There was a much advertised (at least in the UK)  high level UN Security Council meeting on Climate Change on 23 February 2021 (online, chaired by Boris Johnson, UK Prime Minister). It appears  that the meeting has been hastily downgraded to  a non-event after Sir David Attenborough, addressing the Security Council in a short speech, said  something unspeakable:
“We have left the stable and secure climatic period that gave birth to our civilisations. There is no going back – no matter what we do now, it’s too late to avoid climate change and the poorest, the most vulnerable, those with the least security, are now certain to suffer.”
This segment of Sir David’s speech was not included in the BBC video clip  “Attenborough gives stark warning on climate change to UN – BBC News”A,, and reports about the meeting (and Attenborough’s video) almost instantly disappeared from the BBC front page. However, the segment can be found on Sky News,  The entire 8 minute video of Attenborough’s speech  is on the UN site:
On the optimistic note,  Sir David said
“I do believe if we act fast enough we can reach a new stable state. It will compel us to question our economic models and where we place value; invent entirely new industries; recognise the moral responsibility that wealthy nations have to the rest of the world and put the value on nature that goes far beyond money.”
It appears that there is a coherent, and well supported by archaeological and geological evidence, theory behind Sir David’s words; his speech cannot be easily dismissed as fantasy.
I summarise some points of this theory which I happened to learn from various sources  over the last 20 or maybe even 30 years. I am not an expert, and I would much appreciate corrections and further details.
  1. The current period of stable climate which allowed the human civilisation to develop, was about 11 thousand years long, and it was abnormally long on the scale of the last 50 or 100 thousand pretty turbulent years.
  2. The last violent episode which preceded our  golden era was a circulation event triggered by a flood of fresh water from the melting glaciers in North America which directed the Gulf Stream to Africa rather than Europe, correspondingly directing the jet streams in atmosphere over North Atlantic to south of Europe. This meant that temperatures at what now is London were like in nowadays Irkutsk. But Sahara got abundant rain and was covered by forests and lakes with hippos and crocodiles, and flamingos — immortalised in cave paintings (made by us, humans, who happened to migrate there at that time).
  3. All that is dated with surprising precisions by pollen from flowering plants preserved in sea sediments. The most interesting bit  is the length of transitional period between the two climate regimes — just about 10 years.
  4. A similar event can be triggered by collapse of glaciers in Greenland as the result of  melting water accumulating  in under  ice lakes and eventually finding its way to the ocean.  I would not claim, however, that this is to happen tomorrow. But something like that has already happened once.
  5. Of course, climate can mutate in many other directions, including Britain becoming, in hte new stable state of climate,  a subtropical country and Scotland replacing thistle, as its national symbol, by artichoke (Wiki:  a variety of a species of thistle cultivated as a food).  I’ve seen plantations of artichoke in Turkey, they look hilarious.
  6. The Gulf Stream example suggests that switching to the new stable period could be short, but interesting.
  7. My last comment is on artichokes and jet streams. The most poignant artichokes I have seen in Turkey were on what was, two millennia ago, the river bed of Meander, the great river of Asia Minor, now almost dry. The history of glorious civilisations of Mediterranean and Middle East was the history of ecological disasters created by people. It is hard to believe now that North Africa and Turkey were covered by  subtropic forests. Well, England not long ago was covered by mighty oaks, and Scotland — by pines, and the famous heather moors is a secondary landscape, an ecological system which replaced forests destroyed by people, and which is much more fragile than forest.
  8. Meander became a verb  (with the meaning `follow a winding course’)  because the river was bending, creating loops, etc. Last time I’ve seen this word — to meander — was  in relation to jet streams over North Atlantic which started to meander. Water flows against the gradient of altitude of the surface and starts to meander, that is, the flow is becoming unstable, if the gradient is too small. It is claimed that the same happens with jet steams: they flow against the gradient of temperature, and are destabilised by warming up of Arctic which is decreasing the gradient.  There is a possibility that the climate of British Isles can change even without dramatic events in Greenland.
The reasons or triggers for climate change — a becoming a scholastic issue now.  What matters is
A. Ecological systems around the world are weakened and under stress, and their ability  to cope with changes in climate is compromised.
B. The ability of humans, as species, to cope with the change  is also compromised. On one hand, we reached a fantastic level of technological development  (which,  of course, helps) — but on the other, this technology is not equally shared, and, on the top of that:
C. This is the greatest unspeakable issue — the Earth is overpopulated by humans beyond sustainable levels.
D. In point C, the words “beyond sustainable levels” should of course be understood as “at levels unsustainable under the current socio-economic systems prevalent in the world and interconnected within the single global economy”.  However, any  change of the socio-economic systems on that scale is a task perhaps even more challenging than dealing with the climate change itself. It is likely that the number of people killed in the process will be comparable with loss of life in all natural disasters and pandemics triggered by the climate change.
 I do not claim that I am right or correct in every detail. But my message is:
The problems of the Earth and of the human civilisation on the Earth reached the level when they can no longer be entrusted to politicians and journalists — and neither can be left to the social media. The readers of my blog are a few but they tend to be educated people with proven skills of analytic thinking — if you read this, please allocate some of your time to analysis of all that mess.
The 21st century starts now, in 2021 — the same way as the 20th century started in 1914 with World War I.
Posted by: Alexandre Borovik | February 21, 2021

How do illogical proofs or answers “feel” to mathematicians?

My answer to a question on Quora: My high school math teacher often made rudimentary mistakes in her equations. What does that say about her?

This could be turned into a good learning experience for you: watch your teacher and try instantly identify her mistakes and correct them – for yourself. I had this experience in my school days, and remember it fondly. It really helps to start mastering mathematics.

It is up to you to decide whether to try to correct your teacher in front of the class, but remember, it could be very cruel to her, you may regret that later. It would be much more useful to quietly provide your classmates with correct solutions.

And maths teachers in schools are frequently overworked. And what do you know about her home life? She could simply suffer from long term sleep deprivation. Especially in the present crazy times…

Posted by: Alexandre Borovik | February 21, 2021

A look from lockdown at horrors of school mathematics

Kit Yates in The Observer: Home schooling: ‘I’m a maths lecturer – and I had to get my children to teach me’  A few quotes:

A senior lecturer in the department of mathematical sciences at the University of Bath, Yates has a PhD in Maths from Oxford and is the author of The Maths of Life and Death. So when he began home schooling his son Will, five, and daughter Emmie, seven, during lockdown, he was pretty confident he already knew everything they would be expected to learn in maths.

He was wrong. “I’d never heard of a ‘bar model’ or a ‘part-whole model’. I had to get my kids to teach me.” He was shocked by how many of these different, “intimidating” methods and models primary school children are expected to use to solve basic maths problems. “I’ve never needed to use them – you don’t need to know all these different mental models to do maths,” he says. […]

But what he really finds frustrating is the lying. The curriculum is forcing teachers to deliberately teach children lies, he says, which then have to be unpicked later. For example, after years of being taught there are no numbers between zero and one, his seven-year-old is suddenly expected to understand that there are such things as fractions.

Posted by: Alexandre Borovik | February 21, 2021

Why is the constructivist theory applicable in teaching mathematics?

My answer to a question on Quora: Why is the constructivist theory applicable in teaching mathematics?

I suggest to modify the question a bit:

Why do some people find the constructivist theory applicable in teaching mathematics?

This wold allow me to express my surprise: indeed, why?

I have lived in the world of professional research mathematics for almost 50 years now, and I wonder why constructivist theory in mathematics education so blatantly ignores the experience accumulated in the mathematics research community. I feel that the  constructivist theory talks about some different kind of mathematics, not the one known to me and my many friends and colleagues from all around the world. But I am Vygotskian by my philosophy upbringing, and I can see how Vygotsky’s sociocultural approach explains the invention of this mock image of mathematics. I will look for an opportunity to explain that- I hope Quora sooner or later will give me chance to do that.

A very important question. As it was already explained in this thread, this is a well-known and quite usual phenomenon (called childhood amnesia), caused by re-wiring of the brain at the critically important stage of development. The timing is slightly different in different people, and, I feel, in respect to different kinds of brain activity — for example, you cannot forget how to swim or ride a bicycle, if you learnt these skills at the age covered by amnesia. Also, it appears that children do not unlearn how to read or do arithmetic — but they can eventually forget how did they learn to read.

In general, I think non-one should worry about their childhood amnesia — these were natural changes in one’s brain, and they were to one’s benefit.

I collected hundreds of testimonies from people about their very first memories of learning mathematics — and discovered, that it seems that majority of people simply do not remember anything at all about their earliest encounters with school mathematics — including, it appears, many teachers of mathematics. Unfortunately, I had a day job to do and had no time to run a proper statistical analysis. But I think, this is something that should be taken into account in professional education of future school teachers of mathematics.

My answer to a question on Quora: What are suggestions for patterns in daily lives that deal with mathematics?

The first thing that comes to mind is the most mundane: numbering, first of all, house numbers on streets in towns and cities. They make quite an expression on a 4 years old child when first explained to her:

  • house numbers are odd on one side and even on another;
  • they grow in one direction;
  • if look in the direction of increase of numbers, then odd numbers are on the left hand side of the street, even are on the right hand side.

It is useful to bring child’s attention to street signs with street names on them, as well as shops’, cafe’s, barbers’, nail salons’ signs: in some older cities there is a custom to include the name of the street in the name of establishment, so Coronation Butchers are likely to be on the Coronation Street. In short: at the very first opportunity, when child just starts to read and count, explain to her the structure of the street.

Please notice that I am talking about structures, not about patterns.

Mathematics is not a science of patterns, as some people claim,

Mathematics is the science of structures hidden behind patterns.

Structures are much richer and more interesting than patterns.

Let us look at another episode with the same child: he and the adult observe an ant on a trunk of a tree in a city park. Adult invites the child to observe that the trunk for the ant looks like a street, and patches of algae and moss are like lawns and bushes. Child: “And branches are side streets”.

A year later, the child is already able to use a standard city map and confidently guide the adult and a little sister through an unknown to them part of the city. Adult: “And where is our next turn?” Child, glancing at the map: “at this T-junction ahead  of us, to the right”. Adult: “And the name of another street?” Child, after quickly checking the map: “Station Road”.

Map is a mathematical object, and is in a mathematical relation with real streets. etc. in the city; this relation is called scaling (or, in more geometric terms, similarity or homothety). But the exactly the same concept of scaling can be introduced to a child using an ant on a tree trunk as an example.


if you want to see mathematical structures in the world around you, try to see the world through the eyes of a child.


Posted by: Alexandre Borovik | February 15, 2021

Some good answers are already given in this thread, I wish only to hint at the whole class of metaphors which can be used as a quick and cheap answer:

The difference between mathematics and mathematics education is the same as

  • between religion and religious education
  • between ***** and ***** education (you may wish to continue the list using this pattern)

Answering this question, it is very easy to switch into cynicism — one of the responses in this thread, from a former student, is already highly critical of mathematics education. I see myself as working professionally in mathematics education, and it is painful for me to see popular sayings as this one, from American popular culture:

Those who can, do. Those who can’t, teach. Those who can’t teach, teach teachers.

— but I have to live with it. We have to accept that this is a popular attitude to our profession.

My answer to a question on Quora: What are some unusual ways you’ve applied the math you learned in high school to your life?

I once was asked to act as a reviewer of a paper submitted for publication in an academic journal on mathematics education. It was a double blind review: the draft paper sent to me contained no names of authors or their affiliation.

The paper described how the authors set up a website and run online questionnaire among staff at mathematics departments of two British universities on the following issue: what kind of examinations, closed book, or open book, better discriminates between different levels of students’ attainment, and what kind is preferred by the respondents? Three pieces of data were given by the authors:

  • Closed book examinations were selected as the most discriminating or second most discriminating of the assessment methods by 79% of the participants.
  • Closed book examination was selected by 86% of the respondents as their most preferred of the assessment methods.
  • The response rate of the questionnaire was 15%,

What surprised me is that the total number of responses to the on-line questionnaire has not been given in the paper, although omitting the size of the sample from statistical data was unacceptable in published academic research.

However, I calculated the number of responses, and explained in my report to the editors how I did that essentially by mental arithmetic.

This is a cute arithmetic problem; one more general piece of information is needed for solution, but it is something commonsense. Try to think for yourself, it is easy. A solution is given below these warning signs:

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Solution. Indeed, 79% and 86% rates of positive answers to particular questions suggest that 86% – 79% = 7% corresponded to an integer number of people (those who answered positively to one question but not to the other). If 7% consists of 1 person, the number of respondents is 14 or 15. If 7% consist of 2 persons, then the number of respondents is between 28 and 30, but in this case, since the response rate was 15%, the two departments have about 200 mathematics lecturers, which was unlikely in UK universities (here the common sense is used). Hence there were 14 or 15 respondents.

Very conveniently, 11/14 rounds up to 0.79 and 12/14 to 0.86 (here I used a calculator – previous steps had been done by mental arithmetic) 15 respondents would produce not so good rounding of percentages.

I recommended to reject the paper — in my opinion, the paper contained no representative data; a chat in a staff lounge during coffee break, or, even better, on in a pub after a seminar was likely to yield a more representative sample. However, the editors accepted the paper for publication, but asked the authors to reveal the number of respondents – indeed, it was 14.

Posted by: Alexandre Borovik | February 15, 2021

How can one remain a mathematician?

My answer to question on Quora: How can one remain a mathematician?

It is next to impossible to answer your question without knowing your circumstances.

Being a mathematician is a way of life.

The way of life could change for a variety a reasons: for example,

  • external pressures (say, money problems)
  • illness
  • marriage
  • just because life became too boring
  • drug addiction
  • gambling addiction
  • taking certain types of medication
  • and so on …

The list can be expanded, and every situation calls for a different answer. I do not want to take your time and say only that drug addiction is incompatible with mathematics — to remain a mathematician, stop doing drugs. Also taking, for extended periods of time, medication about which you are warned: “when taking this medication, do not make important decisions, do not drive or operate machinery”. If you were given this warning, speak to your doctor and ask for an alternative treatment; explain to your doctor, that mathematics is all about making serious decisions; it is also a mental equivalent of operating heavy machinery.

My answer is based on  many years of my experience as a personal academic advisor to mathematics students.

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