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.

This dictum appears to be increasingly popular on the Internet.

My answer to a question in Quora

At the administrative level, I think it was one of the legacies of Andrey Bubnov, People’s Commisary for Education in 1930’s: commisionning and publishing hard textbooks was a way of setting standards which could not be dilluted in provincial universities and schools. Bubnov’s post was a demotion: prior to that, he was Head of the Main Political Directorate of the Red Army, he was in charge of training of the Army, and, what matters for this thread, educating the officers, setting up military academies and providing them with textbooks. Presence of “canonical” textbooks sets standards obvious not only to teachers but also to students. In England, mathematics in schools and in universities is taught without textbooks (one of the reasons is that university level textbooks are too expensive) , with pretty damaging consequences for the state of education.

The most complete answer from a discussion is the following from Alex Sergeev, PhD in Physics:

As I understand from reading comments, the [discussion is] not school textbooks, but university textbooks, in particular Landau-Lifshitz was mentioned. In such case, I have to disagree with most answers presented.

Firstly, yes, they are indeed noticeably more hardcore than courses of a similar level in the US. Enough to compare two classic courses: Landau-Lifshitz and Feynman Lectures (which are, in turn, not really a walk in a park either, there are plenty of friendlier books). Same can be said about mathematical analysis books which I encountered. Soviet textbooks just go straight to the point and throw lots of definitions and formulas at you, without any preparation. The US textbooks try to explain simple things in more detail, and increase the complexity as they progress.

The reason for it, I think, is the difference in education systems. In the US, the point of education system is to teach students, as well as possible. In the USSR, the point was to get rid of weaker students and have only very good ones left, who would understand the subject no matter how hardcore the approach to it is. It might be more psychological rather than intentional, but in Soviet times it was a general sentiment: if you can’t do it straight-away, you are simply not good enough and should do something else. The US system tries to improve students and then select the best, the Soviet system tried to select the best and then improve them. The US system tries to make geniuses out of average students, the Soviet system tried to select geniuses disregarding average students. I might be a bit too categorical with this, but I don’t think it is too far from truth. Another possible reason, stemming from the above is a lack of competition. In the US, the education system is adapting to students’ need, if the books are not teaching good enough they get replaced or amended. In the USSR, the textbooks were centrally selected and approved, and students had to adapt to whatever they were given.

Edit: I also have just recalled this phrase very widely circulated during Soviet times: “We don’t have irreplaceable people”. (It actually originated much earlier, and was used by Woodrow Wilson, but is widely assigned to Stalin, who in fact never said anything like that. I also believe that the connotation was intended to be different.) This phrase, however, well demonstrates the psychology of Soviet system. No one cared if you fail, there’ll be another person who’d take your place. In the US, if student is struggling, it is partially a teacher’s fault; in the USSR, it is 100% student’s fault.

The title of this wonderful novel by Italo Calvino is already a masterpiece. As soon I saw  the book on s shop shelf, I bought it on impulse because of its title.  To the Russian ear, it had immediate connotations with the famous passage from Chekhov’s “Ionych”:

Then they all sat down in the drawing-room with very serious faces, and Vera Iosifovna read her novel. It began like this: “The frost was intense… .” The windows were wide open; from the kitchen came the clatter of knives and the smell of fried onions… . It was comfortable in the soft deep arm-chair; the lights had such a friendly twinkle in the twilight of the drawing-room, and at the moment on a summer evening when sounds of voices and laughter floated in from the street and whiffs of lilac from the yard, it was difficult to grasp that the frost was intense, and that the setting sun was lighting with its chilly rays a solitary wayfarer on the snowy plain. Vera Iosifovna read how a beautiful young countess founded a school, a hospital, a library, in her village, and fell in love with a wandering artist; she read of what never happens in real life, and yet it was pleasant to listen — it was comfortable, and such agreeable, serene thoughts kept coming into the mind, one had no desire to get up.

In Russia of “the period of stagnation”, the expression “The frost was intense” (“мороз крепчал”) became proverbial and was transformed into a less politically correct, but more politically charged, derivative.

And I was delighted to discover that my instinctive choice was correct and that, indeed, Calvino’s book “did exactly what it said on the tin“!

I discovered an article (in Russian: Герасимова А., Каррик Н. К вопросу о значении чисел у Хармса: Шесть как естественный пpедел) about the special role of number 6 in the works of Daniil Kharms.  Actually, there is a strong feeling that Kharms was writing about the classical subitising/counting threshold. Two samples: one inEglish translation, another is in original Russian (I cannot find a decent translation of the most mathematical of all Kharms’ works).

Falling Out-Old Women

A certain old woman fell out of a window because she was too curious. She fell and broke into pieces.

Another old woman leaned her head out the window and looked at the one that had broken into pieces, but because she was too curious, she too fell out of the window — fell and broke into pieces.

Then a third old woman fell out of the window, then a fourth, then a fifth.

When the sixth old woman fell out, I became fed up with watching them and went to Maltsevsky Market, where, they say, a certain blind man was presented with a knit shawl.

And another one, from the text  “Я вам хочу pассказать…” :

“заметим что:
1. Две pуки, две ноги, посеpедке сапоги.
2. Уши обладают тем же чем и глаза.
3. Бегать – глагол из-под ног.
4. Щупать – глагол из-под pук.
5. Усы могут быть только у сына.
6. Затылком нельзя pассмотpеть что висит на стене.
17. Обpатите внимание что после шестеpки идет семнадцать”.

(A translation of the  last line: 17. Notice the six is followed by seventeen.)