My answer to a Quora question:

What’s something about math that still amazes you, even after knowing it for a long time?

That mathematics is consistent: regardless of how long and complicated are proofs, everything miraculously gets worked out without contradiction.

Mathematics is an ideal world; what strikes is its stability. You may revisit some its corner after being away for 30 years, and discover that everything there is the same as it was when you left it.

I am obsessed with stories of how people learn, and of their motivation for learning.

This is Dr Brian May, and his personal story that appears to be unbelievable: the interesting bit is  2”07 – 3”32 of the BBC film. Aged 7, Brian May got obsessed with stereo photography and very soon started to produce his own stereopictures.

By the time he joined Queen, he was doing PhD in Astrophysics (he formally defended his PhD only years later).

Well, the story is quite believable to me. Once upon a time I knew a boy who, at age 14, was repairing TV sets (primordial by modern standards, black and white, vacuum tube) for all his neighbours in a small provincial town. This job required an oscilloscope; he made one from his family’s TV set by adding an additional circuit and a switch between the two modes of operation: as a normal TV set and as an oscilloscope. In later life, he became a guru and wizard of the black art of fine-tuning of accelerators of elementary particles and was in charge of one of the biggest one in the world.

And, of course, there was Richard Feynman who, as a boy, famously “Fixed radios by thinking“.

Back to Brian May: his PhD thesis is published, and the preface contains this passage:

“I inherited a Fabry-Perot spectrometer and pulse-counting equipment from Prof. Ring, and spent 18 months entirely rebuilding and updating both the optics and electronics, in preparation for obtaining essentially first viable set of radial velocity measuremnents, all around the elcliptic, of the Zodiac Light. The writing of my thesis was virtually complete in 2006, but the submission was deferred due to various pressures.”

It is easy to believe that May, as the lead guitarist of Queen, did not have the same issues with scales of measurement as Nigel Tufnel of Spinal Tap famously had:

This goes to 11…  [watch from 1”16].

My answer to a question in Quora:

Who was a notable person that was originally evil, but eventually regretted their evil and became good later on?

One of more obvious answers is St Paul the Apostle (or Saul, how he was known prior to his inversion on the road to Damascus).  Acts 9:1–6 KJV say:

[1] And Saul, yet breathing out threatenings and slaughter against the disciples of the Lord, went unto the high priest,
[2] And desired of him letters to Damascus to the synagogues, that if he found any of this way, whether they were men or women, he might bring them bound unto Jerusalem.
[3] And as he journeyed, he came near Damascus: and suddenly there shined round about him a light from heaven:
[4] And he fell to the earth, and heard a voice saying unto him, Saul, Saul, why persecutest thou me?
[5] And he said, Who art thou, Lord? And the Lord said, I am Jesus whom thou persecutest: it is hard for thee to kick against the pricks.
[6] And he trembling and astonished said, Lord, what wilt thou have me to do? And the Lord said unto him, Arise, and go into the city, and it shall be told thee what thou must do.

There are conflicting interpretations of this episode, but, in any case, Paul was a changed person since then. An evil man would not write in 1 Corinthians 13:4-7 KJV :

[4] Charity suffereth long, and is kind; charity envieth not; charity vaunteth not itself, is not puffed up,
[5] Doth not behave itself unseemly, seeketh not her own, is not easily provoked, thinketh no evil;
[6] Rejoiceth not in iniquity, but rejoiceth in the truth;
[7] Beareth all things, believeth all things, hopeth all things, endureth all things.

This is the abstract of a talk given by me at the Meeting “Mathematical Academic Malpractice in the Modern Age“, Manchester, Monday 21st May 2018, with the title

Confident students do not cheat: how to build mathematical confidence in our students

I think it could be useful to address the question which, in my experience, is almost never asked: what pushes problem students to cheat by plagiarising work from their peers and, increasingly, from the Internet? Some answer can be found in Denizhan (2014):

“These students exhibit an inability to evaluate their own performances independent of external measurements.”

Plagiarism is one of the psychological defences of a student who does not otherwise know whether his/her solution / answer is correct. Mathematics provides a simple remedy: systematically teach students how they can check their solutions. This will boost their confidence in their answers – and in themselves. I teach linear algebra; I have at least two dozen undergraduate linear algebra textbooks in my office — none of them provides systematic advice on these matters. The same applies, I think, to any other undergraduate subject. In my view, the most efficient methods for checking answers in a particular class of problems usually provided by a more advanced point of view. For example,

• all these elementary problems about systems of linear equations can be effectively checked if the concepts of the rank of a matrix is used;
• the correctness of eigenvalues of a matrix can be checked by using the fact that the sum of eigenvalues is the trace of the matrix, and the product is its determinant, etc.

This retrospective reassessment of previous material can give students a chance to see how actually simple it is — and boost their mathematical confidence. In my talk, I’ll discuss how to incorporate error-correcting aspects of mathematics into course design.

I wish to comment on two specific flaws exhibited by students who encounter proofs first time in their lives.

The first one is

inability to accept the Identity Principle: “$A$ is $A$”, and arguments related to it, as a valid ingredient of proofs.

For many students, a basic observation

For all sets $A$, $A\subseteq A$ ($A$ is a subset of $A$) because every element of $A$ is an element of $A$

is very hard to grasp because of the appearance of the same words about the same set $A$ twice in the sentence: “element of $A$ is an element of $A$”. I have observed that many times and I think that students cannot overcome a mental block created by their

expectation that a proof should yield some new information about objects involved

— and this is the second fundamental flaw.

And, of course, reduction, removal of unnecessary information, is seen by many students as something deeply unnatural.

Every year, I hear from my Year 1 students the same objection:

How can we claim that $2$ is less or equal than $3$, that is, $2⩽3$, if we already know that $2$ is less than $3$, $2<3$?

I think we encounter here a serious methodological (and perhaps philosophical) issue which I have never seen explicitly formulated in the literature on mathematics education:

• a proof of a mathematical statement can illuminate and explain this statement, it may contain new knowledge about mathematics which goes far beyond the statement proved; but
• elementary steps in proofs frequently do not produce any new information, moreover, sometimes they remove unnecessary information from consideration.

A proof can be compared with a living organism built from molecules which can hardly be seen as living entities — and even worse, from atoms which are definitely not living objects.

This is closely related to another issue which many students find difficult to grasp: statements of propositional logic have no meaning, they have only logical values (or truth values, as they are frequently called) TRUE or FALSE. Any two true statements are logically equivalent to each other because they are both true; moreover, the statement

if London is a capital of England then tea is ready

makes perfect sense, and can be true or false, even if constitution of the country has no relation to the physical state of my teapot. [Moreover, the statement is TRUE, because London is NOT a capital of England, it is  a capital of United Kingdom. ]

When my students express their unhappiness about logic which ignores meaning (and I provoke them to express their emotions), I provide an eye-opening analogy: numbers also have no meaning. The statement

The Jupiter has more moons than I have children

compares two numbers, and this arithmetic statement makes perfect sense (and is true) even if Jupiter has no, and cannot have any, connections whatsoever with my family life. Numbers have no meaning; they have only numerical values. Arithmetic, the most ordinary, junior school, sort of arithmetic is already a huge and deep abstraction. We did not notice that because we are conditioned that way.

Learning proofs also involves some degree of cultural conditioning. As a side remark, I suspect (but have no firm evidence) that the role of family — presence of clear rational argumentation in everyday conversations within family — could be important.

I use this example in my lectures when I explain the difference between arithmetic and harmonic means:

A car travelled from A to B with speed 40 miles per hour, and back from B to A with speed 60 miles per hour. What was the average speed of the car on the round trip?

Anatoly Vorobey and Vladimir Kramchatkin made a useful comment on Facebook on this quite standard and well-known problem:

“The answer is obviously 48 [miles per hour]. 95% can not solve this problem the first time. But if they are told in advance that the distance between A and B is 120 [miles], 95% of schoolchildren will easily solve this problem.”

A concrete number, 120 km, serves as a strong hint that students are expected to do something with this number. But, for majority of students, if a magnitude or a quantity is not assigned a concrete numerical value, it does not exist. This is one of the flaws of mathematics education at schools: no-one tells students that they have to be able to see hidden parameters in arithmetic problems. But this is not the only flaw: students are also not told how to check solutions. Checking answers frequently benefits from seeing a problem in a wider context and varying the data. The standard answer that students give to the problem with the car is 50 miles per hour, the arithmetic mean of the two speeds. But this solution immediately collapses if we slightly change the problem: what would happen if the speed of the car on its way back from B to A was 0 miles per hour?

There are two very different types of perfectionism.

Type A: interiorised perfectionism driven by personal, internal criteria. As an apocryphal story goes, one of the presidents of Harvard University was once asked what was so special in teaching at Harvard to justify their extortionate fees. His answer was of just three words: “We teach criteria”. Cambridge appears to be the only university in Britain which teaches criteria. I know a criterion when I see one — I myself was lucky to get my own education at a boarding school and an university which taught criteria. Among my mathematician colleagues (and co-authors) I know a number of Type A perfectionists. Some of my friends teach or spread criteria — by means of art classes, or mathematics circles, or lectures on history of mathematics, or poetry evenings …

Type B: external perfectionism, a Pavlovian dog reflex to meet crude criteria, at the level of metrics, rankings, “likes” on social  media — all of them imposed from outside. In modern world, most perfectionists belong to Type B. IMHO, the best inoculation against the soul-destroying Type B perfectionism is development of a system of deeply interiorised personal criteria. In principle, this is what education should give to every child. It fails. Moreover, the vast majority of schools and universities spread the disease.

There are two very different types of perfectionism.

Type A: Interiorised perfectionism driven by personal, internal criteria. As an apocryphal story goes, one of the presidents of Harvard University was once asked what was so special in teaching at Harvard to justify their extortionate fees. His answer was of just three words: “We teach criteria”. Cambridge appears to be the only university in Britain which teaches criteria. I know — I myself was lucky to get my own education at a boarding school and an university which taught criteria. Among my mathematician colleagues (and co-authors) I know a number of Type A perfectionists. Some of my friends on Facebook teach or spread criteria — by means of art classes, or mathematics circles, or lectures on history of mathematics, or poetry evenings …

Type B: External perfectionism, a Pavlovian dog reflex to meet crude criteria, at the level of metrics, rankings, “likes” on social media — all of them imposed from outside. In modern world, most perfectionists belong to Type B. IMHO, the best inoculation against the soul-destroying Type B perfectionism is development of a system of deeply interiorised personal criteria. In principle, this is what education should give to every child. It fails. Moreover, the vast majority of schools and universities spread the disease.

My answer to a question on Quora:

Mathematicians frequently make mistakes, but they also have instincts and skills to identify them. To locate an arithmetic mistake in a long calculation could be very difficult, but, in surprisingly many cases, it is possible to say that the result is wrong because it does not behave properly under transformation of inputs.

Unfortunately these all-important skills of checking the answers are ignored in the mainstream mathematics education. I am trying to show my students at least some examples. Here is one from a recent lecture.

Problem: A truck travelled from A to B with average speed $60$ km/h and back, on the same road, with average speed $40$ km/h. What was the overall average speed of the truck?

As I expected, students’ answer was $50$ km/h. I asked them: “so you believe that if I put $$\mathrm{u\$}$and $$\mathrm{v\$}$instead of $60$ and $50$, the answer should be $\frac{$\frac{u+\mathrm{v\}\{}}{\mathrm{2\}\$}}$?” Their answer was affirmative “yes”. — “OK”, said I, “but what if if the truck run out of fuel at B and its speed on the way back was $0$ km/h. Your formula produces the answer $\frac{$\frac{60+\mathrm{0\}\{}}{\mathrm{2\}}}=\mathrm{30\$}$ km/h. But the truck will never come back! What is wrong?”

Of course, this example is from school level mathematics, but it gives a simple example of transformation of inputs as a way of checking the answer. In research mathematics, there are many other (quite sophisticated and subject-specific) methods of checking the result without looking into details; they are not a substitution for a proof, but allow to detect 95% of mistakes.