Tony Gardiner died suddenly on 22 January 2024. He will be remembered as a national treasure, a man who made a unique contribution to the development of mathematics education in this country and internationally.

Tony set up, and made significant contributions to the work of, the UK Mathematics Trust, which runs problem-solving challenges taken by over half a million students every year. Tony was the Team Leader of the British IMO team in 1990–95 – and a mentor of many bright young mathematicians who are now the crème de la crème of British academia. For many years, he edited the Problem Solving Journal for Secondary Students, with a circulation over 5,000. He wrote and published more than 15 books on mathematical thinking and mathematical problem solving – as well as on teaching mathematics. He was consulted by several UK Ministers of State for Education, and acted as an advisor on mathematics education to the government of Singapore. More can be said about Tony’s contribution to this world, but there is no need to compete with Wikipedia where the article devoted to him, https://en.wikipedia.org/wiki/Tony_Gardiner, is being feverishly updated.

Tony started his work in mathematics in the 1970s. He was a PhD student of the legendary Bernd Fischer, who had just discovered his three sporadic finite simple groups. It was a very fruitful time, when group theoretic ideas were becoming widely applied in combinatorics. Tony’s further research was very successful and mostly straddled the two areas of combinatorics and permutation groups.

At the same time Tony started to forge an unusual path in combining research in mathematics with a commitment to high school and undergraduate mathematics. His instinct as a researcher led him to investigate the actual working of mathematics education as a system and look at the entire cycle of reproduction of mathematics: from preschool and primary school through all stages of school education to university to teacher training and then back to school as a teacher. This is also augmented by a smaller cycle: BSc – PhD studies and postdoctoral research, then back to university as a lecturer. This breadth of vision was supplemented by his attention to the socio-economic and political background of education and placed him in a very special position among British mathematics educationalists.

Tony had exceptional academic and intellectual integrity. He was very modest. And, above all – he was a very kind man always helping a talented student or a bright school child who needed help, and did so right up to the last days of his life.

A letter to mathematics and computer science colleagues

Dear Colleagues,

Very recently I wrote to a few friends saying that I expected ChatGPT in its next version becoming able to solve every algebra and calculus problem in A Level (the end of school exams in England) and similar school exams in other countries. For that, ChatGPT simply should be shown how to identify what looks as an algebraic, logarithmic, differential etc. equation or a system of equations or inequalities and plug this thing into one of already existing maths problems solvers, for example, the Universal Math Solver, https://universalmathsolver.com/ — it does more than finding an answer, it produces a complete step-by-step write-up of a solution.

But this important symbolic threshold was passed much earlier than I expected. Conrad Wolfram posted on his blog on 23 March an announcement “Game Over for Maths A-level”, https://www.conradwolfram.com/writings/game-over-for-maths-a-level. A quote:

“The combination of ChatGPT with its Wolfram plug-in just scored 96% in a UK Maths A-level paper, the exam taken at the end of school, as a crucial metric for university entrance. (That compares to 43% for ChatGPT alone).”

This means that undergraduate pre-Calculus and Calculus undergraduate exams will follow quickly.
I think it is dangerous to sit and wait while we are overrun by events. I suggest that we have to address the issues on the global scale: changes in the technological and socio-economic environments of education will soon affect hundreds of millions of children in dozens of countries and later become truly global. It is the scale of the problem which is the issue.

There is nothing special in the ChatGPT, it is only one of a dozen AI systems of enhanced functionality which have suddenly appeared on the market. They are pushed by some of the mightiest transnational corporations to the market where, unlike many other markets, the rules of the supply-side economics apply in their full strength (remember the story of iPod? Or selfie sticks?). It does not matter, what we think and feel about the AI: very soon, it will be everywhere around us. It was Marx who said “supply takes demand, if necessary, by force”. A classical example, which is likely to be reproduced in the case of AI, is the multibillion pet food industry: the concept of pet food was invented and forced on people (now called, in TV commercials, “pet parents”) in the late 1950s by the American meat packing industry which by that time completely saturated the American market (for human consumption) and looked for new directions to expand. For billions of people around the globe, AI will become an intellectual pet food for the masses. And we have to take into account that the supply-side push of the AI on people, is likely to be a total assault, in all spheres of human activity, much wider than education.

In many countries, politicians, state bureaucrats, theoreticians of mathematics education, and school teachers led by them, made everything possible to turn students into a kind of biorobots trained for passing school exams. And here comes the moment of truth: if real robots pass exams with much better marks — what is the purpose of the current model of mathematics education?

And we should not be distracted by general philosophical questions of the kind “can machine learning produce sentient beings?” The real, and immediate issue, is the disruption which will be caused by still non-sentient AI in the human society (made of sentient beings).

It is interesting to glimpse a politician’s view of these issues. Please see below some examples of uses of mathematics as given by Rishi Sunak, Prime Minister of the UK, in his speech on improving attainment in mathematics, 17 April 2023, https://www.gov.uk/government/speeches/pm-speech-on-improving-attainment-in-mathematics-17-april-2023 . Interestingly, the speech was given at the London Screen Academy – this is why examples start with “visual effects”, etc.

You can’t make visual effects without vectors and matrices.

You can’t design a set without some geometry.

You can’t run a production company without being financially literate.

And that’s not just true of our creative industries. It’s true of so many of our industries.

In healthcare, maths allows you to calculate dosages.

In retail, data skills allow you to analyse sales and calculate discounts.

And the same is true in all our daily lives…

… from managing household budgets to understanding mobile phone contracts or mortgages.

With a possible exception of the first line (about visual effects¹), all that in 5 (or at most 10) years from now will be done by a combination of AI and specialist mathematics (or maybe accounting) tools — and done much better than 90% of people can do. For example, an app on a smartphone which has access to all financials accounts of the owner – bank accounts, credit cards, tax account, mortgage, etc. and linked to powerful AI servers on the Internet, will be able to take care of household budgets. This app will ask the user, after each contactless payment in the shop, under which heading this payment should be entered in the ledger of the household budget, offering most likely options (maybe deducing them from the shops’ names, like Mothercare or Bargain Booze).

It is widely accepted now that in most areas of human activity ChatGPT and other AI systems are no more than imposters faking answers to questions they do not understand.

However, routine mathematical by their nature tasks of household budgeting, etc. are likely to be important exceptions — because they are intrinsically well structured and less ambiguous. And AI paired with mathematical problem solving software will pass standard school exams better than students or their teachers can do.

I summarise the situation in three bullet points:

• What we see now is a slow motion car crash of the traditional model of mathematics education. Sunak (and practically everyone in the area of education policy) are asleep at the wheel and do not see the road ahead. But in the education policy, we have to look at least 14 years ahead – this is the length of school education (in the UK), from 4 to 18 years of age.
• Most politicians are able to think ahead only on the time scale of the election cycle, 4 or 5 years. They cannot comprehend the scale of quantities and magnitudes (the latter include time) involved in economic and social problems (and even less so in all the mess around the climate change).
• Most politicians lack basic skills of project management and do not understand that work on a serious project should start with the step-by-step reverse planning from the target to the present position.

This why I appeal to professional mathematicians and computer scientists:

Of all people involved in some way in mathematics /computer science education, you are perhaps the only ones free from mental handicaps listed in the three bullet points above. Let us discuss, at first perhaps only in our circle, this fundamental question:

What kind of mathematics education is needed in the era of AI?

Perhaps we have to split the question:

What kind of mathematics should be taught

(a) To future developers, controllers, masters of AI?

(b) To the general public, the users (and perhaps victims) of AI?

If these questions are not answered in our professional communities, we should not expect an answer coming from elsewhere.

Alexandre Borovik

18 April 2023

http://www.borovik.net/selecta

My answer on Quora to the question Are mathematicians gifted people?

I do not know for which my sins Quora bombards me with questions about giftedness, IQ, etc. For several years I tried to ignore them, but finally I realised that I have to formulate my position.

Yes, professional mathematicians possess some mental traits and skills which majority of population do not have. But these traits are not what is called “gift”, “talent”, “ability” in the mass culture; they remained unnoticed, unregistered in the public discourse about mathematics and mathematics education. However, my mathematician friends, when we discuss this topic, know what I am talking about.

IQ is mostly irrelevant to discussion of mathematical “ability”; specific traits of mathematical thinking belong to a much higher cognitive level than skills tested in IQ tests.

A simple example: I had seen once how an eight years old boy was solving some standard puzzle (not of IQ type), with some pattern of hexagons which had to be filled with integers from 0 to 9 so that certain sums were equal — you perhaps had seen this boring stuff . At some point he paused and commented: “Hmm, I have to somehow move information from this corner to that corner”. Moreover, after some thought he had successfully moved the information. This was meta-thinking, ability to reflect on one’s thinking, ability to look at the problem from above. This boy now is quite a successful student in one of the best university mathematics departments in the world, in a pipeline to becoming a professional research mathematician.

Perhaps you have heard this definition:

“Mathematics is the science of patterns”.

IQ tests pay much attention to the speed of pattern recognition. It is a useful skill, but it is not a sign of mathematical abilities. In my life, I had a chance to see a lot of children and teenagers who had an instinct (or maybe it was a trait absorbed in the family?), to look deeper and try to detect the structure behind the pattern — and the boy mentioned above was one of them. Indeed, the simplest description of mathematics is

“Mathematics is the science of structures behind patterns”.

Perhaps my personal experience is outdated, but I was privileged to go through a viciously academically selective system of mathematics education — see my paper “Free Maths Schools”: some international parallels. Aged 14, at a Summer School which was the final step of selection to the specialist boarding school described in the paper, I and my friends were subjected to a battery of IQ tests — which, however, had no relation to admission to the school.

We were tested by professional experimental psychologists who were commissioned by the Soviet Army to study and assess reliability of the IQ tests used by the US Army for assignment of conscripts to particular duties (you see how long ago it was). The psychologists translated real American IQ tests into Russian and tried them on various groups of population. They were excited to discover that our performance refuted a claim that apparently was universally accepted at that time: that practicing IT tests could not improve results.

Indeed our results were quickly improving beyond applicability of tables for conversion of counts of correct answers into IQ scores. Why? Because we did not practice IQ tests — we had access only to tests which we have already taken — but, after every test, we spent hours classifying test questions, analysing them, inventing our own questions and challenging each other to solve them, and we did that in a collective discussion, in brain storming sessions, attacking problems like a pack of enthusiastic young wolves. Perhaps we had already had some specific habits of mathematicians; but there was nothing special about that, even some 8 year old kids might have them, as I have already said.

As I explain in my paper that I mentioned above, in the selection process for my mathematics boarding school, and in the school itself, the use of words gifted, talented, able was explicitly forbidden — they were seen as misleading and divisive.

I am a staunch believer that majority (maybe even all) kids have strong potential for understanding and mastering mathematics. Unfortunately, their mathematical traits are systematically suppressed in the mainstream school mathematics education — mostly because many teacher have no idea what it is about.

You may wish to take a look at my papers, they say more:

A. V. Borovik,  Mathematics for makers and mathematics for users, in Humanizing Mathematics and its Philosophy: Essays Celebrating the 90th Birthday of Reuben Hersh (B. Sriraman ed.), Birkhauser, 2017, pp. 309–327.

A. V . Borovik and A. D. Gardiner, Mathematical abilities and mathematical skills, The De Morgan Journal 2 no. 2 (2012) 75-86.

My answer to a Question on Quora: How do I start studying mathematics from the beginning until I get the gold medal in the Mathematics Olympiad?

I know a number of my fellow mathematicians who won gold medals at the International Mathematical Olympiad ( I think you mean that Olympiad). I doubt that any of them got this medal after starting studying mathematics specifically with this aim. The only fruitful way to learn mathematics is because you are interested in mathematics, because you wish to understand mathematics, because you wish to learn to recognise that specific feeling of joy which comes from understanding mathematics.

I once met a young girl who told me that she loved mathematics because she felt that mathematics briought her closer to God. This is a good reason for studying mathematics, but it is essentially the same as the one just described by me, but expressed from a different viewpoint. But let us think a bit: does winning a gold medal brings you closer to God? I doubt that. Medal is from people, not the God.

My answer to a question in Quora: What is exact definition of mathematics?

I share my small collection of  descriptions of mathematics; some of them sound as attempts to define mathematics.

I. Mathematics is an exact language for description, calculation, deduction, modeling, and prediction — more a systematic way of thinking than a set of rules.

II. Using a legal analogy, mathematics is a language for writing contracts with Nature that Nature accepts as legally binding.

III. The practical importance of mathematics lies in its ability to describe the real world.

The real world consists of what matters. The word “matter” as a noun is used for what the physical world is made of. But if we ask, “What’s the matter with Anne?” we may be asking about a physical ailment, or we may be asking about an idea that is causing Anne to behave strangely. Ideas matter.

The whole point of mathematical education is to make ideas real for students, ideas that were not real for them before. Ideas like fractions, for example. The fact that 2/3 is smaller than 3/4 matters in the real world.

IV. Mathematically educated people are stem cells of a technologically advanced society. Because of the universality of mathematics, mathematicians and well educated users of mathematics are flexible in applying and inventing tools for work in technological environments which never existed before

V. Learning mathematics involves the profound assimilation of intellectual and aesthetic criteria as well as practically orientated ones. The very difficulty in learning mathematics makes it a personality-enhancing experience.

[The above are borrowings from David Corfield, Tony Gardiner, Michael Gromov, Frank Quinn, David Pierce — I do not remember now in which order.]

VI. Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap. — Vladimir Arnold

VII. Mathematics is the music of reason  — James Joseph Sylvester

VIII.  Mathematics is the study of mental objects with reproducible properties. — Philip J Davis and Reuben Hersh

VIII. Finally, my own extension of the thesis by Davis and Hersh:

Mathematics the study of mental constructs with reproducible properties which imitates the causality structures of the physical universe but is expressed in the human language which evolved for social interactions.

My answer to a question on Quora:

Why is math taught differently in school today? What is wrong with the way we learned it twenty years ago?

New technological and economic environment requires a different set of mathematical skills, and taught differently, and — this is the big unmentionable — taught to much smaller number of students. What we see are death throes of the existing system of mathematics education. Politicians are keen to keep the old system of mathematics education for everyone, and for good reason — this is what their electorate expects from them.

This is why there is so much confusion of what, and now, has to be taught at school – in absence of clear economic criteria, anything goes. I heard statements, made by professional mathematics educationalist at meetings at the Department of Education, of the kind: “Why school curriculum should contain fractions? Who of people present here had lately add 3/4 and 7/5? Read more in my papers Mathematics for makers and mathematics for users and Calling a spade a spade: Mathematics in the new pattern of division of labour.

My answer to a question on Quora:

An excellent question. Category theory is important on its own, and has important applications in a number of other mathematical theories; however, the crucial and the most fundamental impact of category theory is invisible and under-reported, it is of cultural nature. It can be compared with the influence of set theory: 99% of mathematicians use only a modicum of naive set theory, ignoring deeply penetrating and frequently very hard results of set theory as the live research discipline which continues to develop and flourish.

There is a telling example: the theory of games of chance was created in the 17th century and gave birth to probability theory; the latter was already quite developed by the time when, in the 20th century, the concept of a deterministic game had finally crystallized – in a paper, of all people, by Zermelo, who proved that chess was a deterministic game: for one of the players, there is a strategy, that is, a function from the set of permissible position to the set of moves, which achieves at least a draw. His paper was published in 1913 (see Zermelo’s theorem (game theory) – Wikipedia). Why did this happen so late? The word “set” in the definition of a strategy came to use only in the second half of the 19th century.

The same is happening with category theory: the vast majority of mathematicians use its ideas and terminology in a very rudimentary and naive form, frequently even without realisation that they are doing so. In the work that I am doing, it had happened to be very important to remember that an algebraic group was a functor from the category of unital commutative rings to the category of groups. Some my colleagues who work in the same theory continue to insist that a group is a fixed set with some operations on it. When my co-author and I recently solved a certain problem which was open since 1999, we were able to do that only because for us a group in question was a functor – not much deeper than that. Why it was not solved by someone else earlier? Because for them a group was just a set.

My answer to a question on Quora:

The answer is very simple: Statistics and Mathematics have different sources of funding. From time to time, mathematical disciplines get separated from mathematics because their end users (industry, the military, the government) wish to pay only for the end product, and do not wish to subsidise its mathematics roots. This has happened with computer science, cryptography, fluid dynamics, etc. — and statistics is just one of many examples. There is no any intrinsic reason for all these disciplines to be separate from mathematics.

Below is a fragment from a book written by my university lecturer in mathematical statistics on the bases of lecture notes taken by his PhD students (they were attending for that purpose our undergraduate lectures). This is the very first chapter, the first definition: The notion of a sample.

As you can see from this example, mathematical statistics can be the purest of pure mathematics.