By V.A. Rokhlin

I was talking about the limit theory. Why, when I look at this theory, do I want to give it such a rough treatment, to banish it from the course of mathematics to non-mathematicians? I don’t want to say that it actually should be banned from everywhere, no. I only want to say the following. The theory of limits at present works as not a tool to introduce the basic notions of calculus, but as a very high and difficult to negotiate barrier that one has to climb over in order to understand anything.

And this barrier is absolutely unnecessary! It is, as a rule, impossible to cross for the students-non-mathematicians, and it is absolutely unnecessary.

Let me give you an example that will demonstrate it. I could have taken as an example the differential or integral calculus, but I will take the simplest of the two cases, or, it’s better say, subjects, the one that can be explained faster, which is the integral calculus.

You have to explain to the beginners what is the integral. Of course, you can begin, and it is probably the right way, simply from the area, as it is usually done.

But why not declare that integrals are simply the areas? Indeed, none of your audience doubt that these areas exist. You would have to spend a lot of time if you wanted to raise any doubts about it in your audience.

Of course, having the area at hand, you can easily construct the integral sums. You will say to your listeners that the area has the well-known properties. Here they are.

If one figure is contained in the other, then the area of the first figure is no more than the area of the second. It is difficult not to agree with that, and everybody will agree with you.

You can say that if you add two figures, put together two figures without any common interior points, then their areas will add, and everybody will agree with that too.

Then you will say that the area does not change if the figure moves around on the plane as a rigid body. They will agree.

And finally, you will say that the area of a unit square is 1.

These four properties, as you well know, uniquely define the area on a wide class of figures, on the class of all exhaustible figures.

The same properties, a little modified, determine the integral. They uniquely determine the integral on a wide class of functions. This way, starting with the area, you can define the integral by its properties that nobody doubts, because we are talking about the area.

Later on you point out that, if you construct the upper and the lower Riemann (or Lebesgue — no difference!) sums, then the area you are interested in will be between these two auxiliary areas. It follows directly from what I have just said. You write the very same inequality that is usually written in the integral calculus texts: the lower sum is no bigger than the integral that is no bigger than the upper sum. To put it shortly, it is the only number between the lower and the upper sums.

All this formulated in the language of areas is quite obvious, it doesn’t cause any doubts and is easy to digest. On the other hand, all this gives us a method to calculate areas. No limits are mentioned. If you want to demonstrate some identity, say, between two integrals or between an integral and a number, you simply notice that both numbers that you want to be equal are between the upper and the lower sums. Therefore they are equal, because the difference between the upper end the lower sum can be made less than any fixed positive number by taking an appropriate partition. This difference is non-negative, and therefore it is zero.

The question about the technique used in any particular case does not arise. All the technique is available. It is described in details in all the available courses, but in these course everything is tuned upside-down.

Very similarly, you can define the derivative in many ways. It will be better, of course, if you start with the intuitive meaning of the derivative, for example with the tangent line or the velocity, how it is usually done. But there is no need for limit theory here. It doesn’t mean that later, when you want, or when the curriculum requires it in a really reasonable way, and when your students really have to get familiar with limits, you can not explain that in fact the derivative is such-and such limit. But at the beginning it is absolutely unnecessary, and many students don’t ever need it.

To put it shortly, I would suggest the following approach, that I will call for convenience naively-axiomatic. The essence of this approach is that you in fact define the notions you are interested in by axioms.

For example, for the integral the axioms are the following.

The integral of a constant is the product of this constant by the length of the interval of integration. Of course, you don’t pull this axiom out of the thin air. At the beginning you will talk about the area. Everything will be prepared. That’s the axiom number one.

Axiom number two: if one function is no bigger than the other at every point, then the integral of the first function will be no bigger than the integral of the second.

And axiom number three: if you integrate a function over the interval that is the union of two smaller [non-overlapping] intervals, the corresponding integral will be the sum of the integrals over these smaller parts.

This is it! It is difficult to think of a less complicated approach. Of course, we can say these things about the area right away, as everybody is used to the area, about integrals we can say it a little later, but, based on these properties, all the integrals can be calculated beautifully. In fact, that’s how they are calculated in all the textbooks.

Moreover, this approach immensely simplifies all the applications of the notion of integral in natural sciences and in mathematics itself. If you want, for example, to show that some volume is expressed as some integral, you simply check that all these tree properties are satisfied. You don’t have to prove anything. Because you have uniqueness, the volume turns out to be an integral, and you get the formula at once, whether you deal with the volume of a body of rotation or the volume in some other situation.

The same applies to the torques, calculations of the centers of gravity, the moments of inertia and all the other mechanical and physical magnitudes. There is no need in

any “passing to the limit” and in general in that long and boring presentation that text books for technical universities are full of, there is no need for it at all.

I am just giving you some examples, as you understand. It would be possible to give much more. But if we want to talk seriously, we have to admit that the course of mathematics for technical universities is simply not put together yet. Not in the sense that there are no curricula and no textbooks, there are. By the course I man something else.

To avoid any misunderstanding, I will tell briefly what I mean by the expression “to put together a course.” Imagine a text that is longer, of course, than the text of the curriculum, but shorter than a textbook, and from which a competent person can see in details what topics to present and how to treat them.

It is not necessary that the students could understand this text. It has to be understandable to their teachers.

Unfortunately, if such a text were written for high schools, it would not be understandable by the teachers. But such a text, written down or kept in one’s head is the course that is put together.

What I want to say is that a course in mathematics that is put together in this sense does not exist yet. Of course, there are many possible approaches to achieving it, I hope that such courses will be put together and the textbooks will be written.

What I have just said about the integral calculus, these few remarks about it, apply not only to it, of course. I think that even the university students of mathematics would benefit from taking a one semester preliminary course of introductory analysis in which the basic notions were introduced not from the point of view of mathematical hairsplitting, so to speak, but were described meaningfully, with the viw towards applications, with a discussion of the geometric and physical meaning and with plenty of material for the exercises. After this, a more systematic and scrupulous treatment of the subject can be given to the students of mathematics.

Such experiments had been done. I don’t know what is the situation now here, say, at the department of mathematics and mechanics at Leningrad State University.

In any case, it looks to me that this naively-axiomatic approach could be useful at the beginning even in teaching professional mathematicians. In any case, I think it is absolutely necessary for teaching mathematics to non-mathematicians.

Now I will say a few words about the more advanced parts of the course. Indeed, some non-mathematicians are taught not only calculus and infinite series. They are taught, for example, about integrals over curves and surfaces, about change of variables in multiple integrals, etc.

These things already are not so straight forward and pleasant, they may present some technical difficulties. What to do about them? Here, of course, our naively-axiomatic method is not always sufficient. Here we have to accept some compromises.

Very recently I had discussed similar matters with one professor from Leningrad who had to explain to his students the change of variables in double integrals. How to do it? Being a mathematician, this teacher doesn’t want to swindle anyone. He is ashamed to swindle anybody, including his students. He wants to give them a proof.

On the other hand, change of variables in double integrals is a rather complicated subject. Several different methods are suggested. We can present our integral as an iterated integral and use the formula for change of variables in a one-dimensional integral. This is one way, there are many other ways. Here, as it looks to me, one habit is at play, professional mathematicians has a habit to give proofs.

But really, what should we instill in a future engineer, in a future physicist, to say nothing about a future philosopher? First and foremost we have to instill understanding. They have to understand the subject. We have to, and this is probably the most important, to stop teaching such things (or teaching them in such ways) that our students can not understand. The same goes for the school children.

It is very common to teach students such things that they don’t understand. What is the use to proof (in a rather restricted sense, of course), the formula of the change of variables in double integrals to the students of a technical university? Isn’t it better if they will understand this formula, even intuitively?

For example, can’t you first explain to the students how the area behaves under a linear transformation? Here is the plane, and a linear transformation is applied to it. How does the area of a triangle behave, or the area of a polygon? You probably can explain that. You can also explain that the area of a small region behaves approximately the same under a smooth nonlinear transformation. So the appearance of the jacobian will not be surprising to anybody. Of course one have to get used to the jacobian first.

[Here the recording was interrupted, probably for a tape change. In the lost piece of the lecture there was a discussion of the implicit function theorem.]

… axioms that establish equivalence of many approaches to, say, defining a surface in space. A surface in space can be defined parametrically, by three equations, it can be defined –implicitly– by one equation, and finally, can be defined as a graph of a function. All these three approaches are equivalent, and this equivalence is established by the theory of implicit functions.

To my surprise, few people know about it, even among the students of mathematics, before they really need it, maybe in some other subject. In the analysis course it is very rarely discussed. I haven’t seen it in textbooks either. The connection of this stuff with the maps is discussed in a very few places. But maybe to the students of a technical university all this can be explained without hairsplitting, so to speak, in such a way that they could understand (by some understandable examples, by some general formulations). To summarize, it looks like, in addition to our naively-axiomatic approach that I talked about, we need also a great freedom of dealing with the subject matter when we teach mathematics to non-mathematicians.

Maybe the teacher should try to recall how he experienced all this material when he was learning. We usually forget such things. For example, I can say about myself that I absolutely don’t remember what and how I learned in school. Maybe I remembered when I was a university student, but then I forgot it little by little. Maybe it is necessary to study this problem also by involving the students in a discussion and by listening to what they have to say. We don’t do enough of that.

As an example, I will mention the following observation. Without any doubts, the students that take this or that class give grades to their professors. They don’t put these grades into professors’ grade books, but they do grade their teachers. More than that, each professor or teacher has a rather stable average grade. It’s like rating in chess, so to speak. Unlike the chess rating, this rating is not published and even usually kept secret. I will not discuss whether it is good or bad. I think everybody knows whether it is good or bad. But undoubtedly, the help of the students is invaluable here.

I think their help is also invaluable in putting together a course that they take. Of course, the generations change, some students will help us to put together a course for their successors. Nevertheless, I think that this activity, undertaken with such arrogance by the professors, can not be successful without help of their students. The students should be consulted first and foremost.

I nave recently met a girl who goes to high school and lives next door, my next door neighbor. I had invited her to visit a while ago. She visited in the evening an said: “Here I am, you had invited me, and I have come. I can’t solve a problem.” (Cheers)

Well, I started taking a look at the problem and was horrified. First of all, it was impossible to solve this problem because you could not understand what the question was. The problem, professionally speaking, had been written incompetently. One could guess, of course, what was asked. We started guessing. But then I discovered, to my surprise, that I was guessing, but she had understood it all from the very beginning. (Cheers) But more than that, when I started solving the problem, it turned out that she knew very well what I was talking about. I thought first that she knew it all. But after we talked a bit more, it turned out that she didn’t understand anything. She knew all the words. But it turned out that these words were quite enough for solving the problem. And she knew all these words.

This kind of experience is invaluable for a teacher and for people that put together a curriculum. Keep in mind that we are talking about universal education, not about teaching the children that go to special schools, who had heard mathematical language and speak it themselves. No, these children are not taught mathematics professionally and have no intentions to work in it. They hear all these words, they learn to pronounce them, but their interpretation of these words is somewhat strange; they either have none at all, or it is strange. In many cases you discover really surprising interpretations.

It turned out that this girl and I understood some words very differently. Well, clearly we should wonder how her teacher understands all these words. And that is the real root of the problem, of course.

Let me draw your attention to yet another peculiarity of teaching mathematics and receiving such teaching. This peculiarity is the following. This teaching is always like magic, so to speak, teaching by occult, as one could put it.

Many years ago I saw a program of an entrance exam to some universities. That program had been approved and signed by some high officials, and in this program I saw something very strange. The topic was solving first degree equations with one unknown, followed by 2 types of equations:

is one, and the other is .

(Laughter) It turns out that these types are different! What was the matter? Some other girl helped me understand this one. From a conversation with her I understood that all the numbers are positive. (Laughter) The negative numbers don’t exist, and indeed, a negative number is the following. It is a positive number in front of which there, for everyone to see stands a minus sign. (Laughter) But if it is so, then, of course, these equations are of different types. Indeed, if you move b to the other side of the equation, you have to change the sign, but all the numbers must be positive! So, a must be positive,

b must be positive, clearly there are two types of equations.

But how had such stuff ended up the program? Well, it got there from the school program. There was such a requirement that the program of the entrance exams should not differ from the school program. If not, then what would happen? How would you take an exam? Now, from the school program it had migrated into the entrance exam program.

And how did it get into the school program? Well, it’s clear how. The school programs are developed by teaching specialists who know how to teach mathematics, and tell us how to do it. Well, they know how to teach mathematics, but they really think that all the numbers are positive.(Liveliness in the hall)

Of course no teacher — well, maybe there are very few, but no ordinary teacher will tell in the class that all the numbers are positive. It is not written in a textbook, so how can he say such a thing? But he thinks so! And if he does, so will do his students. How do they catch it? It’s an interesting question, but it is true beyond any doubts.

The understanding of the subject by the teacher is passed to the students. The understanding of the subject by the lecturer is passed to his audience. It is passed in a mysterious way, but very reliably. We have to keep it in mind. No extra education of the teacher, no proper treatment in the textbook will help if the teacher thinks differently.

The teacher, the instructor is in this sense the central, the decisive figure. I want to repeat again that I absolutely do not believe that you can somehow improve, or change teaching by improving the programs, the textbooks, but without changing, very seriously, the training of the teachers.

True, there are some methods of retraining, various continuous education programs and so on. How effective are these? I don’t have any factual data on that. I have only my personal experience and the personal experience of my friends. And here I have to express a rather grim prognosis. According to my data, any retraining or additional training of the individuals who had learned mathematics at some institution of higher education, such as a pedagogical university, leads to nothing. If they had learned nothing while being students, if during the long following period of teaching they have managed … I am afraid to say forget, it would have been for the better, if they have managed … well, o.k., forget what they had learned, then, of course, an additional training may lead only to cosmetic changes. They will get used to the new words, to the new teaching methods, but it will not change anything that matters.

It seems to me that the universal teaching of mathematics can be improved only one way. It is a slow and difficult way, but it may be the way that is possible. We have to gradually increase the production of competent teachers.

There are absolutely enough capable people available for that. Unfortunately, we don’t teach them properly. The reason is understandable: there are not enough teachers to teach them well.

In good old times, when all the mathematics teachers in high school were the graduates of the top notch universities (it was long ago, before the universal education), the situation was better. We hear it often, read about it, that long ago the mathematics teachers were better. They knew their subject better and taught better. I don’t know if it is true, but if it is, the reason is that these teachers studied not in pedagogical universities, but in a few top quality universities. Now they study in pedagogical universities and many other mediocre universities.

Well, I would not want this lecture to be remembered as such a grim one. I want to say something optimistic at the end. I think that if we need a hundred years to prepare, gradually, layer-by-layer, enough competent mathematics teachers, and organize good mathematical education in our high schools and our institutions of higher education, if hundred years is enough for it — then it is good. (Liveliness in the hall)