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.

Smolny Beyond Borders

Introduction to Liberal Arts Mathematics

I placed the two first lines of this famous John Donne’s poem in my mathematics paper as an epigraph. This is the whole poem:

No man is an Iland, intire of itselfe; every man
is a peece of the Continent, a part of the maine;
if a Clod bee washed away by the Sea, Europe
is the lesse, as well as if a Promontorie were, as
well as if a Manor of thy friends or of thine
owne were; any mans death diminishes me,
because I am involved in Mankinde;
And therefore never send to know for whom
the bell tolls; It tolls for thee.

This is MEDITATION XVII from
“Devotions upon Emergent Occasions”
by John Donne, 1623

The date is unbelievable: 400 years ago.

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.

Added on 19 April 2023:

I got a response to my text from a good colleague of  mine, Bill Lionheart, a brilliant, and really applicable, applied mathematician (inverse problems and mathematical tomography, with a fantastic range of applications, from medical stuff to control of any kind of industrial bulk material flows in pipes, say, grain in grain pumps, to airport security scaners to land mine clearance) .

Bill run  some bits of my my text by ChatGPT and got this reply:

ChatGPT and Wolfram Alpha are two technological innovations that have had a significant impact on mathematics education, particularly in A-level math. These tools have revolutionized the way students learn and solve mathematical problems, providing them with a wealth of information and resources to help them excel in their studies.

ChatGPT is a large language model that has been trained on a vast corpus of data, including textbooks, articles, and other relevant information. It is designed to understand natural language input and generate human-like responses to queries. ChatGPT has made it easier for students to access information about mathematical concepts, theories, and formulas quickly. It can answer questions on any topic related to mathematics, from algebra and geometry to calculus and statistics.

With ChatGPT, students no longer need to spend hours flipping through textbooks or browsing the internet for answers. They can simply ask the tool a question, and it will provide a detailed explanation or solution. This has made math more accessible to a wider audience, including those who previously struggled with the subject.

Wolfram Alpha, on the other hand, is a computational knowledge engine that provides answers to a wide range of mathematical questions. It uses a vast database of algorithms and data to provide accurate and reliable answers. Wolfram Alpha can perform a wide range of mathematical operations, including symbolic and numerical computations, solving equations, and graphing functions.

Wolfram Alpha has made it easier for students to visualize and understand complex mathematical concepts. For example, students can enter an equation, and Wolfram Alpha will generate a graph of the function, making it easier to see its behavior and understand its properties. This has helped students to better understand the relationship between different mathematical concepts and apply them to real-world problems.

The impact of ChatGPT and Wolfram Alpha on A-level mathematics has been significant. These tools have helped to democratize access to mathematics education by providing students with easy access to information and resources. They have also made math more engaging and interactive, allowing students to explore mathematical concepts in a more meaningful way.

Moreover, these tools have also helped to improve student performance in math. With ChatGPT and Wolfram Alpha, students can quickly and easily check their work, ensuring that they have not made any errors. This has helped to reduce the number of mistakes made by students, leading to better grades and more confidence in their abilities.

Alexandre Borovik

18 April 2023

www.borovik.net/selecta

The Electrician, by Boris Eldagsen. This AI-generated image winning a prestigious Sony world photography award, https://www.theguardian.com/technology/2023/apr/17/photographer-admits-prize-winning-image-was-ai-generated.

My answer on Quora: Are mathematicians gifted people?

I do not know for what 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 outdatet, 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) pre-school 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. bit.ly/2qYHtst

A. V . Borovik and A. D. Gardiner, Mathematical abilities and mathematical skills, The De Morgan Journal 2 no. 2 (2012) 75-86. bit.ly/2jTYy4r

Tony Gardiner writes:

The mathematical “house” is fortunate in having “many mansions”, inhabited by a remarkable variety of workers. By any account Geoffrey Howson – who died on 1 November 2022, aged 91 – was a significant player throughout much of the period 1950-2000. However, like so many other workers, he operated effectively, but quietly, so may not have been noticed. Nevertheless his life offers interesting insights into how UK society has changed since 1931 (when he was born as the seventh in a family of seven children), and into how UK mathematics and mathematics education worldwide have evolved since the 1950s.

Geoffrey belonged to the generation, who emerged in significant numbers (perhaps for the first time) in the 1940s. Their families had never been to secondary school – let alone university. Yet – thanks to structural changes and committed teachers – they somehow emerged in small numbers at age 18, ready to take on whatever challenges the post-war world might present.

Geoffrey always remained faithful to his roots (a solid Yorkshireman, from a deprived, but proud, mining community). Yet he came to excel in mathematics, in university politics, and in international mathematics education – as well as in the world of opera, Bauhaus design and embroidery, and medieval church architecture.

Geoffrey went to Castleford Grammar School (founded 1906), and was probably the first from that school to study mathematics. He went on to Max Newman’s department in Manchester, where his teachers included: Max, Walter Ledermann, J.W.S. Cassels, Bernhard Neumann, Graham Higman, Kurt Mahler, Arthur Stone, James Lighthill (MA President 1970), M.B. Glauert, Charles Illingworth, and Bernard Lovell. He was Graham Higman’s second PhD student (proving that the intersection of two finitely generated subgroups in a free group is finitely generated). He also attended Turing’s lectures on morphology – interrupted only by Turing’s death.

Invitations from Reinhold Baer (Illinois) and Saunders Maclane (Chicago) were put aside in order to complete National Service (when he taught RAF trainees about guided missiles). He then moved to the Royal Naval College in Greenwich in 1957 (where he taught the new generation of future naval commanders about similar things).

In 1962 he went to Southampton to manage the School Mathematics Project (SMP). This was the UK equivalent of “new math”, but much more humane and less abstract. At its height SMP materials were “used” (in some sense) in 60% or so of UK secondary schools. But SMP remained a Teachers’ Cooperative, with no government support. Geoffrey’s job was officially to edit and to manage the program of new textbooks. In practice, he had to coordinate the writing (planned and completed by a remarkable group of full-time teachers); the production of draft materials; the revision process; and to deal with the publishers and the exam boards (since no project could survive if there was not a corresponding tailored public examination at age 16 and 18).

Geoffrey became a representative spokesperson for “modern maths” developments in the UK, and so came to interact with those similarly placed in other countries – in both East and West – producing many reports, and editing collections published in the 1960s, 70s, and 80s. He published and edited a huge variety of books and papers – all written in a thoughtful style. His goal was to inform and enlighten, rather than to engage in “theoretical research”. He became a leader in Mathematics Education internationally, but was never really appreciated by the new breed of “Maths Education” researchers. His contributions were mostly pragmatic comparisons, surveys, and analyses, designed to inform and to allow improved judgements to be made. He was also very active in supporting teachers’ colleges and those working in polytechnics.

He helped to salvage ICMI/ICME after it came unstuck around 1980. And it is a mark of the man that he managed this (with Jean-Pierre Kahane) while remaining great friends with those who had been part of the previous regime. He was recognised in other countries but not much within the UK.

He was Head of Department and Dean 1990-92 and may have helped in building up parts of what is now a very strong mathematics department. He also Chaired the LMS/IMA/RSS committee that produced the report “Tackling the mathematics problem”: this was a rare instance of the three scholarly societies acting together on a matter of mutual concern, and then having a significant impact on subsequent policy-making.

They don’t make them like that any more.

My answer to a question on Quora:  Which are the procedures you are following to memorize or learn so fast?

One my favorite procedure is called “understanding”. Prior to memorising, try to understand. Very frequently, the need to memorise disappears – you simply understand.

Another procedure that I try to popularise is called “interiorisation”: making something external a part of yourself.

I remember my conversation with my grandson, at that time 5 years old. I asked him: “What’s new at your new school”. He: “I do not like the school. Too many rules. Hard to remember.” I: “Why should you remember the rules? You simply have to follow them. I’ve seen in the lunch room at your school a poster:”Do not mess with food”. What is so difficult about it?”

My grandson fell in deep thought. We walked along a street on our way to his home, in silence. I didn’t distract him. It should be a law of the land: never distract young children when they think.