I do, on the basis of personal experience in mathematics.

I  retired after 50 years of teaching in universities in 4 different countries with different education systems and pedagogical traditions. This experience shaped my views on our profession. I firmly believe that

1. Teaching is not a science, it is an art, and should be treated as such.
2. Students are not customers (“persons who buy”) – they are clients (“persons who seek the advice of a professional man or woman”).
3. “Good learning experience” means mastering something new and advanced. To help his/her students, a university teacher has to be able to transform and restructure highly complex material from his/her subject area into a form suitable and accessible to the learners.
4. This cannot be achieved without teachers being experts in their disciplines.
5. Successful and inspirational teaching is a highly individual skill. The choice of teaching methods should reflect not only specifics of the target audience, but also the experience, teaching philosophy and individual psychophysiological characteristics of the teacher.
6. Structuring of the learning environment, choice of teaching and assessment methods have to be subject specific.
7. Values, standards, criteria of assessment in learning and teaching have to originate in, and be set by, the professional academic communities of their particular subject areas.
8. The role of managers is to create an environment which helps professional standards to be maintained; however, managers should not interfere in setting the standards.

This Manifesto was published in a an old blog. I wish to transfer from there a comment from one of the leaders of the math circle movement. It describes a creative learning environment flourishing in math circles but absent in universities.

At some point, I have compiled a short list of reasons why I get a lot of satisfaction from teaching a math circle. I love:

-the equality and feeling of mutual respect and attention that develops between me and math circle participants
-the democracy/lack of authority that shows us the “right answer”
-seeing the value alignment and deep intellectual friendship that develops among the participants
-sharing children’s excitement when they realize their own powers
-the feeling of freedom they develop when they get rid of their own mental blocks
-the intellectual stimulation of choosing the problems and personalizing and teaching them to a particular audience
-when children realize that they feel happy from doing a challenging job
-observing their self-discovery
-observing as children come up with amazing solutions and counter-intuitive discoveries
-getting a fresh view of the beauty and awesomeness of the world we observe and create – thus multiplying my own happiness

My answer to a question on Quora:

I do not remember seeing a mathematicians who was embarrassed by their conjectures disproved.

Why? Because making conjectures and refuting them is a normal cycle of mathematics. I think 90% of conjectures die on the same writing desk where they were born, being killed by the same mathematicians who formulated them. In mathematics, it is a daily routine. Refutations are as important as proofs. There is a famous book about the role of refutations in mathematics, Imre Lakatos’ Proofs and Refutations.

And the famous Lewis Carroll’s lines in Through the Looking-Glass capture the spirit:

“I can’t believe that!” said Alice.
“Can’t you?” the Queen said in a pitying tone. “Try again: draw a long breath, and shut your eyes.”
Alice laughed. “There’s no use trying,” she said: “one can’t believe impossible things.”
“I daresay you haven’t had much practice,” said the Queen. “When I was your age, I always did it for half-an-hour a day. Why, sometimes I’ve believed as many as six impossible things before breakfast.”

Proofs and refutations co-exist in the most natural way. Mathematical problems are conjectures. To solve a problem means to prove this conjecture or refute it.

Proofs are frequently done by constructing, in parallel, a counter-example: when a mathematician identifies obstacles for a proof, he/she may wish to try to use them to construct a counterexample; when this attempt at refutation encounters its own difficulties, a mathematician may try to isolate these difficulties and understand their nature – for use in the proof. In this zig-zag movement the aims — to prove a conjecture and refute it — alternate. In a happy outcome , the process converges on a definite answer: either proof or refutation.

But, if you look back at that zig-zag prowl in search of a kill, you may say that half of the time the mathematician believed impossible. Even worse, it is like lions in hunt: ten chases result in one kill; a mathematician normally solves about one problem out of ten that he or she tries.

There is one extreme case of the proof/refutation balance: the original proof of the Classification of finite simple groups. I quote Wikipedia:

The proof consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004.

As a rule, almost each of these “several hundred journal articles” contains a proof of a particular theorem, a special case or an intermediate step of the “global” statement. Since all that is about finite objects, proofs frequently use mathematical induction in a specific form: proof of non-existence of a minimal counterexample to the theorem. As a result, it makes thousands of pages of arguments about non-existent objects. At a first glance, it gives an answer to another question on Quora: What are some aspects of mathematics that are nonsense? But these arguments about eventually non-existent minimal counterexamples are not nonsense — for example, they can be re-used in proving theorems in other branches of group theory.

This text is based on my response to a question on Quora, slightly expanded:

As a mathematician, how would you mentor your child and help her to learn, do and live mathematics in her free time as she is growing up?

I write from the position of a mathematician about what a mathematician can do for her child.

First of all, a mathematician understands and can use the fact of life non-mathematicians are not aware of:

Mathematics is done by the subconscious.

Encourage in your child, and help her, to develop all kinds of intuition, guesswork (with subsequent checking, whenever possible, of the correctness of the guess). Help her to train her vision of the world, see [mathematical] relations in the world, identify mathematical structures present in the world.

As you can see, this immediately moves the discussion away from school and traditional approaches used in the traditional school education. In fact, this is more of an advice for home-schoolers, especially those who themselves have a sufficient mathematics background. The latter does not mean having a B.Sc. in mathematics; for example, reading economics at Cambridge, UK, suffices.

However, mathematical games briefly described here are not a replacement for a systematic course of mathematics.

But this in-family “protomathematics” could perhaps open to a child a door to mathematical thinking well above the demands of standard school curricula.

What follows are a few random examples, chosen from what I did myself with my (grand)children or had seen my colleagues doing with their (young, pre-school or primary school age) children.

Adult spends some quality time with Child, aged between 3 and 4, in a garden, watching insects and ants, and discussing with Child how the world looks from the ant’s viewpoint: that the tree trunk is like a street, and patches of algae and of moss on the bark are like lawns and bushes along the street. Child: “and these branches are like side streets”.

As you can see, a gentle introduction to mathematics can start just by watching the world, in this specific case insects and flowers in a garden.

A child needs to develop a habit of attentively looking at the world; to achieve that, it helps when an adult looks with him, and they discuss what they see.

Mathematics, in one of its many aspects, is a specific vision of the world.

Basic (but mathematically very deep) concepts of similarity and scaling are accessible to a 4 year old child.

The former is erased (with most of geometry) from school teaching, the latter was never properly taught. Euclid’s proof of the Pythagoras Theorem as given in Book 6 of Elements is an example of a scaling argument.

Let us look at one of the simplest examples of fixed point theorems (I have seen it in some popular books on mathematics):

If a (topographic or geographic) map is placed on another map of the same territory, but of larger scale, then it is possible to stick a pin in the both maps which goes through images of the same point (location) of the surface of Earth.

What I’ll give you now is not a proof but an immediately obvious explanation:

Imagine that you an ant and stand on the bigger map which is like a surface of the Earth for you, and you hold the smaller map in your hands; just mark your location on it.

Further development of the theme of maps:

Adult uses every opportunity to explain to Child the structure of actual streets in a big city: street signs, house numbers which go in progression and are odd on one side of the street and even on the other side. A year later, Child is able to confidently guide Adult across a unknown part of the city using a standard AZ map.

Observing an ant on a tree helped. This was an encouraging sign of mathematical development.

Of course, Child’s ability to read is useful. Street names, all kinds of shop signs provide an excellent material for reading and proof that reading gives information about the world.

In a supportive family environment, a child can learn to read by the age of 4 or 5—even in English with its inane orthography. Social class issues creep in here: this is realistic mostly in families of well educated middle class professionals. Sufficient disposable income for buying “quality time” with children (say, by inviting an au pair or hiring a caretaker of some kind for more routine tasks) definitely helps.

And the last episode in development of this theme of maps, a conversation on a street:

Adult: “Look at the name of this street. Does it tell you anything?”

Child: “Yes! Its end meets the street where Grandma lives, just over the corner!”

Adult: “And do you remember, what kind of house numbers there, large or

small?”

Child: “Small!”

Adult: “So, in what direction we have to go?”

Child looks at house numbers around and confidently points: “There!”

More about rearing and writing:

Adult and Child (aged 5) send to each other, from opposite corners of a sofa, small strips of paper with messages written in a substitution cipher: each letter is substituted by the next one (cyclically, z is substituted by a). Suddenly Child exclaims:

“And I invented my own cipher — each letter is replaced by the previous one!”

IMHO, this would later help Child to understand algebraic notation where numbers are substituted by letters. It is worth remembering that François Viète, the inventor of algebraic notation, was the first cryptographer and cryptanalyst known to us by name. He served to King Henry IV of France. Viète’s deciphering of intercepted diplomatic correspondence directly influenced European politics of his time. The King of Spain famously complained to the Pope that Viète sold his soul to the Devil.

Some time later, at the age about 7, Child can handle variables in Scratch, a toy programming language for kids.

Child is invited to guess weight of every household object she can handle by weighing it in hand and check the result by weighing on scales. The same with temperature of water in the bath, checked by a thermometer. Or temperature outside the house.

Further,

All kinds of estimates with subsequent checking: how many steps are in this staircase?

How many steps are to the end of the street?

How to estimate the number of cars in the parking lot without counting them all?

I can add cooking and baking, as a family activity, if Child is entrusted with control of numerical aspects of the recipe: how many spoons of sugar?

Lego:

Playing Lego (with child of 4). Adult encourages Child to pick correct bricks (say, 2 by 3 studs) without looking at them, by touch only. They together follow step-by-step instruction in the manual. And steps are numbered!

Building a symmetric model (say, a plane), Adult builds the left wing, Child builds the right wing by mirroring Adult.

Very soon Child starts picking details of correct orientation even before Adult touches his detail. (Orientation is an exceptionally important concept in geometry which is not even mentioned at school.)

Lego is a great propaedeutic for Scratch, a toy programming language for kids where computer programs are built on the screen from logic blocs which intentionally made looking as bricks in Lego.

And the real unadulterated fun, fun, fun:

Playing Snakes and Ladders with two dices. A player can pick one of the values or their sum. The catch is that, for winning the game, 100 has to be hit without overshooting—for otherwise the player gets back to the beginning, the path is circular, 97+6 = 3.

A fast, furious, and vicious game which trains tactical thinking.

Actually the rules of Snakes and Ladders can be changed in a variety of ways. Adult encourages Child to invent her own rules. Crucially, the new rules need to be agreed and written down before the start of the game.

A useful meta-activity: ask Child to compare two sets of rules: which one produces a more exciting game?

A jigsaw with a clear geometric structure could be very useful. The classical London Tube map is excellent, especially if Adult and Child work together and agree to ignore the white space. But perhaps the London Tube is best for kids who live in London.

I can continue this list, but, I hope, it already gives some idea.

And I have no idea how all that could be done at school.

Within the family—no problem. In small and friendly mathematical circles—it is also OK.

But in school?

And, finally, some general advice:

Never ever distract your child when he/she just sits and thinks.

This post was previously published on Substack.

This was what a colleague wrote today in some discussion of university life:

“I think the biggest problem is that the University effectively values academics time as zero.”

My response:

IMHO, this is indeed the core problem.

This is the Executive Summary of my paper arXiv:2103.04101v1 [math.HO] on 6 March 2021:

Executive Summary. This paper is written in lockdown and can serve as a testimony in support of the apparently self-evident, but largely ignored principle:

*The most important resource for a (pure) mathematician’s research is uninterrupted time for thinking.*

University administrators and research funding bodies systematically ignore it, and the bureaucratic burden imposed by them strangulates mathematics research. A short breathing space provided by lockdown made miracles.

And, in my paper, I provide evidence,  and, I think, pretty convincing (at least to mathematicians), in support of this my claim. I did some of the best work of my life because, thanks to lockdowns and retirement, I took back control of my time.

This is what mathematicians have to fight for: taking back control of our time.

This post is copied from my Substack   article.

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 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.