02/15/21

What are some unusual ways you’ve applied the math you learned in high school to your life?

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:

Spoiler below!

Spoiler below!

Spoiler below!

Spoiler below!

Spoiler below!

Spoiler below!

Spoiler below!

Spoiler below!

Spoiler below!

Spoiler below!

Spoiler below!

Spoiler below!

Spoiler below!

Spoiler below!

Spoiler below!

Spoiler below!

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.

02/15/21

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.

02/14/21

What are some of your favorite basic math hacks?

My answer to questions on Quora:  What are some of your favorite basic math hacks?  and also to What is the greatest trick in mathematics?

I have to apologise: I do not have favorite math hacks. I have never used hacks/tricks. The essence of mathematics is in universal methods for solving all problems in a particular class of problems. Use of “hacks”/”tricks” is a replacement of mathematics by cheap surrogates. Hacks/tricks can work at a certain level, but frequently obstruct students’ progress at the next level of learning of mathematics. Hacks/tricks 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/tricks were the last thing they needed.

02/14/21

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. Have you ever seen an exciting storage facility?

Let us look at mathematics now: it is even more boring than the rest of the school. Why?

  • 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 you to achieve that.

As simple as that.

02/14/21

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.

01/26/21

Why is calculus never used in real life?

My answer to a question 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 an emerging 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.

 

01/2/21

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

01/1/21

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!

 

12/31/20

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.

09/29/20

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

\(1^2=1\)

 

\(11^2 = 121\)

 

\(111^2 = 12321\)

 

\(1111^2 = 1234321\)

 

\(\dots\)

 

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

Part 2.

\(1^2 = 1\)

 

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

 

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

 

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

 

\(\dots\)

 

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.