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The Rudolf Steiner Archive

a project of Steiner Online Library, a public charity

Discussions with Teachers
GA 295

Discussion Thirteen

4 September 1919, Stuttgart

Translated by Helen Fox

Speech Exercises:

Clip plop pluck cluck
clinked clapper richly
knotted trappings
rosily tripled

RUDOLF STEINER: Memorize this before you practice it!

An attempt was made to illustrate the concept of a surface area for nine-year-old children; have the children cut out squares to measure from larger squares and copy them.

RUDOLF STEINER: It is certainly good to make it clear to children that, if the length of one side of a square is 3 feet, the area of the surface is 9 square feet, but this limits us to an area of thought where a whole is built from its parts, and this will not help children to gain a true concept of what a surface area really is. What I meant was: What is the right way to proceed, and at what age, in order to actually discover what a surface really is, and that it is obtained by multiplying length by breadth. How can you manage to awaken this concept of a surface in the child? This depends on when you begin teaching children about surface areas. It doesn’t make sense to teach them about surface areas until after you teach them some algebra. The answer, therefore, is to wait for lessons on surface areas until after we deal with algebra.

Now comes another question: How do you make the transition from ordinary problems with figures to problems with letters — that is, algebra? I will give you a suggestion about how to begin, and then you can work it out for yourselves. Before you move on to algebra you must have already worked on interest with the children; interest is principal multiplied by rate percent multiplied by time, divided by 100.

Interest = Principal × Rate × Time

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

To arrive at this formula, begin with ordinary numbers, and children understand principal, rate percent, time, and so on, relatively easily. So you will try to make this process clear and assure yourself that most of the children have understood it; from there you should move on to the formula, and always make sure that you work according to rule.

\(P\) = principal; \(R\) = rate; \(T\) = time; and \(I\) = interest. What I gave you is a formula I view merely as a basic formula, and with this formula I have taken the first step in moving to algebra. When the children have this formula they merely need to substitute figures for the letters, and then they will always get the right answer.

Now if you have the following formula derived from the first:

$$P = \frac{100I}{TR}$$

you can see that you can change about the 3 letters \(P\), \(R\), \(T\), however you wish, so that the following are also possibilities:

$$T = \frac{100I}{PR}$$ $$R = \frac{100I}{PT}$$

In this way we have taught the children how to work with\(\) interest, and now we can go on to algebra. You can simply say, “We have learned that a sum of \(25\) was equal to \(8\), then \(7\) and \(5\), and another \(5\): that is, \(25 = 8 + 7 + 5 + 5\).” The children will already have understood. Now after you have explained this, you can say, “Here, instead of 25 you could have a different number, and, instead of \(8\), \(7\), \(5\), \(5\) you could have other numbers; in fact, you could tell them that any number could be there. You could have \(s\), for example, as a total, and then you could have \(a + b + c + c\); but if c represents the first \(5\), then \(c\) must also represent the second 5. Just as I put P in place of principal, so in the same place I put the lettecr .

After having shown in a concrete example the transition from number to letter you can now explain the concept of multiplication, and out of this concrete \(g × g\) you can develop \(a × a\), or from \(a × 2\) you can evolve \(a × b\), and so on. This then would be the way to progress from the numbers in arithmetic to algebra with its letters, and from algebra to the calculation of surface areas; \(a × a = a^2\).

Now here is your task for tomorrow. Try to find a truly enlightened way to present to children of ten and eleven the concept of interest and everything associated with it, as well as inverse calculations of rate, time, and principal; then from there demonstrate how to deal with discount — how to teach a child the discounting of bills and the cost of packing and conveyancing, and then continue on to bills of exchange and how to figure them out. That belongs to the twelfth and thirteenth year, and if it is taught at this time it will be retained for the rest of life; otherwise it is always forgotten again. It is possible to deal with it in a simpler form, but it should be done at this age. Anyone who can do this properly has mastered the fundamental method of all computation. Compound interest is not involved at this time. You should therefore go over algebra in an organic way until multiplication, and then continue on to surface area calculation.

Now let’s proceed to the other questions from yesterday, because here it is important also that you should engender presence of mind in the children by assigning them problems.

Someone proposed setting up a little stall with fruit, vegetables, potatoes, and so on, so that the children would have to buy and sell, pay for their purchases, and actually figure out everything for themselves.

RUDOLF STEINER: This idea of buying and selling is very good for the second grade. Also, you should insist that those who have been assigned a problem should really work it out for themselves; you must not allow anyone else do it for them. Keep their interest awake and alive at every point!

Mental arithmetic was discussed.

RUDOLF STEINER related how Gauss 1Carl Friedrich Gauss (1777–1855), German mathematician and astronomer. He demonstrated that a circle can be divided into seventeen equal arcs through elementary geometry and developed a new technique for calculating the orbits of asteroids. He is also the originator of the Gaussian error curve in statistics and is considered the founder of the mathematical theory of electricity, from which derives the gaussmeter. as a boy of six arrived at the following solution to a problem he had to do: all of the numbers from 1 to 100 had to be added together. Gauss thought about the problem and concluded it would be a simpler and easier to get a quick answer by taking the same numbers twice, arranging them in the first row in the usual order from left to right — 1, 2, 3, 4, 5... up to 100, and beneath that a second row in the reverse order — 100, 99, 98, 97, 96 ... and so on to 1; thus 100 was under the 1, 99 under the 2, 98 under the 3, and so on. Then each of these 2 numbers would in every case add up to the whole. This sum would then have to be taken 100 times, which makes 10,100; then, because you have added each of the numbers from 1 to 100 twice (once forward and once backward) this sum would then be halved, and the answer is 5,050. In this way Gauss, to the great astonishment of his teacher, solved the problem in his head.

Along with some other things, two special problems were presented:

1. Calculation of time and distance for locomotives in which the circumferences of the wheels are of different sizes.

2. Exercises involving the filling and emptying of vessels with pipes of various sizes.

RUDOLF STEINER: You can use your imagination in making up arithmetical problems, and you can engender presence of mind through problems that deal with movement. With yesterday's example you can progress to practical life by saying, “I sent an express messenger with a letter. Because of certain circumstances the letter was no longer valid. So I sent another messenger. How quickly must the second messenger travel to arrive before the letter had caused any harm? The children should be able to figure this out at least approximately, which is good for them.

One of the teachers spoke of errors in calculation.

RUDOLF STEINER: These kinds of errors in calculations are usual. It is very common to figure the errors into the whole. There is one such mistake made these days that will at sometime or another have to be corrected. When Copernicus formulated his “Copernican system” he proposed three laws. If all three were to be used to describe the Earth’s course through space we would get a very different movement from what is now accepted by astronomers and taught in schools. This elliptic movement would only be possible if the third law were disregarded. When the astronomer uses the telescope, these things do not add up. Because of this, corrections are inserted into the calculations; through the use of Bessel's equations, corrections are introduced every year to account for what does not accord with reality. 2Friedrich Wilhelm Bessel (1784–1846), German astronomer; he calculated the orbit of Halley’s comet in 1804 and made the first “authenticated” calculation of a star’s distance from Earth. He also calculated the elliptical nature of Earth’s orbit. In Bessel’s corrections there is the third law of Copernicus.

Your method must never be simply to occupy the children with examples you figure out for them, but you should give them practical examples from real life; you must let everything lead into practical life. In this way you can always demonstrate how what you begin with is fructified by what follows and vice versa.

How would you resolve all these problems? (the flow of fluids slowly through small holes, quickly through large holes; rates of circular motion in machines with wheels of different sizes, and so on.)

The best way would be to proceed at this point to the explanation of what a clock is in its various forms — pendulumclocks, watches, and so on.

These are your tasks for tomorrow:

1. Some historical subject related to the history of civilization to be worked out on the lines of the example.

2. The treatment of some subject taken from nature — sunrise and sunset, seasons of the year and so on — whatever may suggest itself to you, something out of the great universe. The point is to show your method of teaching.

3. The principles of music for the first school year.

4. What form would you give to teaching the poetry of other languages? How would you give the children a feeling for what is poetic in other tongues?

5. How can you provide children with an idea of the ellipse, hyperbola, circle, and lemniscate; also the concept of geometrical locus? The children must be taught all this just before they leave our school at fourteen.