Here is a (still rough) translation of the first half of the lecture, the Russian original is available from

В. А. Рохлин Преподавание математики нематематикам

I will hopefully finish the second half in a couple of days.

V.A. Rokhlin

Vladimir Abramovich Rokhlin (1919-1984), a remarkable mathematician, widely known by his ground-breaking work in theory of dynamical systems and topology, who had mentored several top-rate mathematicians, was also a brilliant lecturer. He also thought a lot about the problem of teaching mathematics. His own pedagogical experience was wide and diverse, as he taught at a technical university, at several pedagogical universities, and — during the last 20 years of his life — at Leningrad State University (further details can be found in the book “В.А. Рохлин. Избранные Работы. Воспоминания.” Изд. МЦНМО, 1999).

Here we are publishing some parts of one of the last lectures by V.A. Rokhlin at a meeting of Leningrad Mathematical Society. This lecture took place on November 20 of 1981 at the Leningrad Scientist Hall. This publication is based on the audio tape recording made by B.A. Lifschitz, then transcribed and edited by the efforts of O.Ya. Viro, B.A. Lifschitz and S.V. Ribin.

Unfortunately, the beginning of the lecture (about 15 minutes) has been lost. In these 15 minutes V.A. talked, in particular, about his experience of teaching mathematics to non-mathematics majors at the universities of Arkhangel’sk, Ivanovo and Kolomna in 1950s.

–A.M. Vershik

…At elementary school children are, naturally, not presented with any serious proofs, but they are given some formulations. For example, they are taught how to divide a fraction by a fraction. There is a rule that is formulated, and they have to know this formulation.

Once I had been to a talk by a teaching specialist who explained how to teach children to divide a fraction by a fraction. He said that he formulated the rule and showed several examples of its application. Then children did some examples too, and then they had a written test. And in this written test almost all of them made the same typical mistake. In some cases –the same for all who made the mistake– they for some reason didn’t flip the divider fraction before multiplying it by the dividend; in some cases they did flip it, and in some cases they didn’t.

What was the reason?

The teaching specialist explained that the analysis of the examples had shown the reason for the mistake. The reason was that in the examples given before the test all the divider fractions were proper. The children figured out from the examples what to do, and they did just that.

So, the rule that was formulated in the beginning by this teacher, was not at all the formulation that these children had understood and had been guided by. The intended rule was simply dictated to them as a matter of routine. It had to be dictated, and it was dictated, but it was ignored by most of the class.

The fact that the children at that age had to learn how to understand the formulation of the rule and how to use this formulation, not just the examples shown to them, this fact had totally escaped the attention of that teaching specialist and his audience.

I consider the idea of giving a worked examples immediately afterhe formulation of the rule to be a harmful misconception. I have no doubts that the children should, by following the rule, calculate several examples by themselves first. Of course, they have to become fluent and do it automatically later, but the very valuable observation, the observation that the children, by working on some examples, can learn and should learn how to understand the formulation, was totally lost.

This is only one example. In fact the story that I just have told reflects the basic pedagogical principles. They are wide-spread.

Nowadays one can graduate from high school without in fact solving a single mathematical problem. Children can just emulate the templates and examples given to them to fulfill their school duties.

(*)Probably at the lost beginning of the lecture Rokhlin talked about the harm done by the universally accepted methods of teaching mathematics. Unfortunately only his criticism of the principle that any rule should be immediately illustrated by examples of application, survived. (a footnote by O.Ya. Viro)

When during the recitation sections the first year –and often not only the first year– university students are given a problem to work on, please observe what they do. Of course I am not talking about the math majors.

My subject is different, it is teaching mathematics to non-mathematicians. But, according to my experience, in a class of non-mathematical students only 2 or 3 people actually solve the given problems, the rest of them are sitting and waiting. They actually don’t understand what is wanted of them. They are waiting for the problem to be solved on the chalk board, or for somebody to tell them how to solve this kind of problems. They show no initiative, they are totally unfamiliar with this setup.

It sounds strange, because, if you ask any of these young people to buy some groceries and some medicine and also tell them when the grocery store and when the pharmacy are closed for a lunch break, they will solve this problem beautifully. They will pick the right times to go to the stores and will not go to the pharmacy when it’s closed. Well, there are some strange cases even here (liveliness in the hall).

The problems the students are given on recitations are far simpler than this problem I am talking about, with the grocery and the pharmacy. Nevertheless, the idea of beginning to solve these problems even doesn’t cross the minds of these children, and some of them are 17 or 18 years old. This astonishing situation is, of course, the result of how they were taught mathematics.

(*)Probably at the lost beginning of the lecture Rokhlin talked about the harm done by the universally accepted methods of teaching mathematics. Unfortunately only his criticism of the principle that any rule should be immediately illustrated by examples of application, survived. (a footnote by O.Ya. Viro)

Similarly, the level of knowledge in the area of exact sciences, not only mathematics, is surprisingly low among the adult population, i.e., among people who are no longer students, who had graduated from high school or a university a while ago and are considered educated.

If you take the writers, the musicians, the actors, the film directors, many doctors (naturally, I am not talking about all the people educated in humanities), you will find absolutely surprising things.

They proudly say and write that they are bad at math or physics, they say it with a sneer, and, in general, don’t make much of a difference between the two. If you let me use a rather mundane expression, for them math is some area of technology or physics, something not very deserving of respect, but, in any case, something that should serve their needs.

With the doctors, of course, this situation is changing. These days they teach mathematics even to philosophy students and students in other humanities. All are taught by using the same template, without achieving any insight into or understanding of the subject.

If you read carefully the articles –some of them quite interesting– written by people educated in humanities, you will notice that they enjoy using the expressions from mathematics and physics. It is fashionable, it is modern, but God, what they are writing! Strikingly, they have less than a vague idea of what a factor is and what a divider is, what a degree is, what positive is and what negative is. From the texts that you can come across in newspapers and magazines you can see clearly that they studied all this, for example, the idea that the negative is somehow related to division, to the degree… Well, take, for example, such phrase: “something is negative and therefore it is not zero, but infinity!” (laughter).

I am not exaggerating, I can give you some references where you can read such things. All of them are written by some well-known, even famous people, it is published in the newspapers. These people memorized something in the elementary school and somehow they think that they remember it correctly.

What has to be done about all this? Well, it’s a difficult question.

I don’t believe that this problem, that is the problem of getting a higher level of universal, general education in exact sciences (and mathematics in particular) can be solved fast. It is a difficult problem, and solving it will take a long time and huge efforts.

One may ask whether it is possible or useful to solve this problem. Up till now, till very recent times, people in all the civilized societies were educated in humanities. There were no human society in history where people would be universally educated in exact sciences.

The classical education was wide-spread, very wide-spread among the educated strata of the society, but the exact sciences were never made available even to the educated to any serious degree.

Many modern developing countries have long history, going back centuries and millennia. Many have their own intellectual elite but again, this elite is educated mostly in humanities. They like to study humanities, but the exact science they study with some reluctance, and not very successfully.

Concluding these preliminary remarks, I want to say that nobody really knows, and, of course, I don’t know what would be the result of a serious universal education in mathematics and exact sciences, even if it were possible, whether it would make things better or worse. I don’t know it, and nobody does. We don’t have any experience. Nevertheless, for some reason we want to achieve this goal. We are trying. Somehow we feel intuitively that it will be good if our children and grandchildren become familiar with the logical culture, with the mathematical culture, if they will understand the exact sciences better.

It is very likely that it will lead to some incredible upheaval, some unprecedented results, who knows? I don’t know.

Turning to the more narrow subject of my lecture, I want to say that teaching mathematics to the future mathematicians is infinitely easier than teaching mathematics to non-mathematicians.

After all, we can be earnest with the future mathematicians. Here we have the subject that we know, and we are trying to teach this subject to the future mathematicians. No matter how masterful or how mediocre we are in lecturing or leading the exercise sections, we, by knowing the subject, can transmit our knowledge to the interested people. But how should we deal with those who are not interested and think they have no ability, or just say so.

Very often people who say so are simply stuck in, so to speak, intellectual laziness. This intellectual laziness is a very common phenomenon, and sometimes you can easily find out. By talking to a person who claims to have no ability, that he is infinitely far from all this, by talking to him just a little, you discover that he beautifully understands all you say.

So, the question of ability in this field is a complicated question, you don’t have just to take the word for it when somebody claims to have no inclination to these subjects, claiming to be interested only in humanities.

Before I really turn my attention to teaching mathematics to non-mathematicians, I should first explain who I call a non-mathematician. It’s rather useless to discuss this question in general. I will simply say what I have in mind right now, when I am talking about this topic. I have in mind teaching mathematics to people who have no intention of working in mathematics, who study it either for applications or because they have a non-professional interest in it.

In fact, the presence of interest makes it easier for the teacher. But very often there is no such interest, instead there is a disgust. There are plenty of students now that have to learn some exact sciences, but have no interest in them. Nevertheless, they have to pass the exams, etc. How should we treat such people? What to do about the people who are interested in mathematics but are taught in such a way that makes it impossible to learn? There are many mathematical curricula for the students of technical universities, they vary in their length and their content.

But very often these curricula and the corresponding textbooks are not independent, they are simply watered-down courses for the students of mathematics. There is the same order of presentation, the same limits, the same derivative, the same integral, the same second degree curves, and so on and so forth.

The order is the same, but the treatment of the material is less understandable. There are no proofs that would help to understand the inner workings of the subject. The authors of these books have no talent for writing. It is boring, it is not understandable. The students are lucky if the lecturer gets them interested by explaining something outside of their textbook, so to speak.

This situation is very common, and not only in our country, as I said before. This situation is totally international, and, as I think, the reason is the following. Probably the technical universities, the pedagogical universities and high schools need their own, special, courses of mathematics. Each category of students (if it is big enough, of course) probably needs its own course, and totally different from the one for teaching mathematics students.

I think the main defect of these courses, the main reason for their failures is the following. None of the prominent mathematicians ever worked on putting together a course of mathematics for non-mathematicians.

(*)Maybe because they were too preoccupied with being prominent and had no time for making mathematics understandable by ordinary people? (a note by misha)

I am talking now not about a high school course, but about a university course, and I mean the following.

Usually, before doing differential and integral calculus, the students are taught the theory of limits. It’s the same in high school now. Limits are taught there too.

Nevertheless — and it is a striking example of the current difficulties — the limits is the part of the course that is the most difficult to understand, and, most interestingly, absolutely unnecessary. All the differential calculus, all the integral calculus, and, in general, all the classical mathematics, to say nothing of the finite mathematics, can be treated beautifully without limits. Even more, they are not needed there. They are absolutely extraneous phenomenon, extraneous subject that had been introduced into this area by the people who wanted to build a proper foundation of analysis.

Meanwhile, the purpose of treating the foundations is not achieved in a technical university course, it is not even a purpose of such a course. From this example we can see already that these courses are absolutely thoughtless and are just watered-down university courses in analysis (*)for mathematics students, a note by misha).

Now I will explain my thought about limits in some detail. When I went to high school (maybe it’s still the same now), they explained to me that the area of a circle is some sort of a limit, then they wrote something, said something, and got some formula for the area of the circle. What was said was difficult to understand then, but when I became a mathematician, it became totally clear to me why it was so difficult to understand, the reason being that it was all nonsense.

Nobody of my fellow-students had any doubts that he knew what the area of the circle is. Rather it looked strange to us that for the circle the area was defined for us, but somehow there was no definition for the other figures. It was strange to us that the area of a circle that was totally understandable to us, is defined by using some limits that we could not understand. We felt strange of course (and all the children who thought about it a little bit also feel strange) that some theorems about limits are needed to establish some very clear an simple things that we had never doubted.

But really, why should we define the area of a circle and to prove that it is ? Why not just say that the area of a circle is by definition, what is the difference? It looks like the difference, and a really serious one, is that not only circles have the area, that the area is a general notion, that the area is defined for a wide class of figures, that it has the properties, known and used by everyone, that make the area a useful notion.

So, the attitude towards the area adopted in the high school course was then (and maybe in many cases still remains now) absolutely bizarre.

But in teaching mathematics in technical universities the same attitude is adopted towards integrals, toward derivatives, towards volumes and masses, towards density, towards charges, towards the moments of inertia and in general towards all mathematical and physical quantities of such integral and differential nature.

From the point of view of a person who is not a professional mathematician, all these things exist, they don’t require a definition, they require a computation, and they have to be ready for applications. That’s what is needed.

The point of view that these properties should be defined is not appropriate in teaching here, at least in teaching such a person that doesn’t doubt the existence of area of all the figures, you don’t have to define the area to such a person. Such person has to study the properties of the area, to learn how to calculate this area.

The same is true for the other mathematical notions. Let’s take the notion of the integral. On this occasion I would like to ask a question of historical nature. Tell me, please, did Archimedes have a notion of the integral or not? There are different points of view on it. Some say he did, some say he didn’t. I will express my own opinion. I think that Archimedes, maybe the greatest mathematician of all times, did not have the notion of the integral. Here is why I think so.

Archimedes many times, by many different methods, calculated the integral

He calculated it when he studied the areas bounded by a segment of a straight line and a segment of a parabola. He calculated this integral when he calculated the volume of a ball, and in many other situations. Each time he used a special approach, very ingenious, brilliant. But it looks like he did not know that it was all the same. He probably had a feeling.

The reason for it is totally understandable. The Greeks did not have a notion of the real number. The volume and the area were totally different entities for them, they were the geometrical entities that they did not compare to each other. For example, Archimedes would have protested against such an expression as . He would have said that and can not be added, that it is the same as to add a line segment with a planar figure. But we do it. We have numbers. And it looks like Archimedes didn’t have the numerical notion of the integral. If he had, he undoubtedly would have not calculated the same integral many times.

On the other hand, Archimedes left us the method of exhaustion, that may be very fitting for teaching mathematics to non-mathematicians. Because of this method, instead of all the theory of limits we need only one single fact, I will formulate it now. This fact is very simple. It is that if a non-negative number is less than any positive number, then it is zero. I repeat, if a non-negative number is less that any positive number, it is zero.

This fact is not difficult to prove, but maybe a proof is not even needed. After you are familiar with this fact, you can use it to prove all the equations that are encountered in the differential and integral calculus and in its applications, and more generally in all the analysis, provided that don’t deal with the theorems of existence.

You don’t prove the existence. The limit theory is only needed to prove the existence theorems.

If you don’t have to prove the existence of the area, limits are not needed. If you don’t need to prove the existence of the integral, the limit theory is not needed, and so on. If you only need to calculate, you can get by without the limit theory. This makes the differential and integral calculus infinitely easier right away.

[A BREAK]

January 5, 2011 at 5:23 pm |

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