Puzzles of the Number and Dialogue in the Early Grades of the School of the Dialogue of Cultures

From the author^*
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English translation © 2009 M.E. Sharpe, Inc. The texts translated here, selected by the author, were first published in Arkhe, no. 2 (1996), and no. 3 (1998). Additional detail is provided in footnotes.
^*Irina E. Berlyand, “Ot avtora.” Published with the author’s permission.
Translated by Nora Seligman Favorov.

Irina E. Berlyand has worked at Moscow’s Institute of General and Pedagogical Psychology, where she became acquainted with V.S. Bibler. In 1991 she began working in Bibler’s Dialogue of Cultures group, which, until 1994, was sponsored by the Kemerovo entrepreneur I. Panchishin and later became a research group at the Russian State University for the Humanities, School of Philosophy. She is primarily interested in the phenomena of culture as an object of study in early education (as envisioned within Bibler’s School of the Dialogue of Cultures concept). Her most important publications include Play as a Phenomenon of Consciousness [Igra kak fenomen soznaniia] (Kemerovo, 1992) and Puzzles of the Number [Zagadki chisla] (Moscow, 1996), as well as Bibler’s Zamysly [Conceptions], which was published posthumously and on which she worked with A. Akhutin. E-mail: iberlyand@gmail.com.
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Puzzles of the Number is what imaginary lessons in mathematics are called within the School of the Dialogue of Cultures, SDC, in which imaginary first-graders and their teacher discuss real mathematical problems of number, calculation, and measurement; learn to write numbers and perform various operations with them; and solve problems.
The article’s structure is borrowed from Imre Lakatos’s Proofs and Refutations [Cambridge University Press, 1976]. But the characters portrayed in my Puzzles differ from Lakatos’s characters. His pupils are first and foremost logical personages representing various theoretical and logical aspects of the problem under discussion. The characters in Puzzles of the Number are idealized, imaginary characters, but their dialogue is an attempt not only to exhibit logical and theoretical problems associated with the construction of the concept of number but also to present just how this exhibition can be managed in the early grades within the School of the Dialogue of Cultures by constructing the points of wonder. The prototypes for Lakatos’s characters were the mathematicians Euler, Lagrange, Cauchy, and others; the prototypes for my pupils are not just mathematicians, logicians, and philosophers, but also actual elementary schoolchildren — real girls and boys from Soviet, Ukrainian, and Russian schools in the 1980s and 1990s — primarily pupils from the School of the Dialogue of Cultures. Each of them has a specific “logical role” to play in the dialogue — one has a tendency to imagine a number as a tool of counting, another as a sort of independent entity with a form of its own, a third as a general law for generating numerical series out of units, and so on. Furthermore, almost every one of them has a prototype, and much of what they say and how they reason is based on the statements and reasoning of real elementary schoolchildren.
In keeping with the overall purpose of the lesson, the characters’ statements, to a greater or lesser degree, have been stylized or developed to the extent that they were not fully expressed by the child, but I tried to preserve the children’s approach to the subject matter and their style of reasoning and speaking. The characters portrayed in Puzzles of the Number are more consistent, more intelligible and thorough in their reasoning, and more attentive to the arguments of others than their prototypes, and some of the characters are entirely my invention. However, it seems to me that their manner of comprehension, their approach to the subject, and their style of reasoning and talking are consistent with what modern elementary school pupils could think and say in a School of the Dialogue of Cultures classroom. The names inserted in brackets are the names of my young coauthors — the prototypes of imaginary pupils and the authors of particular statements, and here, with gratitude, I list their teachers, who graciously acquainted me with the transcriptions of their classrooms’ lessons: N.I. Kuznetsova, V.G. Kasatkina, and S.Iu. Kurganov.

Puzzles of the Number: Imaginary Lessons in the First Grade of the School of the Dialogue of Cultures [excerpts]^*

Lesson 1

Teacher: Do you know what a number is?
Alpha: It’s when you count.
Beta: I know how to count. I can count to a hundred.
Teacher: Well, why doesn’t someone try to explain to me1 just what a number is.
Gamma: Numbers are — one, two, three, four . . .
Children: (all together) . . . five, six, seven, eight, nine . . .²
Teacher: That’s good. I can see that you know a lot of numbers. And would you be able, without naming different numbers, to simply explain to me what a number is?
Beta: Without saying a number? No one can do that. Even if you tell me what a chair is, if I haven’t seen a chair, I won’t understand what it is.
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^*Engish translation © 2009 M.E. Sharpe, Inc. From the Russian text: Irina E. Berlyand, Zagadki chisla: voobrazhaemye uroki v 1-om klasse Shkoly Dialoga Kul’tur (Moscow: Akademiia, 1996) [excerpts], 1. The book’s first thirteen lessons were published in 1996 in the second issue of the culturological almanac Arkhe (pp. 307–56). Until 2000, this almanac was compiled and edited by V.S. Bibler and after his death by A.V. Akhutin and I.E. Berlyand. Each issue of the almanac contains a section titled “The School of the Dialogue of Cultures.” The book edition of Puzzles of the Number was published in its entirety in Moscow in 1996 by the Akademiia publishing center. It was also posted at www.bibler. ru and www.culturedialogue.org.
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Teacher: That’s fine. But when you already know what a chair is, can you look at a new chair, one that you’ve never seen before, and know that it’s also a chair?
Beta: Yes, because it’s like other chairs that I’ve seen before.
Teacher: How is it similar?
Beta: It also has a seat, legs, a back . . .
Delta: It’s made of wood.
Eta: It could be plastic.
Kappa: A chair is what you sit on. It could be made of anything.
Alpha: So a number is what you count with like: one, two, three . . . and more.
Kappa: Yea, that’s it! A number is what we count with.³
Teacher: So do we know what a number is now? A number is what we use to count, right?
Beta: I don’t know, a number isn’t exactly what you count with. It’s what we get when we have already counted [Nastya Bykova, first-grader]. If there are a lot of chairs and I count them and say: four. Four is a number.
Kappa: But how did you count? You counted to yourself: one, two, three, four. You used numbers to count each chair, and when you counted the last chair you said: four. Both what you counted with one, two, three, four and the four you wound up with when you finished counting are numbers — both of them are numbers.⁴
Eta: The number four is made in a way as if it has inside it the other numbers: one, two, three [Masha Kvashenko, second-grader].⁵
Teacher: And how are the other numbers made? Do they also contain numbers?
Eta: Oh yea the number five has all the same numbers inside it plus the number four!
Gamma: And big numbers have a lot of numbers inside.
Delta: But I didn’t count: one, two, three, four. I looked at those chairs and I saw right away that there were four.
Alpha: That’s because you probably counted them to yourself quickly, you just didn’t notice that you counted. There aren’t many of them, so you counted them right away.
Kappa: You have known how to count for a long time — you’re used to it, that’s it. And you count fast. But I have a little brother, and when you ask him how old he is, he shows four fingers and then counts them: one, two, three, four. And then he says: four.
Delta: So how does he know how many fingers to show? That means that there were already four before he counted them.
Alpha: So what if there were a lot of them? For example, 128? Isn’t 128 a number? Delta, what do you think? Or is a number just what you can immediately see — you look and you see how many there are?
Delta: One hundred twenty-eight — that’s a number.
Alpha: Well, then. And if there were 128 chairs you would never be able to see right away how many there are; you’d have to count them one by one.
Kappa: So that means that we count with numbers. And we can see right away how many there are but only when there are just a few things. And that’s just because we already counted before.
Delta: That’s true, but what about big numbers? For little ones — one, two, three, four — they’re there before we count them. They’re there from the start.
Eta: And maybe grownups, mathematicians, see big numbers the way we see little ones. They look at a big pile of things and see that there are 128. Or a million. They can tell right away, without counting.
Gamma: (laughing) There’s no way that could happen. How can you see a million right away? That’s an awful lot.
Eta: Maybe it’s a lot for us, but not for mathematicians. Maybe for them a million is like three or four for us.⁶ Kappa was talking about his brother, he’s little, he can’t even count four fingers right away, but we can. And he counts one at a time. Maybe when we grow up and learn to count, we will be able to see a million right away.
Gamma: Well, fine, let’s say you’ll see a million right away. But what about 100 million?
Delta: Let’s ask the teacher how he counts a million — right away or one at a time.
Teacher: Of course I can’t see a million objects all at once. I think even real mathematicians wouldn’t be able to either. But small collections of objects — three, four, seven — like Delta, I can also see all at once, without counting.
Delta: Maybe big numbers aren’t made the same way as small ones? Maybe there are different kinds of numbers.
Teacher: Maybe. We surely will talk about that some more. But we can all agree that big numbers are also numbers, right?
Children: (all together) Yes they’re numbers!
Teacher: That means that when we explain what numbers are we have to keep in mind both big ones and little ones.
Alpha: What’s to explain? It’s already clear. Well, we can’t explain numbers in words, but we’re all able to count, and we’ll learn to do it even better [Masha Kvashenko, second-grader].
Kappa: No, you’re not right, Alpha: We can count without understanding what we’re doing. And maybe we’ll even count correctly. But in order to know that we counted correctly, we have to understand what numbers are.
[The school bell rings signaling the end of the lesson.]
Alpha: I’m not sure that you have to understand what a number is in order to count it correctly. There are probably rules for how to count. You have to know them. Machines count even better than people, but do you really think that they understand?

* * *

Lesson 3

Teacher: During the last lesson Gamma said that first of all man thought up the number one, and without counting anything, he just saw that he was alone and he said: one.
Kappa: So, was that really a number? A number can’t be alone.
Teacher: Why can’t a number be alone?
Kappa: Because one number isn’t a number. You can’t do anything with it — you can’t add or subtract as long as it’s alone. You can’t even count things. After all, Gamma himself said that this first “number” didn’t come from counting, but someone just immediately saw: one. When a number is alone, it’s not a number. There have to be other numbers for a number to be a number. And you have to be able to do something with the numbers, and there’s nothing you can do with one number.⁷
Beta: One really is a certain kind of number.⁸ It’s sort of like you can do anything with it that you can with numbers, but at the same time one isn’t the same as they are. You really can see right away: one.
Teacher: And what does that mean, that there is just one thing?
Beta: It means that there’s nothing next to it. And you can see this right away.
Teacher: There is nothing?
Beta: Well, probably that’s not for sure that there is nothing. There are none of the same things. We say: there is one pen on the table. There is something next to it, but it’s not another pen. And since Gamma was talking about a desert where there was nothing at all except a person . . . I don’t know. Maybe there you really couldn’t have number.
Alpha: I can’t get what we’re talking about. Are we talking about how a number was invented, how it came about, or what it is? These are all different things. Maybe at first there was the number one. But by the time other numbers appeared and people thought up how to count, at that point the number one was no different from the others. You could do all the same things with it as you could with the other numbers.
Delta: I agree with Beta. I think that one is not the same kind of number as the others. I don’t need any other things to be able to see that there is just one thing. I am looking at this pen and I see that there is just one of it. I’m not thinking about the fact that there is no other pen, I just see that this pen is the only one. In the same way I can see that it’s red and that there is one of it.
Kappa: I don’t think you’re right. Not just about number, but about color as well. You wouldn’t see that it was red if you didn’t know that there is such a thing as other colors. Newborns probably can’t tell what color the first thing they happen to see is. When we see things of different colors we begin to tell them apart. But one color isn’t a color. Just as one number isn’t a number. Numbers become numbers only when they are next to other numbers.
Gamma: So, Kappa, you think that the only way numbers can be thought of is all at once.
Kappa: Yes.⁹
Delta: I don’t think that’s how it happened. Not all numbers are the same, and not all of them were thought of at once. Alpha says that numbers, maybe, were not thought of at the same time. Maybe, there was some first number, but now that there are a lot of numbers and we are able to count, it does not matter. Now they’re all the same to us. That’s not what I think. I also don’t know which number was invented first. But it doesn’t matter. Whatever happened at first, now numbers are all different. And one isn’t special because it might have been thought of first, but because it’s that kind of number. After all, we count one by one.
Gamma: But remember, Delta. Alpha showed [in the previous lesson — Eds.] that you can also count two at a time.
Delta: I remember. And then you said that we would have to arrange these chairs in twos. We just took two and transformed it into one. Like a pair of shoes. Two shoes that go together — that’s a pair. One pair. So, it winds up we don’t have four shoes, but two pairs, see?
Beta: So tell me, Delta, if I have two pens and you also have two pens, how many do we have together?
Delta: Four, of course.
Beta: So how did you count — one at a time or two at a time? Two plus two is four?
Delta: I don’t know . . . I think I really did count by adding — two plus two is four.
Alpha: And did you care how they were arranged? When you count, the way the things are arranged does not matter at all. You could add Beta’s pens to yours or the other way around, right?
Eta: Well, I think it’s important. For me, four things to be a real four have to be arranged like this : And not like this: • • • •
Alpha: What do you mean? Aren’t there four of them? Will there be more dots if we change the order?
Eta: No, not more, but a foursome — a real four — has to be arrangedlike this:
And that second one is not like a real foursome.
Alpha: I don’t understand that. I think a number shows how many things there are, not how they are arranged.
Lambda: Look, when Eta was talking about four things arranged in a square, he didn’t say four, he said a foursome. It’s not just a number, but something has already been done with it, with the number. It’s been — put into some kind of order. It’s not four, but a foursome.
Delta: Then that would be one foursome. We turn two into a pair and four into a foursome.
Beta: So we have numbers when we count things, and then they are all the same, and we have numbers that we see by themselves, and those are all different.
Eta: Yes, and, like Beta said, when we are not counting objects but looking at numbers, we think about the numbers themselves, and not about how many things there are and how to count them, then they are different. Each one is arranged in its own way. For example, like this:

one two three four five
That way they look beautiful, the right way.¹⁰
Alpha: I really can’t agree that those are numbers. What Eta drew are pictures. They are beautiful; I also like them. But they are not numbers. When we are counting, it’s not important to us how everything is arranged, how it looks. The only important thing is — how many there are. Numbers don’t have a shape.¹¹ You can’t draw them.
Delta: Well I think they have a shape. At least some numbers do.
Teacher: Some of them, Delta? Not all of them? Which ones do, in your opinion?
Gamma: Delta keeps talking about big numbers and small numbers being made differently. He talked about that during the last lesson too. Maybe big numbers don’t have a shape.
Delta: Maybe. Maybe big numbers don’t have a shape; maybe they aren’t arranged in any way at all. Probably big numbers are the numbers that Alpha was talking about: they don’t look like anything, they just tell us how many: more or fewer.
Alpha: What are you talking about — big numbers, small numbers! No numbers have a shape. And what you’re drawing that’s not numbers, that’s a drawing. As Lambda pointed out, you even call them something different: not three, four, five, and so on, but a threesome, a foursome, a fivesome. That’s what I guess you call them, but normal numbers we count we call the normal way: three, four, five.
Eta: Well, I think that big numbers, maybe, are also arranged somehow. We just can’t see it. I agree with Alpha that you can’t look at numbers the way Delta does: big ones one way and little ones another way. But I think that all numbers must have a shape, be arranged somehow. But big numbers probably have a complicated shape and we can’t see it, we don’t understand it.
Teacher: Alpha, tell us please what you think of what Beta said? Remember, he said that when we use numbers to count objects they have no form, and when we think about the numbers themselves, when it’s as if we are looking at them for themselves, then they do have a form.
Alpha: I don’t think we can know what numbers are by themselves. When we use numbers to count we can tell what they are. But what they are all by themselves, I don’t understand, but I don’t think it matters. It’s not interesting to me.
Beta: For me it’s the other way around. Now it’s getting interesting. Before I thought that if we know how to count, we know what a number is. But now. . . . We all count the same way, but we all think about what a number is differently. Now I’m interested not in how to count, but in what is a number for real.¹²
Teacher: Kappa, what do you think? You’re not saying anything. You’re not interested?
Kappa: I’m thinking.
Teacher: About what?
Kappa: I’m thinking about what a number is.
[The school bell rings.]

Lesson 14

Teacher: An important question has emerged: does equality depend on the units we use to count? Kappa thinks that it does not depend on the units, that if we use other units, two equalities will remain equal.
Kappa: Of course, it will. Look here. I’m drawing dots: ••••••••
••••••••
They are the same — eight dots here and here. We’re counting with dots. Our unit is a dot. You could write it like this: here, on the left, we can represent eight dots with a circle around the unit we’re using to count and write how many of such units there are.
We wind up with eight
.
We count on the right and also wind up with 8 = 8.
Now we’ll count using another unit, for example two dots. The unit will be a pair of dots
.
So on the left we have four such units, 4 , and on the right we have four such pairs, 4 . 4 = 4 , or 4 = 4. And it will always be that way if we don’t add anything, if we don’t draw any new dots. In my opinion this is completely obvious, if we count something without adding anything or taking anything away.
Delta: You already knew, before you counted, how many dots there were, here from the start the number was the same on the right and on the left, therefore it didn’t change. But if we were adding, we wouldn’t know whether they would wind up being the same.
Teacher: Explain, please.
Delta: Here’s an example. I’m drawing five dots on the left. And on the right I also have five, but squares instead of dots Then on the left I draw another three dots. And on the right I draw another three squares.
Kappa: And you get eight, of course. On both sides it’s eight, because five plus three equals eight. It will always be that way, no matter what we’re counting and what unit we’re using to count.
Alpha: Delta’s example is different from Kappa’s example. In the first example we didn’t add anything, we just counted the same dots using different units, but in the second example we added. There’s another difference. In the first example we were counting the same things but we chose a different unit. We could count the dots with dots or with pairs of dots. And in the second example we were counting different things. After all, we can’t count dots with squares or squares with dots. Here we wouldn’t be able to choose between units that are different or the same because the things that we were counting were different.
Kappa: That’s right, that’s just what I’m saying, that it makes no different what you’re counting, what things, and with what unit, five plus three will always be eight, whatever we’re counting.
Delta: But how do you know that will always be the case?
Kappa: It works out that way even from your example.
Gamma: But we checked only one example.
Kappa: Well, let’s check another hundred examples. You’ll see that’s the way it will always be.
Teacher: Is it really true that if we check another hundred examples and decide that for these hundred Kappa’s assertion is correct, can we really say that it will always hold true?
Alpha: We could check even more, and it will probably always be that way. But it would take a long time.
Delta: And maybe with some numbers, very big ones, for example, it might not work. After all, we can’t test every example with all numbers.
Teacher: Delta, you’re always talking about big numbers. For you they seem to be very different from small ones. . . .
Delta: Yes, we can see small ones right away. When there are five or six things, we can see that right away. And even if we are talking about the numbers themselves, rather than things, like Eta, you can see small numbers right away, how they are arranged.
Beta: If you believe that numbers come from counting things, then it’s hard to see how that works with big numbers. Probably nobody has ever counted a million things. And it’s hard to understand how you could get very big numbers. And if you understand a number the way Gamma does, that all numbers come from units, even then you won’t ever get to big numbers, to very big numbers. And it’s hard to understand what’s going on with them. Whatever you think about numbers, it’s always going to be that there’s something different about big numbers. Except maybe if you think about them the way Kappa does. . . .
Kappa: Yes, if you look at them the way I do, then all you have to do is learn to add up big numbers and do everything else with them that you can do with small numbers. If you can do all that with big numbers, that means that they are the same sorts of numbers.
Delta: But still — we can’t check how they act, big numbers.¹³
Alpha: Well, can somebody give me an example, even with very big numbers, in which this isn’t true?
Beta: I can’t give such an example, but I think that doesn’t prove anything.
Alpha: What do you mean, that doesn’t prove anything?
Beta: Well, if something hasn’t happened yet, does that really prove that it can never happen? Everything that has happened at some point happened for the first time.
Kappa: What? Do you really think that some day it could happen that five plus three could equal something other than eight? Or that one plus one might not add up to two?
Beta: I don’t know. I don’t think that could happen. But we have to know for sure and not just think.¹⁴
Eta: I think I’ve come up with an example.
Teacher: What example?
Eta: Well, for instance, I have a piece of clay. And I have another piece of clay. One plus one equals two, right? But if I squish the pieces together then they will join into one and I’ll have one instead of two.
Beta: Are you trying to say that in your example one plus one equals one?
Eta: I don’t know.
Delta: You’re not just adding these two pieces of clay, you’re squishing them, sticking them together. You’re doing what you were talking about before: making one out of two. But if you just place them side by side, one piece and then another, then there will be two pieces.
Eta: Fine, I won’t do anything, no squishing, no sticking. I’ll just dribble some water on the table. How many drops of water are there?
Delta: One.
Eta: And now I’ll add another drop to it, look. How many are there now?
Delta: One drop, but bigger than the first one.
Eta: But it’s one, not two.
Delta: Yes, one drop [this idea belongs to V.G. Kasatkina and N.I. Kuznetsova].
Kappa: You can’t count things like that. If the drop changes — first it’s small, then it’s big — that means it’s not the same, it can’t be a unit; we can’t count with it.
Gamma: What do you mean we can’t? It’s clear that there’s one drop here. You could drop two drops, and three and four. However many you want. I think you can count drops.
Kappa: You can’t count drops.
Teacher: Why not?
Kappa: Because you can’t count things like that, where it might turn out that one plus one equals one.
Teacher: And what sorts of things are those?
Kappa: Well, I guess, things that change.
Beta: Let’s say you have two people — you and I, for example. We’re growing — that means we’re changing. But there are still two of us. Can you count us?
Kappa: I guess you can, because, even though you’re changing, you’re still you. You are you and I am me. So I guess we can be counted. But if something is a single thing, then it should always stay one thing, so you can count it.
Beta: So a person lives and lives and then dies. So there aren’t always as many of him.
Kappa: So then you can’t count.
Teacher: And when can you?
Alpha: First of all, you can count individual things that don’t merge with others, like our drops. And second, things that don’t die, that last forever. If there is one thing, then there will be one thing, and it won’t disappear. If there are two things, then there will be two [Petya Filatov, second–grader].
Teacher: So, you can count things that don’t disappear, that last forever, and those things that don’t merge together and that always exist as separate things, is that right?
Beta: It seems clear what you can count and what you can’t. But when you try to explain it, everything gets confused.
Teacher: Kappa said that you can count those things for which one plus one is always two.
Beta: But that isn’t an explanation. That’s like saying that you can count those things that you can count.¹⁵
[The school bell rings.]

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Notes
¹ Eugene Matusov: “The teacher wants to know how the children really understand the word ‘number’ and wants to help the children themselves understand what their understanding is and uncover and ‘bounce off each other’ various ways of understanding number, while at the same time not serving as the bearer of knowledge, but rather as someone who, together with the children and for the very first time, is establishing a ‘point of wonder,’ which should serve both as a motive for building a comprehension of number and a means of comprehending. The children can sense the difference between questions of various sorts: genuine, information-seeking, and fake questions.” Irina Berlyand: “But for me the most important thing is not that some questions are ‘genuine’ while others are rhetorical, or testing, but rather their differences of substance — some questions have to do with so-called knowledge and allow for an unequivocally correct answer, while others have to do with comprehending, which inescapably involves inquisitiveness and doubt, that is ‘learned ignorance’ (Nicholas of Cusa’s term, of which Bibler was very fond).”

² The children’s idea of number here is founded on a certain intuition of sequence and rhythm, that is, it is associated with time. This subject will be discussed in Lesson 16. See also F. Klein, Elementarnaia matematika s tochki zreniia vysshei, vol. 1 (Moscow, 1987), pp. 26–27; idem, Elementary Mathematics from an Advanced Standpoint. Arithmetic, Algebra, Analysis (Mineola, NY: Dover, 2004), pp. 10–11.

³ Alpha and Kappa are proposing an understanding of number that is similar to the understanding of number within the developmental instruction mathematics program for the V.V. Davydov School: a number is a method to measure magnitudes; counting is a particular case of measurement when the magnitude being measured is a multiple of the measurement unit. See V.V. Davydov, Problemy razvivaiushchego obucheniia (Moscow, 1986); idem, Problems of Developmental Instruction: A Theoretical and Experimental Psychological Study, trans. P. Mochay (Hauppauge, NY: Nova Science, 2008).

⁴ Cf. “While counting or enumerating, we mentally associate each new object of the aggregate under consideration with each of the words that follow one another in our numerical phrase (or sequence); the last number pronounced indicates the number of objects in the aggregate. This number is seen as summing up the experimental operation of enumeration, since it gives a complete accounting of it” (G. Lebeg [Henri Lebesgue], Ob izmerenii velichin [Moscow, 1938], p. 15; H.L. Lebesgue, Sur la mesure des grandeurs [Geneva: L’enseignement mathйmatique, 1956]). [We were unable to confirm an English title, or whether the work has been translated into English. — Eds.]

⁵ Compare Eta’s idea with ordinal number theory, which holds that every member of an ordinal number is an ordinal number, and every ordinal number represents a class of all previous ordinal numbers, “(2.3) If y ∈ α then y is an ordinal. (Hence α is the set of all ordinals smaller than α)” (Α.Α. Fraenkel and Y. Bar-Hillel, Foundations of Set Theory [Amsterdam: North-Holland, 1973], p. 93).

⁶ It was said of the mathematician Ramanujan that each of the positive integers was one of his personal friends. See the review of Ramanujan’s collected works in Dzh. Litlvud [J. Littlewood], Matematicheskaia smes’ (Moscow, 1990), p. 86 ff.; J.E. Littlewood, A Mathematicians’ Miscellany (London: Methuen, 1953).

⁷ There is a way of defining a number and a corresponding set of axioms where every number is defined individually, outside the sequence of other numbers and operations — by assigning a natural number as a class of equinumerous finite sets. See, for example, how Frege constructs the system of natural numbers, where “the natural numbers were the cardinal numbers [Anzahlen] of certain concepts, with ‘the cardinal number of the concept F’ defined as short for ‘the extension of the concept equinumerous-with-the-concept-F’; finally, the statement ‘the concept G is equinumerous with the concept F’ was regarded as short for ‘there exists a one–one correlation between the objects falling under the concept F and the objects falling under the concept G.’ This last expression, Frege was able to show, could be reduced to purely logical expressions.” See Fraenkel and Bar-Hillel, Foundations of Set Theory, p. 183. On this foundation, Frege also constructed the theory of real numbers, however, he had only just completed his most important work after a decade of intense effort when Bertrand Russell told him about his own discovery (the antinomies on which set theory is based). “In the first sentence of the appendix, Frege admits that one of the foundations of his edifice had been shaken by Russell” (ibid., p. 3).
⁸ The word “number” and the concept of number in classical Greek mathematics relates only to natural numbers greater than one. Defining number as Euclid’s set of units precludes an understanding of units as a number. For Euclid, a unit is the basis, the cause of the number, but in and of itself, it is not a number. “A unit [μονάζ] is that by virtue of which each of the things that exist is called one” (Euclid’s Elements, VII, Def. 1, trans D.E. Joyce). See N. Bourbaki, Elements of the History of Mathematics, trans. John Meldrum (Berlin, 1994), p. 147.
⁹ Historically, evidently, at first only natural numbers were “invented.” See Bourbaki, Elements of the History of Mathematics. By numbers, the children mean specifically natural numbers.
¹⁰ Eta’s idea about numbers arranged in a beautiful shape is close to the idea of the figurate number, which was developed in antiquity. A number was understood not merely as an abstraction of quantity, but as having its own quality. Compare Aristotle, “[N]umbers have a certain quality, e.g. the composite numbers which are not in one dimension only, but of which the plane and the solid are copies (these are those which have two or three factors); and in general that which exists in the essence of numbers besides quantity is quality; for the essence of each is what is once, e.g. that of 6 is not what it is twice or thrice, but what it is once; for 6 is once 6” (Aristotle, Sochinenie, vol. 1 [Moscow, 1976], pp. 165–66 [English translation from The Works of Aristotle, trans. J.A. Smith and W.D. Ross (Oxford: Clarendon Press, 1908), vol. 8, p. 1020]. This topic is further addressed in Lessons 5, 10, and 22–26, among others.
¹¹ To use Aristotle’s words, Alpha feels that only quantity exists in the essence of numbers, not quality.
¹² The argument between Alpha and Beta is the argument between the “pragmatic” and “theoretical” approaches to number. The pragmatic approach considers a number to be “defined by the possibility of obtaining approximate values and introducing those into the calculations” (Bourbaki, Elements of the History of Mathematics, p. 147). In ancient Greek mathematics, in the words of Bourbaki, “rigor” and “theoretical preoccupations” came before computational needs.
¹³ Felix Klein, in talking about the development of the concept of number, presumes that understandings of “little” and “big” numbers rest on different types of intuition, specifically different understandings of space. “One the sensibly immediate, the empirical intuition of space, which we can control by means of measurement. The other . . . consists in a subjective idealizing intuition, one might say, perhaps, our inherent idea of space. . . . It is immediately clear to us what a small number means, like 2 or 5, or even 7, whereas we do not have such immediate intuition of a larger number, say 2,503. Immediate intuition is replaced here by the subjective intuition of an ordered number series, which we derive from the first numbers by mathematical induction” (F. Klein, Elementary Mathematics from an Advanced Standpoint: Arithmetic-Algebra-Analysis, trans. E.R. Hedrick and C.A. Noble [New York: Macmillan, 1945]), p. 35. Big numbers have a special meaning in the physical sciences as well. Pointing to the fact that logical definition of an irrational number belongs to precise mathematics and has no significance for the physical sciences, Klein writes: “This may seem, to be sure, to be contradicted by the fact that, in crystallography, one talks of the law of rational indices, or by the fact that in astronomy, one distinguishes different cases according as the periods of revolution of two planets have a rational or irrational ratio. In reality, however, this form of expression only exhibits the many-sidedness of language: for one is using here rational and irrational in a sense entirely different from that hitherto used, namely, in the sense of mathematics of approximation. In this sense, one says that two magnitudes have a rational ratio when they are to each other as two small integers, say 3/7; whereas one would call the ratio 2021/7053 irrational. We cannot say how large numerator and denominator in this second case must be, in general, since that depends upon the problem in hand” (Klein, Elementary Mathematics, p. 36). In other words, big numbers can even have the value of irrational ones, since their nature in some regards is so different from that of small ones. See also the problem of “inaccessible,” cardinal numbers (Frenkel and Bar-Hillel, Foundations of Set Theory, pp. 95–98).
¹⁴ For the first time, the pupils are encountering the problem of proof, not just of verifying whether a particular case is correct. This problem is discussed by Henri Poincarй in his book Science and Hypothesis (New York, 1952). After examining Leibnitz’s demonstration of 2 + 2 = 4 he writes: “This is not a demonstration properly so called; it is a verification. . . . Verification differs from proof precisely because it is analytical, and because it leads to nothing. It leads to nothing because the conclusion is nothing but the premises translated into another language. A real proof, on the other hand, is fruitful, because the conclusion is in a sense more general than the premises. The equality 2 + 2 = 4 can be verified because it is particular. Each individual enunciation in mathematics may be always verified in the same way. But if mathematics could be reduced to a series of such verification it would not be a science. . . . It may even be said that the object of the exact sciences is to dispense with the direct verification” (p. 4). An attempt to demonstrate a general proposition (any) that has to do with, for example, operations with natural numbers based on “verified” specific instances, leads to the problem of (mathematical) induction (see Poincarй, Science and Hypothesis, pp. 8–13). The impossibility of equating proof and verification — since only particular instances can be verified and for something to be demonstrated it must be a general proposition (which could cover an infinite number of particular instances) — is noted by Delta during our lesson when he says, “After all, we can’t test every example with all numbers.”
¹⁵ The question of the applicability of count is truly very complicated. Consider, for example, the following: “As far as arithmetic is concerned, it uses just a small number of experiments, each of which has been repeated a tremendous number of times by man for as long as people have existed. We therefore know in exactly which cases arithmetic is applicable and in which cases it is not. In the latter cases, we do not even try to apply it. We are so accustomed to using arithmetic only when it is applicable that we forget about the existence of cases where it is inapplicable. We assert, for example, that two plus two will be four. I pour two liquids into one glass and two liquids into another; then I pour everything together into one vessel. Will it contain four liquids? ‘That’s a trick question,’ you say, ‘That’s not an arithmetic problem.’ I place a pair of animals in a cage and then another pair; how many animals will be in the cage? ‘Now it’s really obvious that you’re being unfair,’ you will now claim, ‘since the answer depends on the species of animal. It could happen that one creature will eat another. You also have to know whether the count will be performed immediately or in a year, over the course of which the animals might die or have offspring. In essence, you are talking about aggregates about which it is not known whether or not they are unchanging, whether every object of the aggregate will maintain its individuality, and whether there are objects that might disappear and then reappear.’ More than anything else, what you are saying signifies that the ability to apply arithmetic depends on certain conditions. As for the rule that you have given me for recognizing whether or not it is applicable, it is certainly excellent in a practical sense, but it has no theoretical value. Your rule boils down to an assertion that arithmetic is applicable when it is applicable. That is why it cannot be proved that two and two will be four, which nevertheless remains an undeniable truth, since applying it has never failed us” (Lebeg, Ob izmerenii velichin, p. 18, retranslation from Russian). The formulation that “arithmetic is applicable when it is applicable” is essentially the same as the objection Beta raises with Kappa, “That’s like saying that you can count those things that you can count.”

Dialogue in the early Grades of the School of the Dialogue of Cultures*
(на русском языке эту статью можно прочесть здесь - Диалог в начальных классах ШДК)
This brief work represents a preliminary attempt at a phenomenology of classroom educational dialogue. It is based to a certain extent on the dialogue lessons conducted at various times by S.Iu. Kurganov, N.I. Kuznetsova, and V.G. Kasatkina; however, here I will not describe or comment on actual lessons that took place, but carry things a bit further. In other words I will attempt to imagine how the idea of the points of wonder might be realized and embodied in the dialogue lessons.¹⁶ Two things must be noted in advance.
First, in this context, that is, in the context of educational dialogues in an early grade School of the Dialogue of Cultures classroom, dialogue is not, strictly speaking, dialogue of logics and dialogue of cultures because in these learning dialogues, the elementary school children discuss “more particular” (rather than more fundamental) issues such as what is number, what is word, how we count, and so on; they are not able to discuss the founda-
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*English translation © 2009 M.E. Sharpe, Inc., from the Russian text: Irina E. Berlyand, “Dialog v nachal’nykh klassakh Shkoly Dialoga Kul’tur,” first published in Arkhe, 1998, no. 3, pp. 255–63.
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tions of culture. Not yet dialogue of cultures and dialogue of logics, it could be said. But this “not yet” is not meant to imply that the point of wonder situation in our analysis and comprehending — or in the course of development of the upper-grade classroom — is “sublated” (in the Hegelian sense), having attained the level of a dialogue of logics. No — this point of wonder situation is constantly reproduced. This is what we mean when we say that the age of the point of wonder point is preserved.¹⁷
Second, the dialogue that we are talking about is not simply a teaching method where (monologic) content is taught through discussion and a single, correct knowledge, the same for everybody, comes into being and where the parties to the dialogue (its subjects) represent its (dialectical) sides. Approaches to education that — while preserving the monologic academic content — focus the attention on interactive forms of cooperation and communication between teacher and pupil as an educational method are now common and are even more common in the declarations of various pedagogical schools. Our conception goes farther. In the School of the Dialogue of Cultures this is not simply a teaching method [among other teaching methods — Eds.], but something that determines the very content, the very object of learning, on the one hand, and the very organization of the soul, speech, thought, and person, on the other. These two aspects, as I will attempt to demonstrate below, are essential to one another and can be understood only through one another.
1. The pedagogic task in the first and second grade is to present subject matter (number, word, phenomena of nature . . .) as something to be understood, that is, something surprising, causing wonder; to form an attitude [in the students — Eds.] that is oriented not toward using, not toward being aware of, not toward an emotional response to the [ready-made] subject matter, but specifically toward understanding — and understanding demands that the given curricular subject be reproduced as something not yet understood, as a problem. An understanding-oriented attitude is not characteristic of the preschool child yet, and the formation of such an attitude is something that involves special, very difficult work. To understand means to see an object as not understood, as something that can not be fully reduced to my own notion or representation of it, to my own picture of it. Naturally, children’s emerging image of the world is self-sufficient. Children are not inclined to ask questions of themselves. Their preschool experience comes together into a certain integrated picture founded on consciousness (rather than understanding). Interrogativeness and doubt are not characteristic of the young child. This has been described as insensitivity to contradictions. The objective of the early grades is to form that interrogativeness, self-questioning, that sense of wonderness.¹⁸
One of the central tenets of our psychological conception is that in the youngest school-age children, consciousness [i.e., awareness without the child’s self-reflection, when something becomes an object of a child’s consciousness without the child being aware of this awareness — Eds.] relinquishes its position of dominance to thinking. Consciousness reproduces a thing (and the world as a whole) as something existing, whole, and equal to itself. Consciousness attests to the existence of the thing and my own coexistence with it. Thinking raises questions about that existence.¹⁹ The fact that the preschool child is capable of and inclined toward surprise and to find everything strange does not contradict my conviction that an orientation toward understanding needs to be specially formed, and does not simply take shape in the child on its own, and that “the points of wonder” must be specially and with great effort arranged during lessons. In one of my works I attempted to draw a fine line between strangeness and wonderness. Children, on the one hand, notice strangeness in places where an adult might not see it. Children are prepared for strangeness because of their play experience. On the other hand, they see this strangeness and notice it as strangeness, but it does not puzzle them. As Virginia Woolf so aptly observed, commenting on the character Alice of Louis Carroll’s tale, children find “everything so strange that nothing is surprising.”²⁰ This is the situation of consciousness. The world is reproduced as strange, becomes strange. The attitude toward that which is taking place wavers between “everything is stranger and stranger” and “nothing seems quite impossible” (incidentally, Carroll’s Alice is seven, the age of our “points of wonder”).
When children reach a point of wonder, they are on the bound ary between consciousness and thinking — fundamentally different psychological attitudes; within the SCD, this is viewed as the psychological specific nature of the beginning of SDC learning.
2. In order to achieve this objective, we select content that must meet specific criteria. Certain basic, simple, but at the same time foundational points–foci–problems must be found that fulfill the following conditions: (1) they feature problems that are important for modern thought; (2) every culture develops its own particular approach to comprehend it; (3) at the level of consciousness, young children, even before school, have already had some developed experiences with the matter of discussion; and (4) from these “points” or puzzles it must be possible to “grow” subject matter that is less fundamental, for example, academic or scientific subject matter. In the area of the word, for example, the problem of how a word sounds and what it means, language and speech, how words and utterances are addressed, the problem of naming, the problem of the word as the minimal unit of speech, and so forth, fulfill these conditions. In the area of elementary arithmetic these conditions are met by the concept of number, the skill of counting, intuitive sense of magnitude and sequence (out of this material algebra, geometry, analysis, topology, and so on, can then be grown).
It is particularly important that subject matter be selected for discussion that relates to children’s own experience, so that they can ostranit’ [estrange]²¹ and otstranit’ [estrange] it.²²
Rather than a collection of information and skills, the object of discussion becomes a focus, a point through which the entire world becomes a puzzle — the world of mathematics, the world of language and speech, the world of nature, and so on. Instead of presenting these worlds through a “unified cell-concept” (see Hegel, Marx, and Vygotsky, but especially Ilyenkov and Davydov), which goes on to develop and become concrete (as in Developmental Instruction, developed by V.V. Davydov and his colleagues), from the start the stage is set for a clash, a dispute between different possible understandings, between different approaches to a problem.
3. Let us try to analyze such dialogues and demonstrate just how dialogue — through the points of wonder — comes to define thoughts and the very subject matter of the conversation.
Dialogue at the points of wonder resembles Socratic dialogue.²³ The teacher is not conveying specific information or knowledge to the child. The task of dialogue is not only (and, perhaps, not so much) to explain something that is not clear, but primarily to puzzle, to make an object of dialogue that may at first seem plain and simple to the pupils strange, unclear, and, therefore, requiring understanding. Simple, elementary objects are selected that the children encounter every day, that they know how to deal with, and that they feel that they know — and these objects are questioned: what is a number, a word, time, a living being, and so on. The children know (assume that they know) what they are dealing with. For them, these things are not subject to question. For these objects to become puzzling, three conditions must be present. First, there must be different notions about one and the same thing. For now this is not different logics, cultures, and the like, but simply different ways of answering the questions “what is a number?” “what is alive?” “what is a word?” and so on — different opinions and views of different children. Second, these different answers, opinions, and interpretations must — and this is critical, but very difficult to achieve — relate to one and the same object, define one and the same object — otherwise you simply wind up with homonymy, different meanings of one and the same word. Third, each one of these different understandings, conceptions, opinions must pretend to universality.²⁴ Not one of them can be derived from another or qualified as incorrect, insufficient — only thus can they clash with one another on the same territory, on the same object.
The first condition — having different conceptions about one and the same thing — is rather easy to achieve in practice. As soon as children arrive in the classroom, it is easy to see that they have many different ways of looking at things. Some children see numbers as a means of counting (“a number is what you count with”), and some see it as a figure or as something that relates to magnitude; a word is understood as the name of an object, as an utterance, or as a way of addressing others. It is much harder to demonstrate, both to myself (a teacher) and the children, that they relate to one and the same object and, consequently, that dialogue, argument, is needed.
Children (and not only children) are inclined toward relativism and are not particularly susceptible to contradiction. (They exhibit something that appears to resemble what is sometimes thought of as “multiculturalism.” For example, I say that the tram is moving because it is alive, and you say it is because of electricity. Fine — both are correct in their own way. The most important thing is to live in harmony. There are multitudes of similar examples in Piaget, in Kornei Chukovsky (From Two to Five [Ot dvukh do piati]), and in Kurganov’s dialogues. [Bakhtin criticized this relativism as a version of extreme monologism — Eds.]) Children often avoid an argument of substance. Different opinions and viewpoints expressed by the children exist on their own and do not bump up against one another or enter into relationships with one another. Vanya has his number and Katya has hers — they simply coexist side by side and do not enter into argument with one another, as if they denote different things. To use Hegel’s terminology, it could be said that as soon as a common object appears, Vanya’s or Katya’s interpretation is “sublated” into a common interpretation, and they turn out to be “sides or aspects” of the same concept. (For example, in discussing “the puzzles of the number,” one of the dialogue participants says: your explanation works for big numbers, and mine for small ones.)²⁵ To construct and retain a situation where conceptions simultaneously appear to be both different and universal, is very difficult. The difficulty is not only pedagogic. Even in the imaginary school this is achieved only rarely and with great difficulty.
The question arises whether or not dialogue is even possible when it comes to a subject like, for example, number. Perhaps dialogue in the narrow sense of the word (adequate to Bibler’s conception of dialogics) can only take place inside logic, that is, dialogue can only be about what is being. From the perspective [Bibler’s — Eds.] of dialogics, only when dialogue becomes a dispute among logics can it be understood as defining our very thought and being, and not just a random psychological phenomenon characteristic of human communication.²⁶ But on the other hand, dialogue of logics must be rooted in dialogue of cultures, in dialogue between consciousnesses, or it will also degenerate. This aspect has particular significance for the school. The school has been named for the dialogue of cultures — rather than the dialogue of logics — for good reason.
The object of our points of wonder must be understood as dialogic with an intention toward the dialogic nature of logic, but along the boundary of logic — not within logic itself. In other words, what is being discussed is “what a number is,” “what a living thing is,” “what a word is,” and so forth, and not “what being is.” The latter, being itself, is the object of philosophical logic.
4. Who are the presumed subjects of such dialogue? In terms of phenomenology, dialogue presumes a conversation between two or more subjects about something else, something beyond the bounds of the dialogue. Does such a thing as number (or anything else) really exist outside our conversation? Or does it only exist in our discourse about it? Or does it exist in the text of textbooks? Are we talking about something, or are we just passing the time in pleasant conversation with one another? In the second case, “we’ll do as we please.” Whatever we come up with will be just fine, so long as it seems to fit and everyone is happy. In the third case, all we have to do is memorize the textbook. But if we presume an affirmative answer to the first question, then there is some object and we now have something akin to the dialogue we are talking about. The objectivity of dialogue, the presumption that what is being discussed exists outside of dialogue (i.e., “how things are in the matter of facts”), is very important, especially in educational dialogue. Otherwise all we have is a “communion of souls” and the object of the conversation is just a pretext. (There are many examples of conversations where the parties are more interested in interaction than in the object of conversation, from high society chitchat to a man reading poetry to a woman he is courting.) Such is often the case with communication, and there are pedagogues who direct their primary focus on the interaction between the children and the teacher as such. In contrast, the School of the Dialogue of Cultures focuses primarily on objectivity.
But the subjective aspect of dialogue is equally important. If we belittle its importance, seeing it as random, reducing it to the children’s empirical features, with which, of course, the teacher is forced to contend, then we deprive subjectivity of its own independent significance and wind up with developmental instruction (see Davydov), which could be called dialectical (different statements by different empirical subjects represent aspects of a dialectical unity that belongs to a single logical subject).²⁷ Things are different in the School of the Dialogue of Cultures. Pupil’s individual empirical features of thinking and discourse represent the possibility for another way of understanding, another culture, so rather than annoying hindrances, these features become something that opens up new possibilities for dialogue.
Here is an example. The SDC teacher Sergei Kurganov had a pupil by the name of Misha Pletenko (pseudonym), whom a traditional teacher might have labeled “a real numbskull” [or, using the U.S. educational political correctness, “a slow learner” — Eds.]. If the goal [that of a traditional monological teacher — Eds.] had been to bring all the pupils as quickly and smoothly as possible to mastery of preset common knowledge and a set of skills, Misha Pletenko would have slowed the progress of his class considerably. But in Kurganov’s class Misha played an important role in the construction of the points of wonder: his inability to comprehend, first of all, slowed things down and in so doing “estranged” the other children’s conceptions, which were uncomplicated and seemingly consonant with commonly accepted ideas, and incited their wonder. Second, Misha maintained for the entire class the understanding of number and value associated with Piaget’s phenomena, and in so doing kept the preschool vision alive in the classroom, a vision based on unfiltered, unreflected consciousness. Third, Misha was inclined not toward sign- and symbol-based understanding, but rather, one could almost say, toward an eidetic understanding, and through his “monstrous” constructions hinted at the possibility of an alternative understanding of number (possibly close to the Pythagorean view), and worked, so to speak, future “classes of Antiquity Culture.” (Among my imaginary schoolchildren, Gamma somewhat resembles him. Many of Misha’s statements are spoken by this Puzzles of the Number character.)²⁸
What preconditions does our dialogue place on the subject (participant) of dialogue (from the point of wonder perspective)? First and foremost, dialogue demands two or more different subjectsparticipants. ²⁹ The subjects-participants must be different, and see themselves as different, and not just randomly, empirically different, but fundamentally different. Subjective difference is rooted not in the pupils’ psychological idiosyncrasies, but in the complexity and paradoxicality of the object of discussion. For example, one pupil proves the formula (a + b)2 = a2 + 2ab + b2 geometrically and another algebraically. This is important for the SDC not because we respect and value the psychological inclination of one pupil toward visual-image thinking and another toward symbolic thinking (although, it could be a case), but because this difference in the participants’ subjectivities highlights a real, substantive problem of the foundations of mathematics, an argument between algebraic and geometric approaches that has been going on throughout the history of mathematics, with different images of mathematics arguing with one another, different ideas about the nature of proof. This, of course, must also be understood, brought out into open public consideration — we should not simply say to one another, “Look how different we are, and we’re both so clever!” The algebraic formula example is, of course, for older pupils. But similar examples for the early grades can, I believe, be found in the text of Puzzles of the Number.
The subjects of dialogue must also hear one another, that is, they must reproduce the speech of others in their own inner speech. This demands that subjects not be monolithic, integral, and clear to themselves. Rather subjects of dialogue must have an internal rift, must doubt themselves, must be a kind of problem for themselves. Nevertheless, they must reproduce themselves as this, single, integral subject, not dissolve in the dialogue, uphold their own voices, their own positions, reproduce their own approach not out of stubbornness, but revealing their substance — this is a precondition for dialogue. Participants in a dialogue reproduce their own integrity and identity within the dialogue itself, encountering some resistance (this will be addressed below).
5. In empirical terms, first-graders do not meet these conditions. As a rule, first-graders do not hear their classmates, do not doubt themselves, and for them their own ideas are obvious, holistic, and not surprising.
The preschool child’s attitude toward the world and things in the world is mainly determined by conciseness, not by thinking and reasoning, thus to perceive an object as something amazing, something baffling, as an object of understanding, of thinking, contemplating the very question of the possibility of an object, is a complex task. The organization of educational activity in the earliest grade is focused on performing this task. In the points of wonder, an object for the first time rises up before the child as something surprising and baffling — something demanding understanding. At the same time, a child’s own self-awareness becomes for him or her an object of understanding. His or her conciseness-based picture of the world becomes estranged from him or her, and for the first time it takes on a certain wholeness, but at once a child begins to doubt it, question it, and it ceases to be something self-evident. In this sense, it is in school that the preschool age first appears as an integral age, as quasi culture; this does not occur where a preschool child exists naturally within his or her own age.³⁰ This is achieved through the organization of educational dialogue during which views that are natural for children regarding familiar objects — numbers, words, time, phenomena of nature, and the like — clash with something else, encounter resistance. In order for children to become (a) aware of their own idea about something and (b) capable of hearing — reproducing another’s speech within themselves — they must encounter some obstacle that provides resistance. Three types of such resistance can be found and constructed in the classroom:
The first type of resistance is the resistance of an alien point of view. Children find out that their friends do not understand what the word or the number is the same way that they themselves understand the matter. Different points of view, explanations, opinions and suppositions are expressed about what would appear to be one and the same familiar object for the child. The child is forced to see his or her own perspective as if from the side, first as an integral single picture, and second as only one of the possible pictures.
The second type of resistance is that created by empirical observation. Objects behave differently from the way they should according to the child’s own understanding. The classical example of this type of resistance is experiments with floating bodies (children assert that all wooden bodies float and all metal ones sink until the teacher shows them a floating razor blade or needle). Such
experiments also force children to estrange their own viewpoint, their vision, their perspective, to see it from the side (surely not to refuse it).
The third type of resistance is tied to the resistance of a cultural text or artwork. The work presents the reader with a particular type of communication, a specific reader’s position. For example, an impressionist picture literally puts a spectator in a particular, certain position — it does not allow the spectator to come closer to the picture than a certain distance. Similarly, absurdist prose resists the reader’s emotional empathy for a character. Any artwork has its own musculature and actively resists inappropriate reader or spectator behavior.
This third type of resistance demands that the reader already have a formed position. First-graders are not susceptible to this kind of resistance. Experience shows that even the second type of resistance is very difficult for children to appropriately perceive. Children can react rather unexpectedly to empirical experience that contradicts their ideas. For example, in the needle experiment described above, children sink the needle that has ruined their mental picture with its “wrong” behavior. Often children simply make a mental note of such empirical evidence, and this note somehow exists in them separately from their own view of things and does not clash or come into contact with it. Nevertheless, in first grade, children are more susceptible to this sort of resistance than to resistance of the third type, and such situations can be managed in the classroom sometimes.
In terms of logic, all three types of resistance are manifestations of the same thing — resistance exerted by the ideal object. The reason an object resists an arbitrary interpretation is that it allows
for alternative ways of understanding. The reason a classmate’s interpretation resists my interpretation is that it relates to the same (my!) object. But at first, empirically, this appears to children to be different things, and children are more susceptible to resistance of the first type. It is not obvious to children that the very same object is being discussed. Exposing the fact that different types of resistance are the same thing is one of the main objectives of the points of wonder.
Children react earlier and more eagerly to resistance from their classmate’s interpretations than they do to resistance from an empirical object and much earlier than to resistance from a cultural text. Only much later do children learn to adequately deal with resistance of the second and third type.
Logically, I repeat, this is all the same thing — resistance exerted by an ideal object. Take, for example, an object such as a number. Number — the concept of number — resists arbitrary treatment, it does not allow itself to be spoken and thought of however one pleases. But in the first grade, children do not suspect the existence of the modern concept of number. For them, its resistance takes different empirical forms. Resistance of the first type is the resistance of an alien viewpoint (be it the viewpoint of an authoritative teacher, textbook, or classmate) that does not fit with my viewpoint. Number here articulates itself through the resistance of a viewpoint that is alien to my own. The second type of resistance is the resistance of the empirical object that behaves differently than it would behave if my interpretation were correct. This is not necessarily something material; a number can also exert this sort of resistance (e.g., if someone were to come up with an artful “theory” that would, if it were correct, result in two times two equaling five — and all children know that is not true — this would also be resistance by an object).
And the third type is resistance by a text understood as a product of culture, be it mathematical, literary, or philosophical. Resistance by something as ideal as number is empirically articulated for pupils and the teacher in these three forms. These are the forms in which the ideal object itself gives us the opportunity to feel its resistance, it lets us know of its existential essence, of its existence outside our concepts, judgments, and actions. Why is it able to resist? Because we did not come up with it ourselves, at will, however we felt like it — if we had, it would not be able to resist us. The fact that it resists expresses the fact that its own existence is something beyond our thinking, our understanding.³¹
Children whose statements encounter such resistance begin not only to hear others and see objects that require understanding, but they begin to hear themselves. Estranging, delving deeper, arguing their interpretations, they reproduce themselves as participants in dialogue.
6. We see that in talking about the preconditions for the existence of dialogue subjects we arrive at a definition of the object of dialogue — and this does not happen by mere chance. Bibler’s dual definition of dialogue — as argument with oneself and as argument about the objective essence of being — works in analyzing educational dialogue in the points of wonder. Dialogue with and dialogue about turn out to be two facets of the same phenomenon.
This is what makes my dialogue with another person become truly profound dialogue specifically with that person; this person becomes important to me as an interlocutor when the object we are discussing profoundly engages me and he or she, the other, reproduces in this object unavoidable aspects of this object that cannot be reduced to my thinking. Certain aspects of the object that interest me become important when behind them I see (perhaps I merely presume, perhaps I myself devise the possibility) a persona, an interlocutor, a subject with a means of understanding the world overall (and not just the given object) that differs from mine. In this sense, Bibler’s understanding, strictly defined — of dialogue as something more than conversation with someone about anything at all, as dialogue between universal, although different, logics and between ways of understanding — dialogue about something and
dialogue with someone turn out to be different definitions of one and the same thing.

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Notes
¹⁶ On the points of wonder see Shkola dialoga kul’tur. Osnovy programmy (Kemerovo, 1992).

¹⁷ The SDC psychological conception holds that, over the course of mental development, each developmental stage is not sublated during the following developmental stage, but is preserved and reproduced as a nonsublated voice.
Adults reproduce within themselves the voice of the preschooler at play, the voice of the first-grader finding himself at a “point of wonder,” and the voice of the teenager acquiring the mode of comprehending peculiar to antiquity or middle ages. Within the program design, this is reflected in the inter-age lessons that the SDC idea provides for — dialogues where first-graders and sixth-graders interact with one another as equal parties to dialogue — as well as the constructing (reproducing), in different grades and using different subject matters, of “the points of wonder” situations, which sets the stage for the beginning of learning and the age that corresponds to it to be reproduced as something current, and not already superseded (“sublated” in Hegel’s terminology).

¹⁸ For a more detailed treatment of this, see my article, “K problemam pedagogicheskoi psikhologii nachal’nogo obucheniia v Shkole dialoga kul’tur” [Approaching the Problems of Early Education Pedagogical Psychology Within the School of the Dialogue of Cultures School] in the collection Shkola dialoga kul’tur. Idei. Opyt. Problemy [The School of the Dialogue of Cultures: Ideas, Experiences, Problems] (Kemerovo, 1993). What is usually taken for inquisitiveness in preschoolers, manifested in the countless questions they ask adults, is not tied to an understanding-oriented attitude in the sense that I use that word.
When a child asks, “Why is the moon round?” and is satisfied with the answer “Because it is like a melon” (or they themselves come up with an analogous answer), it becomes clear that “why” and “because” for him have completely different meanings than they do for an adult. Many such examples can be found in Piaget. The meaning of these children’s questions, it seems to me, is twofold: on the one hand, questions are a new and interesting means of interacting with someone they care about (and also a new game: “Why does the tram move?” “Because it’s alive.” “Why is it alive?” “Because it is filled with fuel.” “And why is it filled with fuel?” “Because because.” “Why because?” “Because you’re a pesky little because-meister.” “And why pesky?” “Because you’re a wisenheimer.” “And why.
. . .” That dialogue between brothers age four and ten overheard by me is clearly a game. Children do not take hold of a subject, are not interested in understanding it, but are only interested in the dialogue itself). On the other hand, the meaning
of children’s (preschoolers’) questions is to deepen their picture of the world (a picture that, for the most part, is determined by consciousness, not by thinking, cognizing or understanding), but not to question it, to doubt it.

¹⁹ See V.S. Bibler, “Soznanie i myshlenie (Filosofskie predpossylki),” in Filosofsko-psikhologicheskie predpolozheniia Shkoly dialoga kul’tur (Moscow, 1998), pp. 13–87.

²⁰ Virginia Woolf, “Lewis Carroll,” in The Moment and Other Essays (London: Mogarth Press, 1947), p. 82.

²¹ The neologism ostranenie [estrangement] (from the adjective strannyi [strange]) was first used by V. Shklovsky. In his O teorii prozy (1925), he calls estrangement a device of art that slows our perception, permitting us to see a familiar thing as if for the first time, taking us away from recognition and returning us to seeing. [In his “Translator’s Introduction” to Shklovsky’s book, Benjamin Sher discusses the difficulty of translating Shklovsky’s neologism and justifying his final decision to resort to an English neologism, “estrangement” (Viktor Shklovsky, Theory of Prose, trans. B. Sher [Normal, IL: Dalkay Archive Press, 1998]), p. xix — Trans.]

²² This requirement strikes me as exceptionally important for the following reason. When children first go to school they are by no means a tabula rasa. They already know a lot and can do a lot, they are filled with diverse information, preconceptions, “everyday concepts” (a term from L.S. Vygotsky’s Thought and Language), they have experience of preschool play under their belt, and, most important, they have their native language (it would be equally just to say that “their native language has them). A variety of strategies can be used to deal with children’s own experience acquired before school outside organized learning. Within the School of the Dialogue of Cultures this experience is systematically, using specially organized procedures, made into an object of expression, interpretation, estrangement, reflection, and arrangement. It could be said that preschoolers first become integrated whole subjects who are capable of becoming their own object, an object of their own consciousness for themselves, exactly in the first grade, but not before school, where they are merged into themselves. Through the educational activity of the first-grader, his or her consciousness for the first time takes shape as an integrated subject, reproduced in thinking and resistant to this thinking. Educational activity forms and eternally retains this voice of the preschool child at play. Two poles — consciousness and thinking — oppose one another, negate one another — and presuppose one another. The very thing that permits children to “compile,” to organize and retain their “preschool” consciousness is the dominance of thinking.
The voice of consciousness will never be given form and justification, and will never be eternally retained as a nonsublated subject without the other, thinking voice. This is further addressed in “K problemam pedagogicheskoi psikhologii nachal’nogo obucheniia” and “Uchebnaia deiatel’nost’ v shkole razvivaiushchego obucheniia i v shkole dialoga kul’tur,” Diskurs, 1997, nos. 3–4.

²³ I am referring, of course, not to Plato’s dialogues and not to the historical dialogues Socrates conducted, but to the Socratic dialogue as a distinct genre, which M.M. Bakhtin described in the following way: “At the base of the genre lies the Socratic notion of the dialogic nature of truth, and the dialogic nature of human thinking about truth. The dialogic means of seeking truth is counterposed to official monologism, which pretends to possess a ready-made truth, and it is also counterposed to the naпve self-confidence of those people who think that they know something, that is, who think that they possess certain truths. Truth is not born, nor is it found inside the head of an individual person; it is born between people collectively searching for truth, in the process of their dialogic interaction. Socrates called himself a ‘pander,’ he brought people together and made them collide in a quarrel” (from Bakhtin’s Problems of Dostoevsky’s Poetics, trans. Caryl Emerson [Minneapolis: University of Minnesota Press, 1999], p. 110). In Platonic dialogues, especially the later ones, this sometimes takes on a purely formal aspect, and the content often acquires a monologic character in contradiction to the laws of the genre.

²⁴ Seen as a philosophic logic, Bibler’s dialogue of cultures conception is tied to the idea that every culture is universal to the extent that it is understood as culture and as being in dialogue with other cultures. This least of all resembles relativism, “multiculturalism,” and similar things that recognize everyone’s own small truth and presume that all these truths coexist peacefully (all truths, in other words, are relative). Bibler’s main discovery and argument with Hegel is in the notion that several different universals are possible, that is, several logics. Again, I cite Bakhtin: “It should be pointed out that the single and unified consciousness is by no means an inevitable consequence of the concept of a unified truth. It is quite possible to imagine and postulate a unified truth that requires a plurality of consciousnesses, one that cannot in principle be fitted into the bounds of a single consciousness, one that is, so to speak, by its very nature full of event potential and is born at a point of contact among various consciousnesses” (Problems of Dostoevsky’s Poetics, p. 81).

²⁵ See I.E. Berliand [Berlyand], Zagadki chisla (Moscow, 1996).

²⁶ See V.S. Bibler, “Dialektika i dialogika,” Arkhe, no. 3 (Moscow, 1998), pp. 13–23. I use the word dialogue here in a very particular sense, within the context of Bibler’s conception of dialogics, or the logic of dialogue of logics. This is not simply a discussion, communication between two people about anything at all, but — in the extreme — dialogue among different universals, different logics. I offer a quote from Bibler’s work cited above: “Dialogics presumes . . . three dimensions.
The first dimension. The initial dialogue with myself; thinking that is dialogic from the start. . . . This splitting of me into ego and alter ego that is essential for thinking lies at the foundation of the first definition of dialogics. However this definition remains purely psychological . . . until it is admitted that the second I possesses its own logic. Dia-logics, in other words, presumes that it is not just a matter of communication between me and myself, but of communication between logic and logic, communication between two mutually presumed and complementary logical subjects. Such is the second definition of dialogics. However the concept of logic as something that does not coincide with itself . . . presupposes yet another — third — definition. This ambivalence, this noncoincidence with oneself is revealed when logic goes beyond the bounds of another logic, another universal culture” (pp. 14–15). Dialogue is the revelation of paradoxality, of an unsealable crack in the very matter of dispute and in my very thought about this matter. The one presenting the other viewpoint must not only be heard from the outside and contested, but reproduced inside me (1) as an indestructible voice, where I can reproduce his argument as my own in internal discourse, more profoundly and seriously than I hear it from him and (2) as the subject of a whole culture, of logic, of the universal. Not only children, but the vast majority of adults are empirically incapable of such dialogue, and to be completely honest, I know (in explicated form) only one dialogue of this sort, and that is Bibler’s “Dialog Monologista s Dialogikom” [Dialogue Between a Monologist and a Dialogician], Arkhe, no. 1 (Moscow, 1993), pp. 9–70). This dialogue is a sort of super-problem, a horizon, a regulative idea, to use Kant’s term, and not something we have in hand or that is a realistically achievable aspiration. Our “points of wonder” are an attempt to trace a few steps along the horizon of that idea.

²⁷ See Bibler, “Dialektika i dialogika.”

²⁸ Kurganov’s lessons discussed here have been published, with my commentaries; see I.E. Berliand [Berlyand] and S.Iu. Kurganov, Matematika v Shkole dialoga kul’tur (Kemerovo, 1993).

²⁹ This in no way contradicts the assertion affirming the necessity for unity of object. This is true not only within Bibler’s dialogics, but in Bakhtin as well. Compare the quote in note 27

³⁰ See I.E. Berliand [Berlyand], “K postroeniiu psikhologicheskoi kontseptsii shkoly dialoga kul’tur,” in Shkola dialoga kul’tur. Osnovy programmy (Kemerovo, 1992); “Uchebnaia deiatel’nost’ v shkole razvivaiushchego obuchenia i v shkole dialoga kul’tur.” I am of course using the word “naturally” metaphorically. Perhaps it would have been better to say “quasi-naturally.” Here by “naturally” I mean that which arises outside of specially and intentionally organized, goal-oriented,
systematic influences. In this sense we could say that children learn their native language “naturally” and foreign languages “artificially.” “The native language” is also, after all, a sort of metaphor, as it is not innate in a person biologically.
But we have in mind something completely understandable when we speak of a “native language.”

³¹ This is addressed in greater detail in my work titled “Uchebnaiadeiatel’nost’ v shkole razvivaiushchego obuchenia i v shkole dialoga kul’tur.”