ‘easier’ aged 16-18 than aged 14-16. This resonates with me. For too many children for too long we rush to cram abstract concepts and excessive facts too soon into their minds. Yes, we know that new, deeper knowledge is hooked onto existing knowledge but so much of the (semantic – conceptual and factual) knowledge so many children are exposed to aged 8 – 16 is never deeply embedded, weakly held at best. Even the ‘greatest minds’ forget most of what they are taught at school if it is never applied in adult life, or thought about again. Sure, the brain of a ‘good mind’ is trained by bombarding it with information and application of that information but the efficiency or purpose of extrapolating this obvious fact to the whole cognitive ability range of all humans seems grossly inefficient, and consequently alienating. However, once we emerge from the deepest mire of adolescence and bat an eyelid towards the adult world, the benefits of ‘why are we learning this’ or ‘gaining a qualification’ and the increase in some sense of delayed gratification – may mean that teaching some version of GCSE maths aged 14 – 18, instead of aged 12 – 16 may well have some merit.
Obviously the mathematically or cognitively gifted can learn it faster than this and do not warrant being held back or slowed down, while there are others who will always benefit from a narrower, more functional maths curriculum.
Like science at school, the GCSE maths curriculum does not make you a mathematician or a scientist. And I would argue than much of the content is not a pre-requisite to becoming a competent mathematical, scientific, economic or statistical thinker. The breadth of the curriculum is particularly bonkers for those who really struggle to master it.
I have taught maths in year 5 and 6 before. Just as some concept of place value is weakly attained we’ll whizz off into some arithmetic, or shape. Shape?! At a deep, mathematical level, reasoning when a 2D shape is or isn’t a rhombus is some kind of fascinating but honestly shape, triangles, circles, blah – do these mathematically beguiling little beasties really need to be played with. Oh, but its fun! And it breaks up the mundanity of trying – and failing – to understand fractions or basic algebra again.
I don’t have the answers. But I would suggest number bonding to death, times tabling to death, number patterns and scales to death, measuring units to death and then later percentages and simple probability and statistics to death would be a good idea. When I say ‘to death’ I probably mean 3 x 20 minute bursts everyday from 5 – 7; 3 x 30 minute bursts everyday from 8 – 10 and 3 x 1 hour + 5 x 20 mins per week from 11-14.
We dig too deep into too many mathematical concepts before basic number is mastered.
Recently, I’ve become drawn into mental math Monday on Twitter. There was a question how would you do 48 + 49 in your head. I answered make both 50 and then take away 3, so 97. Using the nearest ten (or multiple thereof) to round numbers and adjust from there is – to me – a simple and logical mathematical trick. A lot of people liked this on the twits. But one person objected claiming it was too complicated stating that they did it in one move (96 + 1 = 97), not realising that this was two moves (at least) as they already mentally knew that 48 + 48 = 96. There were a lot of people who did it the “school way” of adding the tens, then adding the units, then adding the two (so, 40 + 40 = 80; 8 + 9 = 17; 80 + 17 = 97). Anyway, there is no right or wrong way to do these things; some have more number bonds stored in their head than others too but it is fair to say that some methods are more elegant or efficient than others. But doing an awful lot of this stuff, daily, in very short bursts, seems to me a really good way of improving our individual and collective mathematical functioning. And it is clearly more useful to all than pythagorus theorem; trigonometry; shape naming and reasoning; solving algebraic equations and long division.
On the topic of algebra. I love it. It explains how maths and number and problem solving works at a fundamental level. It underpins engineering, physics, physical chemistry, economics, cosmology and problem solving in any number of complex scenarios. But there is too much of it, too early, for too many in the GCSE maths curriculum. Hell, even quadratics is in the Grades 1-5 foundation GCSE maths specification that my youngest child is currently battling with (the elder two were fortunate to see the logic in it all with minimal effort). Before Dominic Cummings became the Downing Street devil he wrote some interesting and thoughtful blogs about maths education. One stated that there is too much algebra (as there is too much fundamental top down, onward application which most don’t do, too early in the mandatory science specification at GCSE too) but not enough probability in the national maths curriculum. He reasoned that probability is all about likelihood of occurrence and therefore is about risk analysis and therefore about decision making. We all, whatever we grow up to become, need to make better, more effective decisions. And learning lots of (simple) probability would enable us to do that better, in theory. This was sensible reasoning from Cummings and one I hope is applied if a coherent maths curriculum until 18 actually comes to fruition.
At the core of the debate triggered by Sunak’s speech / intent, in the education sphere, is what is the purpose of education and by extrapolation, what is the purpose of maths education? There are, of course a range of needs and purposes from the personal, to the fulfilling, to the brain / mind development, to the utilitarian and economically driven – the latter which, inevitably, dominates our world. Many purist educators, particularly a certain brand of powerful knowledge-rich education leaders, consultants and inspectors are scornful of the utilitarian, skills-based drivers on education policy. This is a problem that there needs to be more coherence of in government and Whitehall. The aims and needs and games of the DfE seem almost entirely juxtaposed from those of BEIS and probably DEFRA and the Department of Health too.
After 16 years in industry I was briefly a Director of a small STEM enrichment / brokerage charity where I saw the juxtaposition between then (in 2011/2012) BIS (dept of business, innovation and skills) and the DfE first hand. Let’s encourage and inspire the importance of the sciences and maths to the future economy with an exiting one day talk, lecture, workshop or competition and then bombard the minds of young people with three years of largely alien facts, concepts and abstraction! Then measure them in their ability to remember it. Then, mostly, run away from those beautiful, elegant, empowering, problem solving, open-minded “subjects” for life.
In industry I learnt to collaborate and communicate. And I learnt a load of new science. But the art of being a competent scientist (or a competent anything) is not knowing everything yourself but knowing who to ask or knowing how to find out. I saw this huge flaw in our individualistic, content heavy, memory-not-application-test examination system when interviewing undergraduate placement students and fresh graduates in Pfizer. Their shiny qualifications made many of them arrogant. They often thought reciting the second law of thermodynamics made them a scientist. They often couldn’t reason or analyse their way around a relatively simple or not previously encountered problem. They couldn’t think, they could only memorise or recite. Less is more in mathematical and scientific curricula. Let’s grossly narrow the content but spend much more time applying it, deepening it, grappling with problems, interpreting data, repeating challenging concepts, unpicking them and sticking them back together. Those capable of learning lots and remembering lots will always do well in school and university but these are not necessarily the skills or aptitudes which power much of life.
In school, two subjects dominate. Maths and English. I’ve often said that English makes you clever, but maths make you smart. Clever is reading lots and writing about it in carefully structured essays. Certainly useful in many careers but essential in mastering exam based curricula at school and university across the whole subject spectrum. Smart is thinking, problem solving and making quick – hopefully good – decisions. Smart is way more helpful in life and work than clever. Clever is remembering and applying formulae. Clever is writing an A* essay on anything. Clever is doing really well on Mastermind, University Challenge or the Chase. Clever is compliant in school. Clever is conscientious. Clever is engaged with and accepting of society’s conventions in education and politics.
Smart can become disillusioned or disruptive. Smart questions everything. Smart can look at a graph or chart and go, yeah but why, yeah but you’re choosing what data to present and what not to present; yeah, but you’ve massively expanding or shrunk the axes. Smart is how big was you sample size? Smart is does ALL THE EVIDENCE point in the same direction? Smart questions the validity of educational data with all its uncontrolled variables and conveniently fudged start and end points. Smart questions the clever game playing of politicians and SLT Newbie. Smart is cynical and sceptical. Smart is James Bond spotting Granchester Road among the data noise in Skyfall. Smart is quick to see trends and patterns where most others can’t. Smart is quick at mental maths. Smart probably finds quick ways of stimulating and exercising their mind. Smart is impatient in slow, clever conversations – and dull, bureacratic ones too. Smart wants to cut to the chase.
Smart is also knowing that correlation works both ways. Like global temperatures increase with increasing levels of atmospheric CO2. But maybe, a rise in global temperature could cause a rise in global CO2 – for example. Yes, burning fossil fuels clearly not good. But there’s lots of carbon dioxide dissolved in the oceans. Loads. So, if something else is causing rising temperatures (sun cycle for example) then carbon dioxide would evaporate at a faster rate from the oceans thus increasing atmospheric CO2. So the slow, long game of net zero and reducing emissions may not work, or may not reverse global warming, so the only thing to do is to suck and process and capture the excess CO2 out of the atmosphere is we want to fix the climate crisis, isn’t it? I may be wrong on all this! I’m now keen to go down a rabbit hole on whether there is an inverse correlation between increasing atmospheric CO2 and decreasing dissolved ocean CO2!
I digress above, but in saying maths makes you smart, the real truth could be that those with the cognitive propensity to be smart are naturally “good at maths.” Can we make everyone good at maths? That is the question. And that is presumably the desire of our collective modern world and politicians. To make everyone ‘good’ at maths and consequently ‘good’ at employability and ‘good’ at growing the economy and bolstering the international ego of the dwindling global status of the UK.
So, to become collectively good at maths we need to do more of it, of course. But the smart amongst us know that mindlessly doing more of it won’t really help. What we need to do is do it smarter!
In comparison to the most able pupils I’ve taught and the most brilliant scientists I’ve worked with and the best maths teachers in the world, I’m not that great at maths. I learnt the early rudiments easily, practised my times tables and glided with minimal effort to an early A at GCSE in January 1989 (there were no A* in those days, nor 7/8/9) with no revision, minimal practice and just two months after my father died from terminal cancer (colon and liver). I thought I was a genius. I wasn’t. I did maths A level. I had to do Pure and Applied. I wanted to do Pure and Statistics. I always hated mechanics and school physics. I got an N (below an E and above a U). Killed my medical school aspirations. Fell into Chemistry (I got a B in that with minimal revision, for some reason it comes fairly naturally to me). I used and applied a lot of maths (place value, arithmetic, units of measurement, calculations, percentages, graphs, data analysis, statistics, statistical experimental design) as a product development and pharmaceutical materials scientist in industry for sixteen years. I would argue the only thing I did more of than maths in those sixteen years was go to meetings and drink coffee. Most of the science I did was really maths and problem solving. I exercised my maths brain.
Then in 2013, I trained as teacher, aged 41. PGCE 7-14. So I had primary maths lectures and workshops. Excellent. Loved it, thrived. Taught Y6 maths on placement. Then for first four years of teaching from 2014 – 2018 I taught year 5 and 6 maths. Sharpened up my mental arithmetic and time tables speed. Enjoyed teaching it but as with the science curriculum I was constantly frustrated by how I was obliged to teach fractions and decimals to children who hadn’t learnt their times tables or mastered basic number and arithmetic. Then I realised that the school day and timetable probably isn’t conducive to a great many children who find learning maths hard.
I think maths is beautiful and wonderful and useful and that it enables virtually all other subjects (chronology, time, human geography, dimension and perspective in art, comparison and description in language, design, science, economics, engineering etc). I think the history of maths is fascinating too. Astronomy. Egyptians. Babylonians. Ancient Greeks. All those brilliant mathematician philosophers: Descartes, Pascal, Russell. Base 60. Circles and time. Base 12. Base 10. Why don’t we use base 12, but with 12 becoming 10 and introducing two more integers / symbols for our current 10 and 11?
If you are “good at maths” and / or (though its probably and) “good at science” then when you have your degree the scientific, technological, business, banking or economic world is your oyster. So it’s rare for such an academic high flyer to think, oh, I know I’ll go and teach a clunky, ill thought out education for the sake of education box ticking curriculum to loads of children who find learning this stuff much much harder than I did for little pay, reward or value in society. When you could become an engineer, computer scientist or trader on the money markets instead.
Conversely if you’re “good at English or History, but maybe not quite so “good at maths” and you don’t make it into the relatively (economically) niche worlds of creative writing, journalism or political / humanities / social sciences research then English or Humanities teaching is your oyster.
This brings us back to where we started with Alain de Botton, in the chapter on Education in his book “religion for atheists.” The world of education (in Whitehall and Schools) is largely managed by English and Humanities graduates (which goes some way to explain the madness of educational data and the outmoded [but not dead] two sub-levels of linear progress each year bollocks) as strong maths and sciences graduates disproportionally don’t go into teaching – and if they do, they are gold dust in the classroom, so are less frequently promoted out of it.
So, to conclude, Maths makes you smart – or resonates more with ‘smart’ people. School and education is a clever centric place. Industry and commerce are smart-centric places. This is why the two don’t understand each other. Politics should combine smart and clever but generally, like education, veers towards clever (how many maths, engineering or science graduates – proportionally – are MPs? And, I suspect of those that are, haven’t spent that long in the science of engineering roots of industry). Sunak is certainly clever, but is he smart enough to realise that what maths education needs, and that of our whole education system, is a complete re-evaluation, and not just more, more, more.
Toby Payne-Cook, January 5th 2023.







