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The Rudolf Steiner Archive

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Discussions with Teachers
GA 295

Discussion Fourteen

5 September 1919, Stuttgart

Translated by Helen Fox

The principles were developed for teaching music to the first and second grades.

RUDOLF STEINER: Children should be allowed to hear an instrument, to hear music objectively, apart from themselves. This is important. It should be a matter of principle that well before the ninth year the children should learn to play solo instruments, and the piano can be added later for those for whom it is considered advisable. What matters most is that we make a right beginning in this sphere.

Further remark on the concept of interest, proceeding to algebra:

If \(A =\) amount, \(P =\) principal, \(I =\) interest, \(R =\) rate of interest, \(T =\) time, then \(A = P + I\).

Since

$$I = \frac{PRT}{100}$$

then

$$A = P + \frac{PRT}{100}$$

RUDOLF STEINER: It would never be possible to describe capital in this way these days; this formula only has real value if \(T\) equals a year or less, because in reality two cases are given: Either you remove the interest each year, in which case the same initial capital always remains, or else you leave the interest with the capital, in which case you need to figure according to compound interest. If you omit \(T\) — that is, if you figure it for only one year, then it is an actual thing; it is essential to present realities to the children. Do not fail to observe that the transition to algebra as we have spoken of it, is really carried out — first from addition to multiplication, and then from subtraction to division. This must be adhered to strictly.

RUDOLF STEINER explained the transition from arithmetic to algebra with the following example: First you write down a number of figures in which all the addenda are different:

$$20 = 7 + 5 + 6 + 2$$

Some of the addenda could also be equal:

$$25 = 5 + 5 + 9 + 6$$

Or all the addenda could be the same:

$$18 = 6 + 6 + 6$$

If you proceed, as described in our previous discussion, to replace numbers with letters, then you could have the equation:

\(S_1 = a + a + a\); that is, three \(a\)’s, or three times \(a = 3a\).

then

\(S_2 = a + a + a + a + a\); five times \(a = 5a\);

then

\(S_3 = a + a + a + a + a + a + a\); or seven times \(a = 7a\)

and so on. I can keep doing this; I could do it \(9\) times, \(21\) times, \(25\) times, I can do it \(n\) times:

\(S_n = a + a + a ... n\) times \(= na\)

Thus, I get the factor by varying the number of the addenda, while the addendum itself is the other factor. In this way multiplication can easily be developed and understood from addition, and you thus make the transition from actual numbers to algebraic quantities:

\(a × a = a2\), \(a × a × a = a3\).

In the same way you can derive division from subtraction. If we take b away from a very large number a, we get the remainder \(r\):

\(r = a – b\)

If we take b away again, we get the remainder:

\(r_2 = a – b – b = a – 2b\)

If b is taken away a third time we obtain:

\(r_3 = a – b – b – b = a – 3b\) and so on.

We could continue until there is nothing left of number \(a\): suppose this happens after subtracting \(b\) \(n\) times:

\(r_n = a – b – b – b ... n\) times \(= – nb\) When there is nothing left — that is, when the last remainder is \(0\), then:

\(0 = a – nb\)

So a is now completely divided up, because nothing remains:

\(a = nb\)

I have taken b away n times, I have divided \(a\) into nothing but \(bs\), \(a/b = n\), so the \(a\) is completely used up. I have discovered that I can do this \(n\) times, and in so doing I have gone from subtraction to division.

Thus we can say: multiplication is a special case of addition, and division is a special case of subtraction, except that you add to it or take away from it, not just once, but repeatedly, as the case may be.

Negative and imaginary numbers were discussed.

RUDOLF STEINER: A negative number is a subtrahend [the number subtracted] for which there is no minuend [the number from which it is subtracted]; it is a demand that something be done: there being nothing to do it with, thus it cannot be done. Eugen Dühring rejected imaginary numbers as nonsense and spoke of Gauss’s definition of “the imaginary” as completely stupid, unrealistic, farfetched nonsense. 1Karl Eugen Dühring (1833–1921), German positivist philosopher and economist. Wrote Kapital und Arbeit (Capital and Labor), and Logik und Wissenschaftstheorie (Logic and Epistemological Theory).

From addition, therefore, you develop multiplication, and from multiplication, rise to a higher power. And then from subtraction you develop division, and from division, find roots.

addition — subtraction
multiplication — division
raising to a higher power — finding roots

You should not proceed to raising to a higher power and finding roots until after you have begun algebra (between the eleventh and twelfth years), because, with roots, raising to a power of an algebraic equation of more than one term (polynomial) plays a role. In this connection you should also deal with figuring gross, net, taxes, and packing charges.

A question about the use of formulas.

RUDOLF STEINER: The question is whether you should avoid the habitual use of formulas, but go through the thought processes again and again (a good opportunity for practicing speech), or whether it might be even better to go ahead and use the formula itself. If you can succeed, tactfully, in making the formula fully understood, then it can be very useful to use it as a speech exercise — to a certain extent.

But from a certain age on, it is also good to make the formula into something felt by the children, make it into something that has inner life, so that, for example, when the \(T\) increases in the formula \(I = PRT/100\), it gives the children a feeling of the whole thing growing.

In effect, this is what I wanted to say at this point — that you should use the actual numbers for problems of this kind — for example, in interest and percentages — in order to make the transition to algebra, and in doing so, develop multiplication, division, raising powers, and roots. These are things that certainly must be done with the children.

Now I would like to ask a question: Do you consider it good to deal with raising to a higher power and finding roots before you have done algebra, or would you do it later?

Comment about raising to a higher power first and finding roots after.

RUDOLF STEINER: Your plan then would be (and should continue to be) to start with algebra as soon as possible after the eleventh or twelfth year, and only after that proceed to raising to a higher power and finding roots. After teaching the children algebra, you can show them in a very quick and simple way how to square, cube, raise to a higher power, and extract the root, whereas before they know algebra you would have to spend a terribly long time on it. You can teach easily and economically if you take algebra first.

A historical survey for the older children (eleven to fourteen years) was presented concerning the founding and development of towns, referring to the existence of a “Germany” at the time of the invasion of the Magyars.

RUDOLF STEINER: You must be very careful not to allow muddled concepts to arise unconsciously. At the time of Henry, the so-called “townbuilder,” there was of course no “Germany.” You would have to express what you mean by saying “towns on the Rhine” or “towns on the Danube” in the districts that later became “German.” 2This could as well apply to speaking of “America” in regard to events and places prior to the time of Columbus and the European settlers.

Before the tenth century the Magyars are not involved at all, but there were invasions of Huns, Avars, and so on. But after the tenth century you can certainly speak of “Germany.” When the children reach the higher grades (the seventh and eighth grades) I would try to give them a concept of chronology; if you just say ninth or tenth century, you do not give a sufficiently real picture. How then would you manage to awaken in the children a concrete view of time?

You could explain it to them like this: “if you are now of such and such an age, how old are your mother and father? Then, how old are your grandfather and grandmother?” And so you evoke a picture of the whole succession of generations, and you can make it clear to the children that a series of three generations makes up about 100 years, so that in 100 years there would be three generations. A century ago the great grandparents were children. But if you go back nine centuries, there have not been three generations, but \(9\ x\ 3 = 27\) generations. You can say to the child: “Now imagine you are holding your father’s hand, and he’s holding your grandfather’s hand, and he is, in turn, holding your great-grandfather’s hand, and so on. If they were now all standing together side by side, which would be Henry I, which number in the row would have stood face to face with the Magyars around the year 926? It would be the twenty-seventh in the row.” I would demonstrate this very clearly in a pictorial way. After giving the children this concrete image of how long ago it was, I would present a graphic description of the migrations of the Magyars. I would tell them about the Magyars’ invasion of Europe at that time, how they broke in with such ferocity that everyone had to flee before them, even the little children in their cradles, who had to be carried up to the mountaintops, and how then the onrushing Magyars burned the villages and forests. Give them a vivid picture of this Magyar onset.

It was then described how Henry, knowing he had been able to resist the Magyars in fortified Goslar, resolved to build fortified towns, and in this way it come about that numerous towns were founded.

RUDOLF STEINER: Here again, could you not present this more in connection with the whole history of civilization? It is only a garbled historical legend to say that Henry founded these towns. All these tenth century towns were built on their original foundations — that is, the markets — before then. But what helped them to expand was the migration of the neighboring people into the towns in order to defend themselves more easily against the Magyars’ assaults, and for this reason they fortified these places. The main reasons for building these towns were more economic in nature. Henry had very little part in all this.

I ask you to be truly graphic in your descriptions, to make everything really alive, so that the children get vivid pictures in their minds, and the whole course of events stands out clearly before them. You must stimulate their imagination and use methods such as those I mentioned when I showed you how to make time more real. Nothing is actually gained by knowing the year that something occurred — for example, the battle of Zama; but by using the imagination, by knowing that, if they held hands with all the generations back to Charles the Great, the time of their thirtieth ancestor, the children would get a truly graphic, concrete idea of time. This point of time then grows much closer to you — it really does — when you know that Charles the Great is there with your thirtieth ancestor.

Question: Wouldn’t it also be good when presenting historical descriptions to dwell on the difference in thought and feeling of the people of those times?

RUDOLF STEINER: Yes. I have always pointed this out in my lectures and elsewhere. Most of all, when speaking of the great change that occurred around the fifteenth century, you should make it very clear that there was a great difference between the perception, feeling, and thought of people before and after this time. Lamprecht too (whom I do not however especially recommend) is careful to describe a completely different kind of thinking, perceiving, and feeling in people before this time. 3Karl Gottfried Lamprecht (1856–1915), historian who developed the theory that history is social-psychological rather than political. The documents concerning this point have not yet been consulted at all.

In studying the books written on cultural history you must, above all, develop a certain perceptive faculty; with this you can properly assess all the different things related by historians, whether commonplace or of greater importance, and so gain a truer picture of human history.

Rudolf Steiner recommended for the teachers’ library Buckle’s History of Civilization in England and Lecky’s History of Rationalism in Europe.

RUDOLF STEINER: From these books you can learn the proper methods of studying the history of human progress. With Lamprecht only his earlier work would be suitable, but even much of this is distorted and subjective. If you have not acquired this instinct for the real forces at work in history, you will be in danger of falling into the stupidity and amateurism of a “Wildenbruch” for example; 4Ernst von Wildenbruch (1845–1909), German writer, author of Spartacus. he imagined that the stories of emperors and kings and the family brawls between Louis the Pious and his sons were important events in human history.

Gustav Freytag’s Stories from Ancient German History are very good; 5Gustav Freytag (1816–1895), German writer, promoter of German liberalism and the middle class. His books included a series of six historical novels. but you must beware of being influenced too much by this rather smug type of history book (written for the unsophisticated). The time has come now when we must get out of a kind of thinking and feeling that belonged to the middle of the nineteenth century.

Mention was made of Houston Stuart Chamberlain’s Foundations of the Nineteenth Century. 6Houston Stuart Chamberlain (1855–1927), British publicist, naturalized German citizen. Wrote on the superiority of the Western Aryan race.

RUDOLF STEINER: With regard to Chamberlain also you must try to develop the correct instinct. For one part of clever writing you get three parts of bad, unwholesome stuff. He has some very good things to say, but you must read it all yourselves and form your own judgements. The historical accounts of Buckle and Lecky are better. 7Probably refers to Henry Thomas Buckle (1821–1862), English historian, who wrote the incomplete History of Civilization in England. William Edward Hartpole Lecky 1838–1903), Irish historian who wrote on modern European history. Chamberlain is more one of these “gentlemen in a dinner jacket.” He is rather a vain person and cannot be accepted as an authority, although many of his observations are correct. And the way he ended up was not particularly nice — I mean his lawsuit with the “Frankfurter Zeitung.”

Kautsky’s writings were mentioned. 8Karl Johann Kautsky (1854–1938), German Marxist theorist, journalist, and secretary to Friedrich Engels in London, he opposed Bolshevism and the Russian revolution.

RUDOLF STEINER: Well yes, but as a rule you must assume that the opposite of what he says is true! From modern socialists you can get good material in the way of facts, as long as you do not allow yourselves to be deceived by the theories that color all their descriptions. Mehring too presents us with rather a peculiar picture; 9Franz Mehring (1846–1919), German Socialist historian and journalist. because at first, when he was himself a progressive Liberal, he inveighed against the Social Democrats in his book on Social Democracy; but later when he had gone over to the Social Democrats he said exactly the same things about the Liberals!

An introduction was presented on the fundamental ideas in mathematical geography for twelve-year-old children, with observations on the sunrise and the ecliptic.

RUDOLF STEINER: After taking the children out for observations, it would be very good to let them draw what they had observed; you would have to make sure there is a certain parallel between the drawing and what the children saw outside. It is advisable not to have them do too much line drawing. It is very important to teach these things, but if you include too much you will reach the point where the children can no longer understand what you are saying. You can relate it also to geography and geometry.

When you have developed the idea of the ecliptic and of the coordinates, that is about as far as you should go.

Someone else developed the same theme — that is, sunrise and sunset — for the younger children, and tried to explain the path of the Sun and planets in a diagrammatic drawing.

RUDOLF STEINER: This viewpoint will gradually lose more and more of its meaning, because what has been said until now about these movements is not quite correct. In reality it is a case of a movement like this (lemniscatory screw-movement):

Enclosing a cross

Here, for example, [in position 1] we have the Sun; here are Saturn, Jupiter, Mars, and here are Venus, Mercury, and Earth. Now they all move in the direction indicated [spiral line], moving ahead one behind the other, so that when the Sun has progressed to the second position we have Saturn, Jupiter, and Mars here, and we have Venus, Mercury, and Earth over there. Now the Sun continues to revolve and progresses to here [position 3]. This creates the illusion that Earth revolves round the Sun. The truth is that the Sun goes ahead, and the Earth creeps continually after it.

The ancient Egyptian civilization was described.

RUDOLF STEINER: It is most important to explain to the children that Egyptian art was based on a completely different method of representing nature. The ancient Egyptians lacked the power of seeing things in perspective. They painted the face from the side and the body from the front. You may certainly explain this to the children, especially the Egyptian concept of painting. Then you must point out how Egyptian drawing and painting was related to their view of natural history — how, for example, they portrayed men with animal heads and so on.

In ancient times the habit of comparing people with the animals was very common. You could then point out to the children what is present in seed form, as it were, within every human face, which children can still see to a certain extent. 10See The Foundations of Human Experience, lecture 9. The Egyptians still perceived this affinity of the human physiognomy with animals; they were still at this childlike stage of perception.

Question: What should one really tell children about the building of the Egyptian pyramids?

RUDOLF STEINER: It is of course extraordinarily important for children too that you should gradually try to present them with what is true rather than what is false. In reality the pyramids were places of initiation, and this is where you reach the point of giving the children an idea of the higher Egyptian education, which was initiation at the same time. You must tell them something about what happened within the pyramids. Religious services were conducted there, just as today they are conducted in churches, except that their services led to knowledge of the universe. Ancient Egyptians learned through being shown, in solemn ritual, what comes about in the universe and in human evolution. Religious exercises and instruction were the same; it was really such that instruction and religious services were the very same thing.

Someone described the work of the Egyptians on the pyramids and obelisks, and said that several millions of people must have been needed to transport the gigantic blocks of stone, to shape them, and to set them in place. We must ask ourselves how it was possible at all, with the technical means available at that time, to move these great heavy blocks of limestone and granite and to set them in place.

RUDOLF STEINER: Yes, but you only give the children a true picture when you tell them: If people were to do this work with the physical strength of the present day, two and a half times as many people would be necessary. The fact is that the Egyptians had two and a half times the physical strength that people have today; this is true, at least, of those who worked on the pyramids and so on. There were also, of course, those who were not so strong.

Question: Would it be good to include Egyptian mythology?

RUDOLF STEINER: Unless you can present Egyptian mythology in its true form, it should be omitted. But in the Waldorf school, if you want to go into this subject at all, it would be a very good plan to introduce the children to the ideas of Egyptian mythology that are true, and are well known to you. 11See Rudolf Steiner, Egyptian Myths and Mysteries, Anthroposophic Press, Hudson, NY, 1990.