The news reports that the Government will try to promote more ‘specialist maths schools’ similar to the King’s College and Exeter schools.

The idea for these schools came when I read about Perelman, the Russian mathematician who in 2003 suddenly posted on arXiv a solution to the Poincaré Conjecture, one of the most important open problems in mathematics. Perelman went to one of the famous Russian specialist maths schools that were set up by one of the most important mathematicians of the 20th Century, Kolmogorov.

I thought – a) given the fall in standards in maths and physics because of the corruption of the curriculum and exams started by the Tories and continued by Blair, b) the way in which proper teaching of advanced maths and physics is increasingly limited to a tiny number of schools many of which are private, and c) the huge gains for our civilisation from the proper education of the unusual small fraction of children who are very gifted in maths and physics, why not try to set up something similar.

Gove’s team therefore pushed the idea through the DfE. Dean Acheson, US Secretary of State, said, ‘I have long been the advocate of the heretical view that, whatever political scientists might say, policy in this country is made, as often as not, by the necessity of finding something to say for an important figure committed to speak without a prearranged subject.’ This is quite true (it also explains a lot about how Monnet created the ECSC and EEC). Many things that the Gove team did relied on this. We prepared the maths school idea and waited our chance. Sure enough, the word came through from Downing Street – ‘the Chancellor needs an announcement for the Budget, something on science’. We gave them this, he announced it, and bureaucratic resistance was largely broken.

If interested in some details, then look at pages 75ff of my 2013 essay for useful links. Other countries have successfully pursued similar ideas, including France for a couple of centuries and Singapore recently.

One of the interesting aspects of trying to get them going was the way in which a) the official ‘education world’ loathed not just the idea but also the idea about the idea – they hated thinking about ‘very high ability’ and specialist teaching; b) when I visited maths departments they all knew about these schools because university departments in the West employ a large number of people who were educated in these schools but they all said ‘we can’t help you with this even though it’s a good idea because we’d be killed politically for supporting “elitism” [fingers doing quote marks in the air], good luck I hope you succeed but we’ll probably attack you on the record.’ They mostly did.

The only reason why the King’s project happened is because Alison Wolf made it a personal crusade to defeat all the entropic forces that elsewhere killed the idea (with the exception of Exeter). Without her it would have had no chance. I found few equivalents elsewhere and where I did they were smashed by their VCs.

A few points…

1) Kolmogorov-type schools are a particular thing. They undoubtedly work. But they are aimed at a small fraction of the population. Given what the products of these schools go on to contribute to human civilisation they are extraordinarily cheap. They are also often a refuge for children who have a terrible time in normal schools. If they were as different to normal kids in a negative sense as they are in a positive sense then there would be no argument about whether they have ‘special needs’.

2) Don’t believe the rubbish in things like Gladwell’s book about maths and IQ. There is now very good data on this particularly in the form of the unprecedented SMPY multi-decade study. Even a short crude test at 11-13 gives very good predictions of who is likely to be very good at maths/physics. Further there is a strong correlation between performance at the top 1% / 1:1,000 / 1:10,000 level and many outcomes in later life such as getting a doctorate, a patent, writing a paper in *Science* and *Nature*, high income, health etc. The education world has been ~100% committed to rejecting the science of this subject though this resistance is cracking.

*This chart shows the SMPY results (maths ability at 13) for the top 1% of maths ability broken down into quartiles 1-4: the top quartile of the top 1% clearly outperforms viz tenure, publication and patent rates. *

3) The arguments for Kolmogorov schools *do not translate to arguments for selection in general* – ie. they are specific to the subject. It is the structure of maths and the nature of the brain that allows very young people to make rapid progress. These features are not there for English, history and so on. I am not wading into the grammar school argument on either side – I am just pointing out a fact that the arguments for such maths schools are clear but should not be confused with the wider arguments over selection that involve complicated trade-offs. *People on both sides of the grammar debate should, if rational, be able to support this policy*.

4) These schools are not ‘maths hot houses’. Kolmogorov took the children to see Shakespeare plays, music and so on. It is important to note that teaching English and other subjects is normal – other than you are obviously dealing with unusually bright children. If these children are not in specialist schools, then the solution is a) specialist maths teaching (including help from university-level mathematicians) and b) keeping other aspects of their education *normal*. Arguably the greatest mathematician in the world, Terry Tao, had wise parents and enjoyed this combination. So it is of course possible to educate such children without specialist schools but the risks are higher that either parents or teachers cock it up.

5) Extended wisely across Britain they could have big benefits not just for those children and elite universities but *they could also play an important role in raising standards generally in their area* by being a focus for high quality empirical training. One of the worst aspects of the education world is the combination of low quality training and resistance to experiments. This has improved since the Gove reforms but the world of education research continues to be dominated by what Feynman called ‘cargo cult science’.

6) We also worked with a physicist at Cambridge, Professor Mark Warner, to set up a project to improve the quality of 6th form physics. This project has been a great success thanks to his extraordinary efforts and the enthusiasm of young Cambridge physicists. Thousands of questions have been answered on their online platform from many schools. This project gives kids the chance to learn *proper problem solving* – that is the core skill that the corruption of the exam system has devalued and increasingly pushed into a ghetto of of private education. Needless to say the education world also was hostile to this project. Anything that suggests that we can do much much better is generally hated by all elements of the bureaucracy, including even elements such as the Institute of Physics that supposedly exist to support exactly this. A handful of officials helped us push through projects like this and of course most of them have since left Whitehall in disgust, thus does the system protect itself against improvement while promoting the worst people.

7) This idea connects to a broader idea. Kids anywhere in the state system should be able to apply some form of voucher to buy high quality advanced teaching from outside their school for a wide range of serious subjects from music to physics.

8) One of the few projects that the Gove team tried and failed to get going was to break the grip of GCSEs on state schools (Cameron sided with Clegg and although we cheated a huge amount through the system we hit a wall on this project). It is extremely wasteful for the system and boring for many children for them to be focused on existing exams that do not develop serious skills. Maths already has the STEP paper. There should be equivalents in other subjects at age 16. There is nothing that the bureaucracy will fight harder than this and it will probably only happen if excellent private schools decide to do it themselves and political pressure then forces the Government to allow state schools to do them.

Any journalists who want to speak to people about this should try to speak to Dan Abramson (the head of the King’s school), Alison Wolf, or Alexander Borovik (a mathematician at Manchester University who attended one of these schools in Russia).

It is hopeful that No10 is backing this idea but of course they will face determined resistance. It will only happen if at least one special adviser in the DfE makes it a priority *and* has the support of No10 so officials know they might as well fight about other things…

This is the most interesting comment probably ever left on this blog and it is much more interesting than the blog itself so I have copied it below. It is made by Borovik, mentioned above, who attended one of these schools in Russia and knows many who attended similar…

‘There is one more aspect of (high level) selective specialist mathematics education that is unknown outside the professional community of mathematicians.

The schools have nothing in common, with the exception of being unique, each one in its own way.

The visual processing module in the brain crunches 10,000,000,000 bits per second.

I offer a simple thought experiment to the readers who have some knowledge of school level geometry.

The best kept secret of mathematics: it is done by subconsciousness.

Mathematics is a language for communication with subconsciousness.

* frequently use language which is very fluid and informal;

* improvised on the spot;

* includes pauses (for a lay observer—very strange and awkwardly timed) for absorbtion of thought;

* has almost nothing in common with standardised mathematics “in print”.

However, specialist mathematics schools form a continuous spectrum from just ordinary, with standard syllabus, but good schools with good maths teachers to the likes of Louis-le-Grand and Fazekas. My comments apply mostly to the top end of the spectrum. I have a feeling that the Green Paper is less ambitious and does not call for setting up mathematics boarding schools using Chetham’s School of Music as a model. However, middle tier maths school could also be very useful — if they are set up with realistic expectations, properly supported, and have strong connections with universities.’

A Borovik

Absolutely – it’s a no-brainer, Dominic!

IMO we cannot afford the typical UK false-pride in maths-ignorance, nor the PC resistance to excellence, even if it does get labelled “fingers-elitism-fingers”.

Relatedly: I was somewhat taken aback by a BBC2 TV series on China a few years ago, in which it was stated (I have no independent confirmation but no reason to doubt either) that entry into the top Chinese universities requires the equivalent of A Level maths regardless of the student’s main subject – non-STEM as well as STEM.

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The false pride in mass ignorance of maths is certainly a problematic attitude in UK. But I don’t see how creating a set of institutions for the highest attainers will address that. If anything it will do the opposite.

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I know this maynot be relevant but my son is a Maths teacher who lived abroad and taught in senior schools for 14 years. He loved teaching. However he decided to return to the UK did one year in a SEN school, and then started at a Main stream school in Leicester. He hated it so much he has given up teaching. He said some of the kids were “feral” rude, no interest in learning, and he found it very stressful. Some were so disruptive they ruined the lesson for those who did want to learn. That and the amount of work he had to do out of school hours was totally demoralising him. He then taught at a Further Ed. College for abit, and felt it a waste of time to be trying to teach maths to those who were having to do resits as they had failed GCSE before- and were totally disinterested. Until these problems can be addressed and pupils realise how lucky they are to have all the facilities in schools here, until their parents back up the teachers and take some interest in their own kids education, until teachers and schools have more power and discipline, the UK will continue to be low down in Ed. standards in the world. In poorer countries people value their education- it’s a route out of poverty, but here our welfare system is too generous…why should they bother. In Mauritius (where he lived) the level of Maths of children of the same age, was much higher than it is here. There is too much pressure on teachers and for what they do- they are not paid enough.

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Dear Lydia, you write, and rightly so: ” Until these problems can be addressed and pupils realise how lucky they are to have all the facilities in schools here, until their parents back up the teachers …” IMHO, it could happen that this time will never come. I have a whole paper on that issue — and, with apologies for self-promotion, here is a link: goo.gl/TT6ncO

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Thanks for your insightful and encouraging post, Dominic.

I see the distinction you make in this post between the justification for elite Maths/Phys/Comp schools vs elite Humanities schools. Nevertheless, have you written elsewhere on what a Humanities curriculum for such “future leaders” might look like? Your (excellent) Odyssean essay had some interesting thoughts about govt/politics reading, but what would your literature / history / ethics / other arts curriculum consist of? Do the top private schools offer a useful model, or could something more specific and thoughtful be devised?

Will

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There is an aspect of mathematics / physics education of “mathematically able” children which is almost never mentioned: “mathematically inclined” (my preferred term) children have high capacity to learn by absorption. This trait remains dormant in normal school environement but gets activated when kids find themselves surrounded by children *like them*. My university has a large and vibrant community of maths PhD students, and it is a place where learning by absorption can be observed “in the wild”. It is less known that the same could happen with a certain kind of 14-16 years old kids when they are put together in the same learning environment. This, I think, gives a partial answer to your question: “what would your literature / history / ethics / other arts curriculum consist of?” IMHO, the curriculum has to be diverse, should allow different children to be successful in different subjects / arts and learn from each other by absorbtion. Why? Because mathematicians and physicists are stem cells of a technologically advanced society, they have to be re-educatable, able to change their role, metamorphose — and inevitaly be autodidacts in the process. Indeed, who will teach them in their professional future? They have to teach themselves and learn from each other.

I wonder if Dominic would agree with me that perhaps his physicists exhibited this specific stem cell behaviour.

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I strongly agree with this and with your stem cell analogy – your guess is right that the physicists who worked on the campaign operated exactly like this!

Best wishes

d

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There is one more aspect of (high level) selective specialist mathematics education that is unknown outside the professional community of mathematicians.

I am not an expert on “gifted and talented” education. On the other hand, I spent my life surrounded by people who got exclusive academically selective education in mathematics and physics, whether it was in the Lavrentiev School in Siberia, or Lycée Louis-le-Grand in Paris, or Fazekas in Budapest, or Galatasaray Lisesi (aka Lycée de Galatasaray) in Istanbul — the list can be continued.

The schools have nothing in common, with the exception of being unique, each one in its own way.

I had research collaborators and co-authors from each of the schools that Ilisted above. Why was it so easy for us to find a common language?

Well, the explanation can be found in the words of Stanislas Dehaene, the leading researcher of neurophysiology of mathematical thinking:

“We have to do mathematics using the brain which evolved 30 000 years ago for survival in the African savanna.”

In humans, the speed of totally controlled mental operations is at most 16 bits per second. Standard school maths education trains children to work at that speed.

The visual processing module in the brain crunches 10,000,000,000 bits per second.

I offer a simple thought experiment to the readers who have some knowledge of school level geometry.

Imagine that you are given a triangle; mentally rotate it about the longest side. What is the resulting solid of revolution? Describe it. And then try to reflect: where the answer came from?

The best kept secret of mathematics: it is done by subconsciousness.

Mathematics is a language for communication with subconsciousness.

There are four conversants in a conversation between two mathematicians: two people and two their “inner”, “intuitive” brains.

When mathematicians talk about mathematics face-to-face, they

* frequently use language which is very fluid and informal;

* improvised on the spot;

* includes pauses (for a lay observer—very strange and awkwardly timed) for absorbtion of thought;

* has almost nothing in common with standardised mathematics “in print”.

Mathematician is trying to convey a message from his “intuitive brain” directly to his colleagues’ “intuitive brain”.

Alumni of high level specialist mathematics schools are “birds of feather” because they have been initiated into this mode of communication at the most susceptible age, as teenagers, at the peak of intensity of their socialisation / shaping group identity stream of self-actualisation.

In that aspect, mathematics is not much different from arts. Part of the skills that children get in music schools, acting schools, dancing school, and art schools is the ability to talk about music, acting, dancing, art with intuitive, subconscious parts of their minds — and with their peers, in a secret language which is not recognised (and perhaps not even registered) by uninitiated.

However, specialist mathematics schools form a continuous spectrum from just ordinary, with standard syllabus, but good schools with good maths teachers to the likes of Louis-le-Grand and Fazekas. My comments apply mostly to the top end of the spectrum. I have a feeling that the Green Paper is less ambitious and does not call for setting up mathematics boarding schools using Chetham’s School of Music as a model. However, middle tier maths school could also be very useful — if they are set up with realistic expectations, properly supported, and have strong connections with universities.

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^There is one more aspect of (high level) selective specialist mathematics education that is unknown outside the professional community of mathematicians.^

Well not entirely. I’m not any kind of professional mathematician or directly involved in education, but I do have a 13-year-old daughter who is one of the “couple of children” in an eight-form-entry year (~230) at the local Middling Comprehensive.

Like many parents in my shoes, I began by fretting about stretch & challenge and not enough maths- stretched across five years to the GCSE in Y11. Then I gradually realised she loved maths lessons with or without much serious challenge. That is because she sits and works with the other child and they understand each other very well. Character matters too given more ‘quirky’ natures at that end of the curve, but these two are similar enough. I regularly praise the gods for putting this pair together in the same top set class in a school with a parallel set structure and two top sets (no sign of their ilk in the other one). We could easily have had two quite isolated, bored children in different top sets instead of the synergy and that fragmented/intuitive way of discussing maths

Waiting until 16 is too long, but boarding schools or significant travel will filter out too many. I’ve converged on virtual classrooms as an imperfect ‘quick win’ that should reach more of them. Look around today and the right kind of tech is sitting there now e.g. Microsoft gave collaborative working Teams to education a few weeks ago. It could just as easily be Google. Then look at children of my daughter’s age, especially at this end of the maths/physics curve and they are significantly more comfortable with tech than children who are just a few years older. If this has been tried then I think it’s worth trying again in the brave new mobile/cloud world, but only if the system accommodates consistent classes and between-child comms.

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Many thanks Professor Borovik. I work in education, and have observed the same behaviour – I wonder if this trait could be harnessed by auto-didactic platforms like Khan Academy?

I’m still interested in whether anyone has given thought to an ethics course specifically designed for the cognitive elite? I know Charles Murray has some interesting thoughts (centred on humility and rhetoric – p109 onwards here: http://emilkirkegaard.dk/en/wp-content/uploads/Charles-Murray-Real-Education.pdf) but have not come across much else.

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I would recommend papers in The De Morgan Journal (ISSN 2049-6559):Volume 2, Issue 2: “Specialist Mathematics Schools and Education of “Mathematically Able” Children”, they provide some insider information from advanced level specialist mathematics schools around the world.

A. D. Gardiner, Introduction, pp. 1-4. bit.ly/2iWdyih

M. Lemme, Utter elitism: French mathematics and the system of classes prépas, pp. 5-22. bit.ly/2jJDYRs

A. V. Borovik, “Free Maths Schools”: some international parallels, pp. 23-35. bit.ly/2jhXHcA

D. Yumashev, ZFTSh: A specialist correspondence school, pp. 37-41. bit.ly/2j5Rxsz

P. Tanovic, Matematicka Gimnazija, pp. 43-46. bit.ly/2jib8cr

P. Juhász, Hungary: Search for mathematical talent, pp. 47-52. bit.ly/2iW8mLg

F. Truong and G. Truc, ‘Studying in a prépa as surviving in hell’: untold episodes from a mythical media tale, pp. 53-61. bit.ly/2k94vHQ

D. Pierce, St. John’s College, pp.63-73. bit.ly/2jTVhSG

A. V . Borovik and A. D. Gardiner, Mathematical abilities and mathematical skills, pp. 75-86. bit.ly/2jTYy4r

A. D. Gardiner, Nurturing able young mathematicians, pp. 87-96. bit.ly/2jR3nLo

Acceleration or enrichment: Report of a seminar held at the Royal Society

on 22 May 2000, pp. 97-125. bit.ly/2jpdHqW

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Anybody interested in the detail should definitely read these papers.

The ‘utter elitism’ one on French maths schools is particularly interesting / well-written / amusing…

Thanks A

d

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Alison Wolf is still involved with the school as governor and Committee Chair, but the Head Teacher is Dan Abramson and the chair is now David Benello.

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Interesting stuff, thanks for sharing. Could you expand on point 5, that they can raise standards generally in their local area. How does removing high attainers from schools help improve standards? In terms of training, I would expect the experience of teaching at one of these schools to be very different from teaching at a mainstream school. (I’m basing that on one colleague who taught at Kings). So I’m not sure how valiable that experience would be.

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Considering that Nick Gibb’s proposal to test Yr 6 pupils for automatic recall of number bonds for multiplication has been shelved, I don’t think the omens are good. We recently tested 23 Yr 10 pupils in Burnley comp (rated ‘good’ by Ofsted) and found that none of them had achieved automaticity in number bonds for addition. The only one who came close was Polish. Yet after 10 training sessions of 15 minutes each with flashcards and worksheets, 7 of them achieved a response time of 1.5 seconds or less, and a further three were almost there with response times of under 2.0 seconds. Even the least able pupils made impressive gains. Ironically, this was carried out by a Chemistry teacher–the Maths department didn’t want to know.

As for our most able pupils, I expect that private funding of supplementary education for exceptional pupils is the only thing that stands a chance, simply because no one could actually stop it. Private school vouchers in the US–which were heavily supported by the Walton Family Foundation–are a precedent of sorts.

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The west would benefit from lowering the official status of math so social justice types don’t get upset if high IQ students learn math at their own pace. If we could pretend that doing math is more like playing an instrument then nobody’s feelings would be hurt if a few kids get really good at it. If you claim that math is really important the left will fight against some kids getting more than others, whereas if you say math is a specialized subject not dependent on general intelligence (which of course is false ) then the left with fight for educational equality of outcome elsewhere.

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I’d like to get a group of like-minded people together to write a proposal for a multi-academy trust with a national maths boarding school as the flagship school, with regional maths/physics/com sci boarding schools in the U.K. Anyone who is interested in this or similar ideas please email me: paul.youlten@gmail.com

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I fully agree with your prescription here. However, if maths held the key to development, how do you explain the relative performance of the French and UK economies over recent years. French education at all levels is much more focussed on maths – but France does not seem to be doing much better in economic terms. The same can also be said for Russia and the UK.

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In my final year at school the maths class was only three of us, taught in the Oxford tutorial style. Two of us were jolly good at maths but the third was The Real McCoy. We should have been split into two streams.

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If I read the chart correctly, most of the top 1% of maths aptitude have only an average chance of being in the top 5% of incomes – rising to about 10% for the top 0.25%. Those figures would, I am sure, compare very badly with those of economics graduates. Obviously money is not everything, perhaps particularly to this group (Perelman declining the 1million prize!) but still speaks to recognition and incentives.

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Very interesting piece and thought the comment about children who are unusually gifted being special needs was absolutely right. When I had to wade through a PCAP course at the last university where I taught I did my mini research project on this topic and pointed out that our library had nine shelves of books on teaching those who fell below educational standards but just three books (the most recent published in 1965) on how to deal with unusually gifted children. With utter predictability I was told I was being elitist.

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