Posted by: Alexandre Borovik | February 14, 2021

What are some of your favorite basic math hacks?

My answer to a question on Quora:  What are some of your favorite basic math hacks?

I have to apologise: I do not have favorite math hacks. I have never used hacks. The essence of mathematics is in universal methods for solving all problems in a particular class of problems. Use of “hacks” is a replacement of mathematics by cheap surrogates. Hacks can work at a certain level, but frequently obstruct students’ progress at the next level of learning of mathematics. Hacks are frequently favoured by teachers who always taught students at a certain level but never at higher levels. These teachers tend not to care much about their students’ progress beyond their class.

In my opinion, teachers have to be assessed not by their students’ marks in their classes, but by their students’ success at the next stages of education. In that environment, any desire to teach “hacks” somehow disappears.

I apologise to be so firm in my opinion on that matter, but for at least a decade I taught mathematics at a Foundation Studies programme in a very big university: up to 400 students in a lecture theatre who failed mathematics at secondary school and had to be brought up to the level where they could start studies in (relatively) mathematically intensive degree programmes at the university level. Hacks were the last thing they needed.

Posted by: Alexandre Borovik | February 14, 2021

Why is math in school boring? Is it the curriculum’s blind spot?

My answer to a question on Quora: Why is math in school boring? Is it the curriculum’s blind spot?

My answer:

  • Math in school is boring because mathematics education lost its purpose and meaning.

Lockdowns during this pandemic laid bare the true social and economic role of schools:

  • Schools are storage rooms for children — to free their parents’ time for paid employment.
  • Mathematics in schools is boring because teaching mathematics is reduced to coaching children to pass exams.
  • Exams dominate mathematics education because they are loved by employers: exam grades provide very cheep, for employers, way to select future employees, compliant and obedient. And mathematics exams could be made hard.
  • From the point of view of an employer, a good grade in exciting and very interesting subject is of less value than the same grade in an excruciatingly boring subject — because the latter requires much more effort, I would even say, strong will, to get.
  • You know, office work is painfully boring; it is advisable to be preparing for that from childhood. Being boring, school mathematics helps to achieve that.

As simple as that.

Posted by: Alexandre Borovik | February 14, 2021

Intuitively, what is a finite simple group?

My answer to a question on Quora: Intuitively, what is a finite simple group?

There are two ways to describe an object: how it is made and what it is doing. For example, a knife can be desribed as “an elongated flat piece of metal, sharpenenned on one edge, with a handle attached” (it is how it is made), or “a thing to cut bread” (how it is used).

I have not red every answer, but the discussion of groups in this thread so far appears to be restricted to the viewpoint of “how they are made”. But what do finite groups do?

They act. They act on sets of various nature; this sets are made of elements. The same group may have many different actions. For example, the group of symmetries of the cube can be seen as acting on

  1. the set of 8 vertices of the cube
  2. the set of 6 faces,
  3. the set of 8 edges seen as non-oriented segments
  4. the set of 16 oriented edges,
  5. the set of 4 non-oriented “main diagonals”,
  6. the set of 3 non-oriented axes passing through the centers of opposite faces.

Indeed, every symmetry of the cube is moving elements of each of this sets within that set (perhaps actually fixing some of them or even all of them).

Elements of the group, for the purpose of this discussion, can be called actors (I invented that name specifically for this post on Quora). What follows is a description, not a rigorous definition.

Each actor moves elements in the set in some way (and this could be an identity move, when nothing is actually changed in the set) . What follows are properties of actions of a group:

  • There is an actor which does nothing.
  • For every actor there is another actor, which reverses its moves.
  • For any two actors, there is an actor who is doing the combination of movements of the first two actors.

An action of a group is called trivial if every actor does not move anything.

An action of a group is called faithful if different actors do different movements. In the example with the group of symmetries of the cube, actions 1 to 4 are faithful, 5 and 6 are not faithful.

The key point: if an action of a group is not faithful, the same movements of elements can be achieved by an action on the same set of another group with smaller number of actors.

Definition: A finite group is simple if all its actions are trivial or faithful.

In short, a simple finite group cannot be replaced, in its non-trivial action, by a smaller group.

This is why finite simple groups are atoms of finite group theory, and why classification of finite simple groups has tremendous importance for combinatorics.

As we can see, the group of symmetries of the cube is not simple. Moreover, its action 6 above contain only movements of 3 elements which can be done but the symmetric group \(Sym_3\) on three letters, or, which is the same, by the group of symmetries of an equilateral triangle (this triangle can be easily seen within the cube). By contrast, the groups of rotations (symmetries which do not change orientation) of the equilateral triangle or the regular pentagon are simple.

By the way, “how it is made” and “what it is doing” are called, in Hegelian dialectics, essence and phenomenon. Questions “intuitively, what is …” refer to phenomena. Intuitively, a knife is a thing to put butter on bread. Intuitively, a group is a mathematical object that acts, or which can be used to describe action. There are other mathematical objects that also can act, in their own way: rings and algebras, for example. On the other hand, you can use a spoon to put butter on bread.

Posted by: Alexandre Borovik | January 26, 2021

Why is calculus never used in real life?

My answer on Quora: Why is calculus never used in real life?

Quora is on Internet, and since you submitted your question to Quora, you have access to the Internet, hence have something with Internet connection, say, a smartphone. Mathematics hardwired in your smartphone (or even ordinary, non-smart mobile phone) is beyond understanding of 95% percent of graduates from mathematics departments of British universities (and American universities, too, I think). Signal processing of electromagnetic waves in the communication channel between the phone and the mobile company mast is all about some sophisticated form of calculus. I’ve seen a claim that there are more mobile phones in the world than toothbrushes. Of course, there is absolutely no reason why the users of mobile phones should know calculus.

You are absolutely right: calculus is never used by ordinary people in their life. Moreover

a whole country can now exist and be fully functional without anyone in the country having any knowledge of calculus.

Indeed, what for? All stuff which has calculus hardwired or coded in is made elsewhere, in countries like China.

This is a new form of colonialism.

Perhaps it could be called high tech colonialism. A high tech colony is not an independent country, it can be at any moment switched off from outside.

If you are happy to live in this brave new world, it is your choice. If not – perhaps you should start learning mathematics, this would give you some chance to become sighted in the land of blind.


Posted by: Alexandre Borovik | January 2, 2021

The Numerology of 2021

A friend wrote to me:

“I understand the 19th century went from 1789 to 1914 while the 20th century went only from 1914 to 1989, according to the historians.”

Still, maybe

19th century: 1789 — 1913
20th century : 1914 — 2020
21st century : 2021 — for as long as it takes

The digits 2021 look ominous. And interesting things have just started to happen.

Just a thought

Posted by: Alexandre Borovik | January 1, 2021

Is A level further maths meant to be so easy?

My answer on Quora: Is A level further maths meant to be so easy?

This is the result of slow degradation of mathematics curriculum under political and socio-economic pressures. However, for people in charge of school education in England, it is important to continue to pretend that A level mathematics is hard, challenging, and relevant.

Mathematics education is poisoned by centrally set written examinations. They require a new set of exam question every year, new — but somehow familiar to students; this means creation of exceptionally boring problems which differ one from another only by values of numerical parameters. Real mathematics is built around iconic, unique, eternal problems which cannot be used in written exams and for that reason never mentioned to students in England — and in most other counties. Standardised maths exams in China are real hard – but also brain and soul destroying.

If A-level Further Mathematics is easy for you, try problems from Tony Gardiner’s and my book The Essence of Mathematics Through Elementary Problems. A pdf file is downloadable for free.

А если Вы читаете по-руски, посмотрите на учебные материалы Moscow Center for Continuous Mathematical Education

Good luck — and Happy New Year!


Posted by: Alexandre Borovik | December 31, 2020

Unreasonable ineffectiveness of mathematics in biology

This post appeared first 2006 in a now-defunct blog, reposted in 2011 and in 2018.  I repost it again as a source of a quote from Israel Gelfand which appeared in Wikipedia.

Israel Gelfand:

Eugene Wigner wrote a famous essay on the unreasonable effectiveness of mathematics in natural sciences. He meant physics, of course. There is only one thing which is more unreasonable than the unreasonable effectiveness of mathematics in physics, and this is the unreasonable ineffectiveness of mathematics in biology.

I heard that from Israel Gelfand in a private conversations (and more than once. Because of Gelfand’s peculiar style of work (see  Conversations with A. S. Golubitski — it is about him), I was often present during his conversations with his biologist co-authors about structure of proteins. The quote was included by me in an earlier version of the book ‘Mathematics Under the Microscope‘, but is not present in the published version (originally I planned to insert it in Section 11.6).

Besides being one of the most influential mathematicians (and mathematical physicists) of 20th century, Gelfand also had 50 years of experience of research in molecular biology and biomathematics, and his remark deserves some attention.

Indeed biology, and especially molecular biology, is not a natural science in the same sense as physics. Indeed, it does not study the relatively simple laws of the world. Instead, it has to deal with molecular algorithms (such as, say, the transcription of RNA and synthesis of proteins which ensures the correct spatial shape of the protein molecule) which were developed in the course of evolution as a way of adapting living organisms to the changing world. If they solve a particular problem in an optimal way, they should allow some external description in terms of the structure of the problem. Indeed, this is the principal paradigm of physics; it is an experimental fact that the behavior of physical systems is governed by various minimality / maximality principles, and the optimal points have, as a rule, especially nice mathematical properties.

But why should a biological system to be globally optimal? Evolution is blind, and there is no reason to assume that the optimal solution is reached. The implemented solution could be one of myriads of local optima, sufficiently good to ensure survival. Lucky strikes could be so rare that the huge search space and billions of years of evolution produced just one survivable algorithm, which, as a result, dominates the living world, and is perceived by us as something special. But it might happen that there is absolutely no external characterization which allows us to distinguish it from other possible solutions, and that its evolutionary phylogeny is its only explanation.

However, I am not a philosopher and cannot claim that my solution of Gelfand’s paradox is correct. What I claim is that philosophers ask wrong questions. The classical conundrum of relations between mathematics and physical world starts to look very different — and much more exciting — as soon as we include biology into consideration. I will try to continue this discussion.

Posted by: Alexandre Borovik | September 29, 2020

What is it about mathematics that may change your thoughts?

My answer on Quora:

What is it about mathematics that may change your thoughts?

At the level of school mathematics, I am afraid, nothing is likely to change your mind. Changes start when you master proof (this does not happen in school mathematics nowadays, and less and less features in the undergraduate university mathematics). Mastering proofs means being able to solve problems of that kind:

given a statement you have never seen before, prove it or construct a counterexample.

An example of this problem:

Part 1



\(11^2 = 121\)


\(111^2 = 12321\)


\(1111^2 = 1234321\)




You see a distinctive symmetric pattern. Will this pattern continue forever? Prove this or give an example of when pattern breaks.

Part 2.

\(1^2 = 1\)


\((1+x)^2 = 1+2x+x^2\)


\((1+x+x^2)^2 = 1+2x+3x^2+2x^3+x^4\)


\((1+x+x^2+x^3)^2 = 1+2x+3x^2+4x^3+3x^4+2x^5 + x^6\)




You see a similar symmetric pattern. Will this pattern continue forever? Prove this or give an example of when pattern breaks.

Part 3. If you got (and justified by proof or counterexample) different answers in Part 1 and 2, explain why.

Mastering this kind of thinking means that you learn to look at the both alternatives: true / false without knowing in advance which one is correct. This is a skill that most people are lacking in their real life problem solving, they concentrate only on one option. This deficiency of thinking leads them to chose one option on the basis of … well, frequently they cannot explain, if asked, on the basis of what. Quite frequently, their choice was made on the basis of their prejudices and preconceptions.

Posted by: Alexandre Borovik | September 4, 2020

The not so coded language of Ofstead

There is a peculiar estblishment in the administrative system of English school education, Ofsted. Its full name is the Office for Standards in Education, Children’s Services and Skills. It is a non-ministerial department of the UK government, reporting to Parliament (not to the Governement!). Ofsted is responsible for inspecting a range of educational institutions, including state schools and some independent schools. This is the only relatively independent part of the system; Ofsted inspectors are Her Majesty’s Inspectors. I had a chance to meet a few of them, and got great respect to them.

The not-so-coded language of Ofsted’s reports on particular schools (easily available on the Internet) appears to be directed at middle class parents. These examples are from 2017 — compare a few (state) schools in a big English city, just a mile away from each other:

School 1 [Ofsted’s assessment: Outstanding]: “Children enter the Reception classes with skills and abilities that are broadly as expected for their age. Excellent provision enables them to make very good progress and achieve in an outstanding manner. As a result, by the time they enter Year 1, standards are above average, and for a significant number of children they are exceptionally high. This is particularly the case in communication, language and literacy and in personal, social and emotional development…”

Translation of words “personal, social and emotional development”: children come from supportive well educated middle class families, and find in the school their tribe. What is behind the scene: the school is in (architectural) conservation area, parents pay through their noses for an address in the catchment area.

School 2 [Good]:
“Children enter the Nursery class with skills that are below those typically expected for their age, especially their mathematical, social and emotional skills.”

School 3 [Good]: “Pupils […] join Reception with communication and mathematics skills that are low for their age. They then make good progress and enter Year 1 at improved levels but which are still mostly below average.”

School 4 [Good]: “Pupils make good progress in Nursery and Reception from starting points which are below and sometimes well below expected levels. In most years, only half of the children reach the expected levels for their age by the end of the Reception Year.”

The reason of challenges School 4 is facing is explained in the Ofsted’s report: “The proportion supported through school action plus and with a statement of special educational needs is well above average.” “A well above-average proportion of the pupils are eligible for the pupil premium, which provides additional funding for children in local authority care and pupils known to be eligible for free school meals.” “A high number of pupils leave and join the school in all year groups at different times throughout the year.”

Intake makes all the difference, and, I conjecture, an increasing number of parents are desperate to stay above the educational waterline and are prepared to pay premium for an address in the catchment area of a desirable school . Pandemic should only increase the social differences in school education.

Posted by: Alexandre Borovik | April 23, 2020

Poems for Lockdown X: Александр Кушнер

Александр Кушнер

Сторожить молоко я поставлен тобой,
Потому что оно норовит убежать.
Умерев, как бы рад я минуте такой
Был: воскреснуть на миг, пригодиться опять.

Не зевай! Белой пеночке рыхлой служи,
В надувных, золотых пузырьках пустяку.
А глаголы, глаголы-то как хороши:
Сторожить, убежать, — относясь к молоку!

Эта жизнь, эта смерть, эта смертная грусть,
Прихотливая речь, сколько помню себя…
Не сердись: я задумаюсь — и спохвачусь.
Я из тех, кто был точен и зорок, любя.

Надувается, сердится, как же! пропасть
Так легко… столько всхлипов, и гневных гримас,
И припухлостей… пенная, белая страсть;
Как морская волна, окатившая нас.

Тоже, видимо, кто-то тогда начеку
Был… О, чудное это, слепое “чуть-чуть”,
Вскипятить, отпустить, удержать на бегу,
Захватить, погасить, перед этим — подуть.

Предложил Ю.Б.

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