The Rudolf Steiner Archive

Lecture VIII

The first question is as follows:

What is the real difference between multiplication and division in this method of teaching? Or should there be no difference at all in the first school year

The question probably arises from my statement that in multiplication the so-called multiplicand (one factor) and the product are given, and the other factor has to be found. Of course this really gives what is usually regarded as division. If we do not keep too strictly to words, then on the same basis we can consider division, as follows:

We can say: if a whole is divided in a certain way, what is the amount of the part? And you have only another conception of the same thing as in the question: By what must a number be multiplied in order to get a certain other number?

Thus, if our question refers to dividing into parts, we have to do with a division: but if we regard it from the standpoint of “how many times ...” then we are dealing with a multiplication. And it is precisely the inner relationship in thought which exists between multiplication and division which here appears most clearly.

But quite early on it should be pointed out to the child that it is possible to think of division in two ways. One is that which I have just indicated; here we examine how large each part is if we separate a whole into a definite number of parts. Here I proceed from the whole to find the part: that is one kind of division. In the other kind of division I start from the part, and find out how often the part is contained in the whole: then the division is not a separation into parts, but a measurement. The child should be taught this difference between separation into parts and measurement as soon as possible, but without using pedantic terminology. Then division and multiplication will soon cease to be something in the nature of merely formal calculation, as it very often is, and will become connected with life.

So in the first school years it is really only in the method of expression that you can make a difference between multiplication and division; but you must be sure to point out that this difference is fundamentally much smaller than the difference between subtraction and addition. It is very important that the child should learn such things.

Thus we cannot say that no difference at all should be made between multiplication and division in the first school years, but it should be done in the way I have just indicated.

At what age and in what manner should we make the transition from the concrete to the abstract in Arithmetic?

At first one should endeavour to keep entirely to the concrete in Arithmetic, and above all avoid abstractions before the child comes to the turning point of the ninth and tenth years. Up to this time keep to the concrete as far as ever possible, by connecting everything directly with life.

When we have done that for two or two-and-a-half years and have really seen to it that calculations are not made with abstract numbers, but with concrete facts presented in the form of sums, then we shall see that the transition from the concrete to the abstract in Arithmetic is extraordinarily easy. For in this method of dealing with numbers they become so alive in the child that one can easily pass on to the abstract treatment of addition, subtraction, and so on.

It will be a question, then, of postponing the transition from the concrete to the abstract, as far as possible, until the time between the ninth and tenth years of which I have spoken.

One thing that can help you in this transition from the abstract to the concrete is just that kind of Arithmetic which one uses most in real life, namely the spending of money; and here you are more favourably placed than we are on the Continent, for there we have the decimal system for everything. Here, with your money, you still have a more pleasing system than this. I hope you find it so, because then you have a right and healthy feeling for it. The soundest, most healthy basis for a money system is that it should be as concrete as possible. Here you still count according to the twelve and twenty system which we have already “outgrown,” as they say, on the Continent. I expect you already have the decimal system for measurements? (The answer was given that we do not use it for everyday purposes, but only in science.) Well, here too, you have the pleasanter system of measures! These are things which really keep everything to the concrete. Only in notation do you have the decimal system.

What is the basis of this decimal system? It is based on the fact that originally we really had a natural measurement. I have told you that number is not formed by the head, but by the whole body. The head only reflects number, and it is natural that we should actually have ten, or twenty at the highest, as numbers. Now we have the number ten in particular, because we have ten fingers. The only numbers we write are from one to ten: after that we begin once more to treat the numbers themselves as concrete things.

Let us just write, for example: 2 donkeys. Here the donkey is the concrete thing, and 2 is the number. I might just as well say: 2 dogs. But if you write 20, that is nothing more than 2 times 10. Here the 10 is treated as a concrete thing. And so our system of numeration rests upon the fact that when the thing becomes too involved, and we no longer see it clearly, then we begin to treat the number itself as something concrete, and then make it abstract again. We should make no progress in calculation unless we treated the number I itself, no matter what it is, as a concrete thing, and afterwards made it abstract. 100 is really only 10 times 10. Now, whether I have 10 times 10, and treat it as 100, or whether I have 10 times 10 dogs, it is really the same. In the one case the dogs, and in the other the 10 is the concrete thing. The real secret of calculation is that the number itself is treated as something concrete. And if you think this out you will find that a transition also takes place in life itself. We speak of 2 twelves—2 dozen—in exactly the same way as we speak of 2 tens, only we have no alternative like “dozen” for the ten because the decimal system has been conceived under the influence of abstraction. All other systems still have much more concrete conceptions of a quantity: a dozen: a shilling. How much is a shilling? Here, in England, a shilling is 12 pennies. But in my childhood we had a “shilling” which was divided into 30 units, but not monetary units. In the village in I which I lived for a long time, there were houses along the village street on both sides of the way. There were walnut trees everywhere in front of the houses, and in the autumn the boys knocked down the nuts and stored them for the winter. And when they came to school they would boast about it. One would say: “I've got five shillings already,” and another: “I have ten shillings of nuts.” They were speaking of concrete things. A shilling always meant 30 nuts. The farmers' only concern was to gather the nuts early, before all the trees were already stripped! “A nut-shilling” we used to say: that was a unit. To sell these nuts was a right: it was done quite openly.

And so, by using these numbers with concrete things—one dozen, two dozen, one pair, two pair, etc., the transition from the concrete to the abstract can be made. We do not say: “four gloves,” but: “Two pairs of gloves;” not: “Four shoes,” but “two pairs of shoes.” Using this method we can make the transition from concrete to abstract as a gradual preparation for the time between the ninth and tenth years when abstract number as such can be presented.^{1It should be noted that before this transition from the concrete to the abstract dealt with above, a rhythmic approach is used in the teaching of the rudiments of number, e.g. the tables, in the lower classes.}

When and how should drawing be taught?

With regard to the teaching of drawing, it is really a question of viewing the matter artistically. You must remember that drawing is a sort of untruth. What does drawing mean? It means representing something by lines, but in the real world there is no such thing as a line. In the real world there is, for example, the sea. It is represented by colour (green); above it is the sky, also represented by colour (blue). If these colours are brought together you have the sea below and

the sky above (see sketch). The line forms itself at the boundary between the two colours. To say that here (horizontal line) the sky is bounded by the sea, is really a very abstract statement. So from the artistic point of view one feels that the reality should be represented in colour, or else, if you like, in light and shade. What is actually there when I draw a face? Does such a thing as this really exist? (The outline of a face

is drawn.) Is there anything of that sort? Nothing of the kind exists at all. What does exist is this: (see shaded drawing). There are certain surfaces in light and shade, and out of these a face appears. To bring lines into it, and form a face from them, is really an untruth: there is no such thing as this.

An artistic feeling will prompt you to work out what is really there out of black and white or colour. Lines will then appear of themselves. Only when one traces the boundaries which arise in the light and shade or in the colour do the “drawing lines” appear.

Therefore instruction in drawing must, in any case, not start from drawing itself but from painting, working in colour or in light and shade. And the teaching of drawing, as such, is only of real value when it is carried out in full awareness that it gives us nothing real. A terrible amount of mischief has been wrought in our whole method of thinking by the importance attached to drawing. From this has arisen all that we find in optics, for example, where people are eternally drawing lines which are supposed to be rays of light. Where can we really find these rays of light? They are nowhere to be found. What you have in reality is pictures. You make a hole in a wall; the sun shines through it and on a screen an image is formed. The rays can perhaps be seen, if at all, in the particles of dust in the room—and the dustier the room, the more you can see of them. But what is usually drawn as lines in this connection is only imagined. Everything, really, that is drawn, has been thought out. And it is only when you begin to teach the child something like perspective, in which you already have to do with the abstract method of explanation that you can begin to represent aligning and sighting by lines.

But the worst thing you can do is to teach the child to draw a horse or a dog with lines. He should take a paint brush and make a painting of the dog, but never a drawing. The outline of the dog does not exist at all: where is it? It is, of course, produced of itself if we put on paper what is really there.

We are now finding that there are not only children but also teachers who would like to join our school. There may well be many teachers in the outer world who would be glad to teach in the Waldorf School, because they would like it better there. I have had really quite a number of people coming to me recently and describing the manner in which they have been prepared for the teaching profession in the training colleges. One gets a slight shock in the case of the teachers of History, Languages, etc., but worst of all are the Drawing teachers, for they are carrying on a craft which has no connection whatever with artistic feeling: such feeling simply does not exist.

And the result is (I am mentioning no names, so I can speak freely) that one can scarcely converse with the Drawing teachers: they are such dried-up, such terribly “un-human” people. They have no idea at all of reality. By taking up drawing as a profession they have lost touch with all reality. It is terrible to try to talk to them, quite apart from the fact that they want to teach drawing in the Waldorf School, where we have not introduced drawing at all. But the mentality of these people who carry on the unreal craft of drawing is also quite remarkable. And they have no moisture on the tongue—their tongues are quite dry. It is tragic to see what these drawing teachers gradually turn into, simply because of having to do something which is completely unreal.

I will therefore answer this question by saying that where-ever possible you should start from painting and not from drawing. That is the important thing.

I will explain this matter more clearly, so that there shall be no misunderstanding. You might otherwise think I had something personal against drawing teachers. I would like to put it thus: here is a group of children. I show them that the sun is shining in from this side. The sun falls upon something and makes all kinds of light, (see sketch). Light is shed upon everything. I can see bright patches. It is because the sun is shining in that I can see the bright patches everywhere. But above them I see no bright patches, only darkness (blue). But I also see darkness here, below the bright patches: there will perhaps be just a little light here. Then I look at something which, when the light falls on it in this way, looks greenish in colour. Here, where the light falls, it is whitish, but then, before the really black shadow occurs, I see a greenish colour; and here, under the black shadow, it is also greenish, and there are other curious things to be seen in between the two. Here the light does not go right in.

You see, I have spoken of light and shadow, and of how there is something here on which the light does not impinge: and lo, I have made a tree! I have only spoken about light and colour, and I have made a tree. We cannot really paint the tree: we can only bring in light and shade, and green, or,

a little yellow, if you like, if the fruit happens to be lovely apples. But we must speak of colour and light and shade; and so indeed we shall be speaking only of what is really there—colour, light and shade. Drawing should only be done in Geometry and all that is connected with that. There we have to do with lines, something which is worked out in thought. But realities, concrete realities must not be drawn with a pen; a tree, for example, must be evolved out of light and shade and out of the colours, for this is the reality of life itself.^{2The sketch was made on the blackboard with coloured chalks but it has only been possible to reproduce it in black and white.}

It would be barbarous if an orthodox drawing teacher came and had this tree, which we have drawn here in shaded colours, copied in lines. In reality there are just light patches and dark patches. Nature does that. If lines were drawn here, it would be an untruth.

Should the direct method, without translation, be used, even for Latin and Greek?

In this respect a special exception must be made with regard to Latin and Greek. It is not necessary to connect these directly with practical life, for they are no longer alive, and we have them with us only as dead languages. Now Greek and Latin (for Greek should actually precede Latin in teaching) can only be taught when the children are somewhat older, and therefore the translation method for these languages is, in a certain way, fully justified.

There is no question of our having to converse in Latin and Greek, but our aim is to understand the ancient authors. We use these languages first and foremost for the purposes of translation. And thus it is that we do not use the same methods for the teaching of Latin and Greek as those which we employ with all living languages.

Now once more comes the question that is put to me whenever I am anywhere in England where education is being discussed:

How should instruction in Gymnastics be carried out, and should Sports be taught in an English school, hockey and cricket, for example, and if so in what way?

It is emphatically not the aim of the Waldorf School Method to suppress these things. They have their place simply because they play a great part in English life, and the child should grow up into life. Only please do not fall a prey to the illusion that there is any other meaning in it than this, namely, that we ought not to make the child a stranger to his world. To believe that sport is of tremendous value in development is an error. It is not of great value in development. Its only value is that it is a fashion dear to the English people, and we must not make the child a stranger to the world by excluding him from all popular usages. You like sport in England, so the child should be introduced to sport. One should not meet with philistine opposition what may possibly be philistine itself.

With regard to “how it should really be taught,” there is very little indeed to be said. For in these things it is really more or less so that someone does them first, and then the child imitates him. And to devise special artificial methods here would be something scarcely appropriate to the subject.

In Drill or Gymnastics one simply learns from anatomy and physiology in what position any limb of the organism must be placed in order that it may serve the agility of the body. It is a question of really having a sense for what renders the organism skilled, light and supple; and when one has this sense, one has then simply to demonstrate. Suppose you have a horizontal bar: it is customary to perform all kinds of exercises on the bar except the most valuable one of all, which consists in hanging on to the bar, hooked on, like this ... then swinging sideways, and then grasping the bar further up, then swinging back, then grasping the bar again. There is no jumping but you hang from the bar, fly through the air, make the various movements, grasp the bar thus, and thus, and so an alternation in the shape and position of the muscles of the arms is produced which actually has a healthy effect upon the whole body.

You must study which inner movements of the muscles have a healthy effect on the organism, so that you will know what movements to teach. Then you have only to do the exercises in front of the children, for the method consists simply in this preliminary demonstration.^{3A method of Gymnastic teaching on the lines indicated above was subsequently worked out by Fritz Graf Bothmer, teacher of Gymnastic s at the Waldorf School, Stuttgart.}

How should religious instruction be given at the different ages?

As I always speak from the standpoint of practical life, I have to say that the Waldorf School Method is a method of education and is not meant to bring into the school a philosophy of life or anything sectarian. Therefore I can only speak of what lives within the Waldorf School principle itself.

It was comparatively easy for us in Württemberg, where the laws of education were still quite liberal: when the Waldorf School was established we were really shown great consideration by the authorities. It was even possible for me to insist that I myself should appoint the teachers without regard to their having passed any State examination or not. I do not mean that everyone who has passed a State examination is unsuitable as a teacher! I would not say that. But still, I could see nothing in a State examination that would necessarily qualify a person to become a teacher in the Waldorf School.

And in this respect things have really always gone quite well. But one thing was necessary when we were establishing the school, and that was for us definitely to take this standpoint: We have a “Method-School”; we do not interfere with social life as it is at present, but through Anthroposophy we find the best method of teaching, and the School is purely a “Method-School.”

Therefore I arranged, from the outset, that religious instruction should not be included in our school syllabus, but that Catholic religious teaching should be delegated to the Catholic priest, and the Protestant teaching to the pastor and so on.

In the first few years most of our scholars came from a factory (the Waldorf-Astoria Cigarette Factory), and amongst them we had many “dissenting” children, children whose parents were of no religion. But our educational conscience of course demanded that a certain kind of religious instruction should be given them also. We therefore arranged a “free religious teaching” for these children, and for this we have a special method.

In these “free Religion lessons” we first of all teach gratitude in the contemplation of everything in Nature. Whereas in the telling of legends and myths we simply relate what things do—stones, plants and so on—here in the Religion lessons we lead the child to perceive the Divine in all things. So we begin with a kind of “religious naturalism,” shall I say, in a form suited to the children.

Again, the child cannot be brought to anunderstanding of the Gospels before the time between the ninth and tenth years of which I have spoken. Only then can we proceed to a consideration of the Gospels in the Religion lessons, going on later to the Old Testament. Up to this time we can only introduce to the children a kind of Nature-religion in its general aspect, and for this we have our own method. Then we should go on to the Gospels but not before the ninth or tenth year, and only much later, between the twelfth and thirteenth years, we should proceed to the Old Testament.^{4This paragraph can easily be misunderstood unless two other aspects of the education are borne in mind. Firstly: Here Dr. Steiner is only speaking of the content of the actual Religion lessons. In the class teaching all children are introduced to the stories of the Old Testament. Secondly, quite apart from the Religion lessons the Festivals of the year are celebrated with all children in a Rudolf Steiner School, in forms adapted to their ages. Christmas takes a very special place, and is prepared for all through Advent by carol singing, the daily opening of a star-window in the “Advent Calendar,” and the lighting of candles on the Advent wreath hung in the classroom. At the end of the Christmas term the teachers perform traditional Nativity Plays as their gift to the children. All this is in the nature of an experience for the children, inspired by feeling and the Christmas mood. Later, in the Religion lessons, on the basis of this experience, they can be brought to a more conscious knowledge and understanding of the Gospels.}

This then is how you should think of the free Religion lessons. We are not concerned with the Catholic and Protestant instruction: we must leave that to the Catholic and Protestant pastors. Also every Sunday we have a special form of service for those who attend the free Religion lessons. A service is performed and forms of worship are provided for children of different ages. What is done at these services has shown its results in practical life during the course of the years; it contributes in a very special way to the deepening of religious feeling, and awakens a mood of great devotion in the hearts of the children.

We allow the parents to attend these services, and it has become evident that this free religious teaching truly brings new life to Christianity And there is real Christianity in the Waldorf School, because through this naturalistic religion during the early years the children are gradually led to an understanding of the Christ Mystery, when they reach the higher classes.

Our free Religion classes have, indeed, gradually become full to overflowing. We have all kinds of children coming into them from the Protestant pastor or the Catholic priest, but we make no propaganda for it. It is difficult enough for us to find sufficient Religion teachers, and therefore we are not particularly pleased when too many children come; neither do we wish the school to acquire the reputation of being an Anthroposophical School of a sectarian kind. We do not want that at all. Only our educational conscience has constrained us to introduce this free Religion teaching. But children turn away from the Catholic and Protestant teaching and more and more come over to us and want to have the free Religion teaching: they like it better. It is not our fault that they run away from their other teachers: but as I have said, the principle of the whole thing was that religious instruction should be given, to begin with, by the various pastors. When you ask, then, what kind of religious teaching we have, I can only speak of what our own free Religion teaching is, as I have just described it.

Should French and German be taught from the beginning, in an English School? If the children come to a Kindergarten Class at five or six years old, ought they, too, to have language lessons?

As to whether French and German should be taught from the beginning in an English School, I should first like to say that I think this must be settled entirely on grounds of expediency. If you simply find that life is making it necessary to teach these languages, you must teach them. We have introduced French and English into the Waldorf School, because with French there is much to be learnt from the inner quality of the language, not found elsewhere, namely, a certain feeling for rhetoric which it is very good to acquire: and English is taught because it is a universal world language, and will become so more and more.

Now, I should not wish to decide categorically whether French and German should be taught in an English School, but you must be guided by the circumstances of life. It is not at all so important which language is chosen as that foreign languages are actually taught in the school.

And if children of four or five years do already come to school (which should not really be the case) it would then be good to do languages with them also. It would be right for this age. Some kind of language teaching can be given even before the age of the change of teeth, but it should only be taught as a proper lesson after this change. If you have a Kindergarten Class for the little children, it would be quite right to include the teaching of languages but all other school subjects should as far as possible be postponed until after the change of teeth.

I should like to express, in conclusion, what you will readily appreciate, namely, that I am deeply gratified that you are taking such an active interest in making the Waldorf School Method fruitful here in England, and that you are working with such energy for the establishment of a school here, on our Anthroposophical lines. And I should like to express the hope that you may succeed in making use of what you were able to learn from our Training Courses in Stuttgart, from what you have heard at various other Courses which have been held in England, and, finally, from what I have been able to give you here in a more aphoristic way, in order to establish a really good school here on Anthroposophical lines. You must remember how much depends upon the success of the very first attempt. If it does not succeed, very much is lost, for all else will be judged by the first attempt. And indeed, very much depends on how your first project is launched: from it the world must take notice that the matter is neither something which is steeped in abstract, dilettante plans of school reform, nor anything amateur but something which arises out of a conception of the real being of man, and which is now to be brought to bear on the art of education. And it is indeed the very civilisation of today, which is now moving through such critical times, that calls us to undertake this task, along with many other things.

In conclusion I should like to give you my right good thoughts on your path—the path which is to lead to the founding of a school here on Anthroposophical lines.