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The Rudolf Steiner Archive

a project of Steiner Online Library, a public charity

Faculty Meetings with Rudolf Steiner
GA 300

Sixty-Sixth Meeting

30 April 1924, Stuttgart

Dr. Steiner: The first thing I would like to discuss is my discussion today with the present twelfth-grade students. With one exception, the students stated they did not need to take their final examinations at the end of this year, but could wait a year. At the end of the Waldorf School, they would go through a cramming class. It was important to them, however, that this cramming for the final examination be taught by the Waldorf School.

A teacher comments.

Dr. Steiner: The point is that we said we wanted to resolve this matter after meeting with the twelfth-grade students. We cannot handle such things if someone comes afterward and says there is still one more thing. If arguments are always presented about everything after it is done, then we will never finish anything. Things will only become confused. How is it that now there are suddenly two? Where did that come from? The problem is, that was overlooked. It makes no sense that such things occur suddenly. Is the faculty in control, or the children? The results should remain as they were today at noon, and that girl will need to have some sort of private instruction. In general, we should teach the class in a way appropriate to a twelfth-grade Waldorf School class.

The first thing we need to consider for the curriculum is literary history. Yesterday, I mentioned that, in general, they should have already covered the main content of literary history. A cursory survey will have to suffice for the things they have not learned. On the other hand, you should undertake a complete survey of German literary history in relation to things that play into it from outside.

Therefore, you have to begin with the oldest literary monuments and work them all into an overview. Begin with the oldest literary monuments, starting with the Gothic period, then go on to the Old German period and continue into the development of the ,em>Song of the Nibelungs and Gudrun. Do that in a cursory way, but so that they get a picture of the whole. Then, go on to the Middle Ages, the pre-classical period, the classical and romantic periods, up to the present. Give them an overview, but one that contains the general perspectives. The content should enable them to clearly know what they need to know about such people as Walther von der Vogelweide, Klopstock, or Logau. I think you could cover that in five or six periods. You can certainly do that.

I would then follow that with the main things they need to know about the present. You should discuss the present in much more detail with the twelfth grade. By present, I mean you would discuss the most important literary works of the 1850s, 60s, and 70s, then follow that with a more detailed treatment of the subsequent movements, so that they would have some insight into who Nietzsche and Ibsen were, or such foreigners as Tolstoy and Dostoyevsky, and so forth. The result should be that we graduate well-educated people.

Next is history, which you should do in a similar way. Start with a survey of history as a whole, beginning with the history of the East, which then gives rise to Greece and more modern Christian developments. You can surely go into these things without teaching anthroposophical dogma. You can present things that have a genuine inner spirituality. At the workers’ school, for example, I once showed how the seven Roman kings followed the model of the seven principal aspects of the human being, since that is what they are. Of course, you cannot simply say that Romulus is the physical body, and so forth. Nevertheless, Livius’s History of Kings has that in its inner structure. We find that the fifth king, Tarquinius Priscus, is clearly a person of intellect, corresponding to the I. He brings a new impulse, just as with the spirit self, the Etruscan element. You should treat the last one, Tarquinius Superbus, such that the highest we can reach sinks in most deeply, as it, of course, did with the Roman people, where it sunk into the Earth.

In the same way, you can very beautifully develop oriental history. In Indian history, we find the formation of the physical body, in Egyptian history, the etheric body, and in Chaldaic- Babylonian history, the astral body. Of course, you cannot teach it in that form. You need to show how those human beings living in the astral developed astronomy, how the Jews have the principle of the I in the principle of Yahweh, and how the Greeks for the first time developed a true understanding of nature from a human perspective. The viewpoint of the earliest peoples was still within the human being. You could give them an overview you can be proud of. Historical events form a complete series.

Geography class will also consist in giving them an overview. In both history and geography, what is important is to give them an overview. They can then search out the details by themselves.

You could divide aesthetics and art class as we discussed yesterday: into symbolic, classical, and romantic art. You could also treat not only the science of art by saying that in Egypt it was symbolic, in Greece classical, and in what followed, romantic, but also, the arts themselves, in that architecture is a symbolic art, sculpture is a classical art, and painting, music, and poetry are the romantic arts. Thus you can view the arts themselves in a way that offers a kind of inner division.

In teaching aesthetics and art, you can treat the elements of architecture so that the young people will have a proper understanding of how a house is constructed, that is, you could include construction materials, the construction of a roof, and so forth, in aesthetics.

Then we have languages. There, it is better if we describe the goals by saying that in English or French the students should get an idea of modern literature.

Now we have mathematics. How far did the eleventh grade come in mathematics?

A teacher: In the eleventh grade we got as far as indeterminate equations in algebra. In trigonometry, aside from spherical trigonometry, they went as far as computing acute-angle triangles. In complex numbers, as far as Moivre’s theorem, then polynomial equations. In analytic geometry, we went as far as working with second-order curves, but we worked in depth only with the circle. In constructive geometry, we did sections and intersections.

Dr. Steiner: Our experience with last year’s class has shown that we cannot do it that way. It is too much for the human soul to do such things.

What is important is to go through spherical trigonometry, that is, the elements of analytical spatial geometry, in a way that is as clear as possible.

In descriptive geometry we have Cavalieri’s perspective. The students should be able to draw a complicated form, such as a house, in Cavalieri’s perspective. The inside as well as the outside.

In algebra, you need only cover the beginnings of differential and integral calculus. They do not need to be able to compute maximums and minimums. They will learn that in college. You should teach them only the basic concepts of calculus, but do that thoroughly.

You should emphasize spherical trigonometry and how it is used in astronomy and geodesy in a way appropriate to their age, so that they have a general understanding of it.

Spatial analytical geometry should be used to teach them how equations can express forms. I would not be afraid to complete this subject by giving them examples of questions like, What curve is represented by the equation

$$x^\frac{2}{3} + y^\frac{2}{3} + z^\frac{2}{3} = a$$

which results in an astroid. The main thing is to make equations so transparent that the students have a feeling for how things are hidden within equations.

You should also do the opposite. If I draw a curve or place a body in space, they should be able to recognize the general form of the equation without necessarily having it correct in all details, but at least have an idea of what the equation would be.

I don’t think the normal mathematical education that connects differential and integral calculus with geometry is particularly useful. I think it should be connected with quotients instead. I would begin with the quotient


then make the dividend and the divisor smaller and smaller, simply as numbers, and then go on to develop differential quotients. I would not begin with the idea of continuity, because you do not really get an idea of differential quotients that way. Don’t begin with differentials, but with differential quotients. If you begin with a series, then go on to geometry only after you have presented tangents, that is, move from the secant to the tangent. Go on to geometry only after the students have completely comprehended differential quotients purely as numbers or through computations, so that they are presented with the picture that geometric visualization is only an illustration of what occurs numerically. You can then teach them integrals as the reverse process. Thus, you will have a possibility of showing them that the computation is not a fixing of geometry, but that geometry is an illustration of the computation. That is something people should consider more often. For example, you should not consider positive and negative numbers as something in themselves, but as a series of numbers such as

$$(5 - 1), (5 - 2), (5 - 3), (5 - 4), (5 - 5), (5 - 6)$$

In the last instance, I do not have enough, I am missing one, and I write that as (-1). Emphasize only what is missing without using a number line. You will then remain within numbers. A negative number is the amount that is not present. It is a deficiency of the minuend. There is much more inner activity in working that way. You can excite some of the students’ capacities in a much more real way than when you do everything beginning from geometry.

A teacher: Where should we begin?

Dr. Steiner: Now that the class is ready for spherical trigonometry, you will need to move from trigonometry to developing the concept of the sphere qualitatively, that is, without starting computations. Instead of drawing on a plane, they need to begin drawing on a sphere, so that they get an idea of what a spherical triangle is, that is, how a triangle lies upon a sphere. You need to make that visible for the children, then go on to show them how the sum of the angles is not equal to 180°, but is larger. They need to really understand triangles on a sphere, with their curved lines, and then begin the computations. In geometry, the computation is only the interpretation of the sphere. I do not want you to begin by considering the sphere from its midpoint, but from the curvature of the surfaces. Then you can go on to a more general discussion of the non-linearity, how you could look at a corresponding figure on an ellipsoid, or how it would look on a paraboloid, where it is no longer completely closed. Don’t begin with the center, but with the distortion of the surface; otherwise you will have difficulties with other solids. In a way, you will need to think of yourself on the surface; in a sense, you will have to form a picture of what you would experience if you were a spherical triangle. You need to ask yourself, What would I experience as a triangle on an ellipsoid?

In that connection, you will also have to show the students what would happen if you used the normal Pythagorean theorem on a spherical triangle. You cannot, of course, use squares for that. Doing things this way has an effect upon the general education, whereas normally they affect only the intellect.

You can cover permutations and combinations quickly, and, if there is enough time, the beginnings of probability theory, for instance, the life expectancy of a human being.

In the eleventh grade, you need to go through sections and intersections, shadows and indeterminate equations, and analytical geometry up to conic sections. In eleventh-grade trigonometry, teach the functions in a more inner way, so that you present the principle relationships in sine and cosine. There, of course, you will have to begin from geometry.

Begin twelfth-grade physics with optics, as we discussed yesterday.

Natural history. We have already discussed zoology. In geology and paleontology, begin with zoology, since only then do they have some inner value. You can begin with zoology, go on to paleontology, and arrive at the various layers of the Earth. In botany, you can begin with flowering plants (phanerogans), and then also go on to geology and paleontology.

Chemistry. We want to consider chemistry in its innermost connections to the human being. In the twelfth grade, our students already have an idea of organic and inorganic processes. It is now important to go on to those processes found not only in animals, but also in human beings. We can speak without hesitation about the formation of ptyalin, pepsin, and pancreatin. You should teach the metallic processes in the human being by developing things from principles, for instance, something we could call the lead process in the human being, so that the students understand them. You need to show that within the human being all materials and processes are completely transformed. In connection with the formation of pepsin, what is important is to begin with the formation of hydrochloric acid, showing that it is lifeless. Then go on to consider the formation of pepsin as something that can occur only within the etheric body, even though the astral body has some effect upon it. In other words, show how the process completely disintegrates and then is rebuilt. Begin hydrochloric acid, with the inorganic process using salt. Discuss all the characteristics of hydrochloric acid, then go on to show how that differs from what occurs in an organic body. The result should be the demonstration of the differences between vegetable protein, animal protein, and human protein, so the students have an idea that there is a progression of protein based upon the various structures of the etheric body. Human protein is different from animal protein. You can also begin with differences by looking at a lion and a cow. In the lion, we find a process that is much more directed toward the circulation than in a cow where the entire process is more directed toward the metabolism. In the lion, the metabolic process is formed together with the breathing, whereas in the cow, the breathing is supported by the digestion. This will enliven the processes more. You need to have an inorganic, an organic, an animal, and a human chemistry. Some examples for children might be hydrochloric acid and pepsin, or blackthorn juice and ptyalin. Then they will get the picture. You could also use the metamorphosis of folic acid into oxalic acid.

A teacher asks whether to include quantitative chemistry.

Dr. Steiner: Well, it is certainly very difficult to explain these things with what you can normally assume. You need to begin with cosmic rhythm to explain the periodic system. That is the way you need to go, but you cannot do that in school. It is complete nonsense to begin with atomic weights; you need to begin with rhythms. You can explain all of the quantitative relationships through harmonics. The relationship between oxygen and hydrogen is, for example, an octave. But, that would go too far. I think you should develop the concepts we mentioned before and that will be enough for the twelfth-grade curriculum.

Eurythmy is not intended for the final examination.

Religion class. In general, the character of religious instruction is already in the curriculum. I can certainly not add much to what you have already presented. There is nothing we really need to change. The question is what to do in the upper grades. In the end, you should be able to give the twelfth grade a survey of world religions, but not in a way that gives the children the idea that some of them are untrue. Instead, you need to show the relative truths in their individual forms. That would be the ninth level. In the eighth level, you need to go through Christianity so that it appears in the ninth level as the synthesis of religions. Develop Christianity in the eighth level, and in the ninth level emphasize world religions so that, once again, their high point is Christianity. In the seventh level, you should present a kind of evangelical harmony, present Christianity in its essence and in the way it appears. By then, the children will all know the Gospels. Therefore, at the seventh level, a harmony of Gospels, at the eighth, Christianity, and at the ninth, world religions.

I will prepare the curriculum for modern languages in the ninth through twelfth grades and give it to you at a meeting about the foreign language classes.

There is a discussion about the university classes in Stuttgart.

Dr. Steiner: I would like to hear whether you think what has been proposed for the courses is too much or not. I would like to hear what you expect. What you thought of for the course that is just beginning and will continue until the next summer vacation? If we want to avoid a terribly chaotic situation, we certainly should not do things more than five days a week. I thought of doing a five-lecture series; Wednesday and Friday are not available. I could give lectures on Monday, Tuesday, Thursday, and Saturday, and two on one day.

I think we should present only five areas. We cannot present social understanding yet. It would also be very good to teach some practical subject, say, geodesy. We don’t want to have any specific themes. I think Dr. Schwebsch could teach aesthetics and literature; Stein, history; Unger, epistemology; Baravalle, mathematics; and Stockmeyer, geodesy.

It seems that one error has been that there is too much lecturing. Sometime we will also need to present something about music theory. We should do that in the course next winter. So that there will be a certain amount of liveliness, I propose that wherever possible, you bring the most recent events into the discussion. It would be good, for example, to work through our perspective on aesthetics as I discussed in the two little essays. Since there is only one lecture per week, you can only give a sketch. You should, for instance, handle the theme “Beauty arises when the sense-perceptible receives the form of the spirit” as I did that in my essay “Goethe as the Father of a New Aesthetic.” You could show that for the various arts, for architecture, painting, and so forth. In literature, I think you should discuss the most recent publications, namely, how Ibsen, Strindberg, and so forth reveal an unconscious movement toward a certain kind of spirituality, and then also, of course, the pathological, like for instance, Dostoyevsky.

Marie Steiner: Shouldn’t we also discuss Morgenstern, Steffen, and Steiner?

Dr. Steiner: You could extend Steffen’s characterization of lyrics.

In history, you could present an overview of the period from 1870 until 1914, stopping at that point. People would leave with rather long faces saying that you have only gotten to the World War and now they need to give some thought to the war itself. Go only to the assassination at Sarajevo.

In mathematics, you will have to orient yourselves by what was presented previously. I think it is important to treat the most important mathematical things. (Speaking to Dr. von Baravalle) You could present the things you have in your dissertation. It would also be very good if you developed mathematical concepts, such as those of normal functions or elliptic functions, in a visual way. Don’t just drone on about formal mathematics. Present how things are qualitatively. It would also be good to use that as a starting point to go into the entirety of relativity theory, how it is justifiable or not. I think people should have an idea of the following: You could present the question of relativity theory through the example of a cannon that is shot in Freiburg. It can be heard at some distance and you can compute the distance. You would then go on to compute how the time would change if you moved toward or away from the noise. The time it takes to hear the noise would lengthen if you moved from Karlsruhe to Frankfurt. If you then moved in the opposite direction the time would shorten until it was zero when you heard the cannon in Freiburg itself. You could then continue past Freiburg, so that you would have to hear the cannon before it was shot. That is the basic error of the theory of relativity. It can’t be so difficult to develop this mathematical concept of movement.

I think the problem with these courses is that they are actually unnecessary. With some differences, you have simply continued what other popular lectures offer, which is unnecessary; there is no real need for them.

What is important in geodesy is to get away from presenting a copy of the Earth. For example, if you begin, as people do, to try to avoid error through differential methods, you will need to explain geodetic methods to a certain extent. You will then have asymptotic methods. You could then discuss to what extent human beings depend upon approaching only certain things. You can show how extremely useful it is not to think in a determined way about some things, such as the character of a human being, but to think in a way similar to the way you measure with a diopter, where there is always some small difference. You can come closer to the truth in that way than you can when you state everything in specific words. We should characterize people only by looking at them from one side and then another. A person can be a choleric and a melancholic at the same time. This is the perspective you should bring to the fore. If you use geodesy as a basis for explaining the problems of the Copernican system, you can achieve a great deal.

You should form the lectures series by using such titles as: “What Can Aesthetics and Literature Add to Life?”; “What Can History Add to Life?”; “What Can Epistemology Add to Life?”; “What Can Mathematics Add to Life?”; and “What Can Geodesy Add to Life?” Under that, you could put “The Board of Directors of the Anthroposophical Society and the Faculty of the University Courses,” and above it, as a title, “Goetheanum and University Courses.”

These proposals are being made to you from Dornach.