Discussions with Teachers
GA 295
4 September 1919, Stuttgart
Translated by Helen Fox
Discussion Thirteen
Speech Exercises:
Clip plop pluck cluck
clinked clapper richly
knotted trappings
rosily tripled
RUDOLF STEINER: Memorize this before you practice it!
From “We Found a Path” by Christian Morgenstern:
Those who do not know their destination,
Cannot find the way,
Will trot their whole life
All his life;
Arrives at the end,
Where he came from,
Has the meaning of the crowd
Only further fragmented.Those who know nothing of the goal,
For one must ask
Can still learn it today;
If only they burn
For the divine truth;
If he is not completely immersed in vanity
And not drunk to the top
With the wine of time.
About the quiet things,
And one must dare,
If one wants to attain light;
Those who cannot seek,
Like only a suitor ever can,
Remains under the spell of deception
Sevenfold veil.
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
$$I = \frac{P \cdot R \cdot T}{100}$$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:
$$P = \frac{100 \cdot I}{T \cdot R}$$you can see that you can change about the 3 letters \(P\), \(R\), \(T\), however you wish, so that the following are also possibilities:
$$T = \frac{100 \cdot I}{P \cdot R}$$ $$R = \frac{100 \cdot I}{P \cdot T}$$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 \(9 \cdot 9\) you can develop \(a \cdot a\), or from \(a \cdot 2\) you can evolve \(a \cdot b\), 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 Gauss1Carl 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.
Dreizehnte Seminarbesprechung
Sprechübungen:
Klipp plapp plick glick
Klingt Klapperrichtig
Knatternd trappend
Rossegetrippel
Rudolf Steiner: Auswendig üben!
Aus «Wir fanden einen Pfad» von Christian Morgenstern:
Wer vom Ziel nicht weiß,
Kann den Weg nicht haben,
Wird im selben Kreis
All sein Leben traben;
Kommt am Ende hin,
Wo er hergerückt,
Hat der Menge Sinn
Nur noch mehr zerstückt.Wer vom Ziel nichts kennt,
Kann’s doch heut erfahren;
Wenn es ihn nur brennt
Nach dem Göttlich-Wahren;
Wenn in Eitelkeit
Er nicht ganz versunken
Und vom Wein der Zeit
Nicht bis oben trunken.Denn zu fragen ist
Nach den stillen Dingen,
Und zu wagen ist,
Will man Licht erringen;
Wer nicht suchen kann,
Wie nur je ein Freier,
Bleibt im Trugesbann
Siebenfacher Schleier.
T. versucht, den Begriff der Fläche für neunjährige Kinder anschaulich zu gestalten. (Quadrate zum Messen von anderen, größeren quadratischen Flächen ausschneiden lassen, Schablonieren.)
Rudolf Steiner: Es ist gut begreiflich zu machen, daß dann, wenn man 3 Meter als Länge einer Quadratseite hat, die Fläche 9 Quadratmeter ist, aber damit bleiben wir immer in der Sphäre, welche aus solchen anschaulichen Stücken etwas zusammensetzt, und es wird trotzdem sehr schwierig sein, da eine richtige Vorstellung der Fläche hervorzurufen.
Gemeint habe ich: Wie geht man richtig vor, und in welches Lebensalter kann solch ein Vorgehen fallen, um tatsächlich herauszubekommen, daß die Fläche Fläche ist und Fläche wird, wenn man die Länge mit der Breite multipliziert? Wie kommt man dazu, diesen Begriff der Fläche beim Kinde hervorzurufen? — Das hängt davon ab, wo man hineinfallen läßt diesen Unterricht über die Flächen. Da muß gesagt werden: Es ist nicht gut, den Unterricht über die Flächen dorthin fallen zu lassen, wo man die Buchstabenrechnung noch nicht durchgenommen hat. Wir können den Unterricht über die Fläche rationell erst vornehmen, wenn wir schon vorgenommen haben die Buchstabenrechnung. So ist die Antwort: Wir warten mit dem Flächenunterricht, bis wir die Buchstabenrechnung vorgenommen haben.
Und nun weiter die Frage: Wie bringen Sie es dahin, daß Sie mit den Kindern übergehen von der gewöhnlichen Zifferrechnung zur Buchstabenrechnung? Ich will Sie darauf leiten, und dann führen Sie es weiter aus. Sie müssen, ehe Sie zur Buchstabenrechnung übergehen, doch schon mit den Kindern durchgemacht haben die Zinsrechnungen: Zinsen sind gleich Kapital mal Prozent, mal Zeit, dividiert durch 100
$$Zinsen = \frac{Kapital \cdot Prozent \cdot Zeit}{100}$$
Kürzt man auf die Anfangsbuchstaben ab, so kann man schreiben:
$$Z = \frac{K \cdot P \cdot T}{100}$$
\(T\) = tempus, lateinisch = Zeit, ist die gebräuchlichste Abkürzung für Zeit.
Sie gehen, indem Sie zu dieser Formel kommen, von gewöhnlichen Zahlen aus, und das Kind begreift verhältnismäßig leicht, was das Kapital ist, welches die Prozente sind, welches die Zeit ist und so weiter.
Also diesen Vorgang werden Sie dem Kinde klarzumachen versuchen und sich überzeugen, daß die Kinder in ihrer Mehrheit die Sache begriffen haben. Und von da würden Sie zur obigen Form übergehen und immer darauf sehen, daß Regel hineinkommt.
\(K\) ist = Kapital; \(P\) ist = Prozent; \(T\) ist = Zeit (Tempus); \(Z\) ist = Zinsen. Dann ist das oben Angegebene eine Formel, die ich mir bloß als Grundformel merke. Dadurch habe ich schon den ersten Schritt gemacht vom Übergang zur Buchstabenrechnung. Wenn das Kind nun diese Formel hat, so braucht es nur die Zahl einzusetzen in diese Formel, und es muß immer das Richtige herauskommen. Haben Sie die dann daraus abgeleitete Formel:
$$K = \frac{Z \cdot 100}{T \cdot P}$$
so können Sie sich mnemotechnisch merken, daß Sie die drei Buchstaben \(K\), \(P\), \(T\) beliebig miteinander vertauschen können, so daß sich noch folgende Möglichkeiten ergeben:
$$T = \frac{Z \cdot 100}{T \cdot P} \qquad P = \frac{Z \cdot 100}{K \cdot T}$$
Auf diese Weise haben wir dem Kinde Kapitalrechnung beigebracht, und jetzt können wir übergehen zum Buchstabenrechnen. Sie können ruhig sagen: «Wir haben gelernt, eine Summe 25 war gleich 8 mehr 7 mehr 5 mehr 5, \(25 = 8 + 7 + 5 + 5\).» Nicht wahr, das hat das Kind einmal begriffen. Jetzt, nachdem Sie ihm das auseinandergesetzt haben, können Sie ihm sagen: «Da (statt 25) kann aber auch eine andere Summe stehen, und da (statt 8, 7, 5, 5) können andere Zahlen stehen, so daß wir auch sagen können, da stünde «irgendeine» Zahl. Also stünde da zum Beispiel: \(S\), eine Summe. Und da stünde: \(a + b +c + c\). Aber, wenn da \(c\) stünde anstelle der ersten 5, so muß es auch anstelle der zweiten 5 stehen. Gerade so, wie ich anstelle von beliebigem Kapital K einsetze, setze ich an dieser Stelle den Buchstaben c ein.»
Nachdem Sie in einem konkreten Fall den Übergang von der Zahl zum Buchstaben gezeigt haben, dann können Sie nun auch den Begriff des Multiplizierens entwickeln, und aus diesem konkreten \(9 \cdot 9\) können Sie entwickeln \(a \cdot a\). Oder Sie können aus \(a \cdot 2\) entwickeln \(a \cdot b\) und so weiter. Also das würde der Weg sein, aus diesen Zahlenrechnungen überzugehen zur Buchstabenrechnung. Und aus dieser zur Flächenberechnung, \(a \cdot a = a^2.\)
Aufgabe für morgen: Zinsenrechnung, recht geistreich einleuchtend entwickeln für Kinder im elften, zwölften Jahr mit dem, was dazugehört, mit der Umkehrung: Prozent-, Zeit-, Kapitalrechnung. — Dann von da aus entwickeln, wie man beleuchtet Diskontrechnung. Dann wie man dem Kinde beibringt Rabatt- und Emballagerechnung, und wie man ihm beibringt den Begriff und die Berechnung eines Wechsels. Das gehört hinein in das zwölfte und dreizehnte Jahr, so daß es für das ganze Leben bleibt; sonst wird es später immer wieder vergessen. Man kann es ja in einfacher Weise nehmen, aber da hinein gehört es. Wenn jemand dieses ordentlich kann, dann kann er die Methodik des ganzen Rechnens. Zinseszinsrechnung gehört nicht in diese Jahre hinein.
Also organisch übergehen in die Buchstabenrechnung bis zum Multiplizieren und von da in die Flächenberechnung.
Nun würde ich bitten, daß wir auf die anderen Fragen von gestern eingehen. Denn auch da ist wichtig, daß Sie die Kinder durch Rechnungstellen zur Geistesgegenwart anregen.
G. schlägt die Errichtung eines kleinen Verkaufsstandes vor mit Früchten, Gemüse, Kartoffeln und so weiter, wobei die Kinder selbständig einkaufen, verkaufen, bezahlen, Geld herausgeben, überhaupt selbständig alles berechnen müssen.
Rudolf Steiner: Dieses Kaufmannsprinzip ist ganz gut für die zweite Klasse. Und es ist gut, darauf zu bestehen, daß derjenige, dem man eine Rechnung gegeben hat, sie auch wirklich selbst löst, und daß man keinen anderen für ihn eintreten läßt. Immer das Interesse aller wachhalten!
Es wird über das Kopfrechnen gesprochen; über das Rechnen, ohne zu schreiben.
Rudolf Steiner erzählt, daß Gauß als sechsjähriger Knabe einmal zu folgender Lösung gekommen sei: Gestellt war die Aufgabe, die Zahlen von 1 bis 100 zu addieren. Gauß überlegte sich, daß es vorteilhafter und einfacher sei, um schnell zu dem Resultat zu kommen, die gleichen Zahlen nochmals zu nehmen, sie aber so zu der ersten Reihe von 1 bis 100 anzuordnen, daß man sich die erste Reihe wie gewöhnlich von links nach recht geschrieben 1, 2, 3, 4,5... 100 vorstellen könne, darunter aber dann in umgekehrter Anordnung die zweite Reihe 100, 99,98, 97,96... 1, so daß zu stehen kommen unter die 1 die 100, unter die 2 die 99, unter die 3 die 98. Dann ergäben jedesmal die beiden untereinanderstehenden Zahlen addiert die Summe 101. Die Summe müsse hundertmal genommen werden, ergibt 10100, und müsse dann nur noch — weil man darin ja zweimal die Zahlen von 1 bis 100 addiert hat, einmal vorwärts, einmal rückwärts - halbiert werden, ergibt 5050. So löste Gauß zum nicht geringen Erstaunen seines Lehrers damals im Kopf diese gestellte Aufgabe.
T. führt unter anderen zwei Arten von Aufgaben an: 1. Zeit- und Streckenberechnung, wenn Lokomotiven mit verschieden großem Radumfang gegeben sind; 2. Aufgaben mit Voll- und Auslaufenlassen von Gefäßen mit verschieden weiter Ausflußröhre.
Rudolf Steiner: Beim Ausdenken von Rechenaufgaben kann man Phantasie verwenden. Man kann Geistesgegenwart erzeugen durch Bewegungsaufgaben. Sie können mit dem gestrigen Beispiel zur Praxis übergehen, wenn Sie sagen: Ich habe einen Eilboten fortgeschickt mit einem Botenbrief. Der Brief ist gegenstandslos geworden. Ich muß einen anderen Boten fortschicken. Wie schnell muß der weiterkommen, um noch vorher anzukommen, ehe der Brief sein Unheil angerichtet hat? Wenigstens annähernd soll das Kind das berechnen können, das ist ganz gut.
Ein Teilnehmer weist auf Fehlerrechnungen hin.
Rudolf Steiner: Solche Fehlerrechnungen sind überhaupt sehr üblich. Es ist üblich, daß man gleich die Fehler miteinrechnet. Nun, in einem Punkte wird heute eine solche Fehlerrechnung gemacht und wird einmal korrigiert werden müssen. Als Kopernikus sein «Kopernikanisches System» aufgestellt hat, stellte er drei Lehrsätze auf. Würde man alle drei benützen, um den Weg der Erde durch den Weltenraum zu skizzieren, so würde man eine ganz andere Bewegung bekommen, als sie jetzt von unseren Astronomen angenommen und auf unseren Schulen gelehrt wird. Diese elliptische Bewegung wird nur dadurch möglich, daß man den dritten Lehrsatz unberücksichtigt läßt. Wenn der Astronom sein Fernrohr hinausrichtet, so stimmen die Dinge nicht. Zu diesem Zweck setzt man auch Fehler in Rechnung; durch die Besselschen Gleichungen werden jedes Jahr Fehler eingesetzt für das, was in der Wirklichkeit nicht stimmt. Die Besselschen Fehlergleichungen, in denen steckt der dritte Satz des Kopernikus.
Methodisch muß man so verfahren, daß man das Kind nicht bloß beschäftigt mit ausgedachten Beispielen, sondern daß man zu praktischen Beispielen aus dem Leben kommt. Man muß alles ins Praktische auslaufen lassen. Dabei kann man immer durch Folgendes das Vorhergehende befruchten lassen und umgekehrt.
In was würden Sie alle diese Bewegungsberechnungen, das Auslaufen von Flüssigkeiten durch kleine Löcher langsam, durch große schnell, die Kreisbewegungsaufgaben an Maschinen mit verschieden großen Rädern - in was würden Sie das auslaufen lassen?
Sie würden am besten dazu übergehen, den Kindern die Uhr zu erklären in ihren verschiedenen Gestalten, als Pendeluhr, Taschenuhr und so weiter.
Aufgaben für morgen:
Erstens: ein geschichtliches Thema zu behandeln nach dem früher gegebenen Musterbeispiel, kulturgeschichtlich. Zweitens: Behandlung von irgend etwas aus der allgemeinen Natur, Auf- und Untergang der Sonne, Jahreszeiten oder dergleichen, das Ihnen naheliegt, etwas aus dem Weltgebäude. Es kommt darauf an, die Unterrichtsmethode geltend zu machen.
Drittens: Über die Prinzipien des Musikalischen im allerersten Schuljahr.
Viertens: Wie ist Poetisches zu gestalten im Englischen und Französischen? Wie ist den Kindern beizubringen das Empfinden des Poetischen in der englischen, der französischen Sprache?
Fünftens: Wie ist es möglich, dem Kinde beizubringen den Begriff der Ellipse, der Hyperbel, des Kreises, der Lemniskate und den Begriff des geometrischen Ortes? — Das alles ist den Kindern beizubringen, unmittelbar bevor sie die Schule verlassen.
Thirteenth Seminar Discussion
Speech exercises:
Klipp plapp plick glick
Sounds like clattering
Clattering and stomping
Horse trotting
Rudolf Steiner: Practice by heart!
From “We Found a Path” by Christian Morgenstern:
Those who do not know their destination,
Cannot find the way,
Will trot their whole life
All his life;
Arrives at the end,
Where he came from,
Has the meaning of the crowd
Only further fragmented.Those who know nothing of the goal,
For one must ask
Can still learn it today;
If only they burn
For the divine truth;
If he is not completely immersed in vanity
And not drunk to the top
With the wine of time.
About the quiet things,
And one must dare,
If one wants to attain light;
Those who cannot seek,
Like only a suitor ever can,
Remains under the spell of deception
Sevenfold veil.
T. tries to illustrate the concept of area for nine-year-old children. (Have them cut out squares to measure other, larger square areas, use stencils.)
Rudolf Steiner: It is easy to understand that if you have a square side length of 3 meters, the area is 9 square meters, but this keeps us in the realm of putting together such concrete pieces, and it will still be very difficult to form a correct mental image of area.
What I mean is: What is the right approach, and at what age can such an approach be used to actually show that area is area and becomes area when you multiply length by width? How can we bring about this concept of area in children? — That depends on where you fit this teaching about areas into the curriculum. It must be said that it is not good to introduce lessons on area before the children have learned to count using letters. We can only teach lessons on area rationally once we have already taught counting using letters. So the answer is: we wait with lessons on area until we have taught counting using letters.
And now on to the next question: How do you manage to transition the children from ordinary arithmetic to letter arithmetic? I will guide you, and then you can continue. Before you move on to letter arithmetic, you must have already gone through interest calculations with the children: Interest is equal to capital times percentage times time, divided by 100.
$$Interest = \frac{Capital \cdot Percentage \cdot Time}{100}$$
If you abbreviate to the initial letters, you can write:
$$Z = \frac{C \cdot P \cdot T}{100}$$
\(T\) = tempus, Latin = time, is the most common abbreviation for time.
By arriving at this formula, you start with ordinary numbers, and the child understands relatively easily what capital is, what percentages are, what time is, and so on.
So you will try to explain this process to the child and make sure that the majority of the children have understood it. And from there, you would move on to the above formula and always make sure that the rule is included.
\(K\) is = capital; \(P\) is = percent; \(T\) is = time (tempus); \(Z\) is = interest. Then the above is a formula that I simply remember as a basic formula. This is the first step I have taken in the transition to letter calculation. Once the child has this formula, they only need to insert the number into this formula, and the correct answer will always come out. Do you have the formula derived from this:
$$K = \frac{Z \cdot 100}{T \cdot P}$$
you can use a mnemonic device to remember that you can swap the three letters \(K\), \(P\), \(T\) with each other as you wish, resulting in the following possibilities:
$$T = \frac{Z \cdot 100}{T \cdot P} \qquad P = \frac{Z \cdot 100}{K \cdot T}$$
In this way, we have taught the child capital calculation, and now we can move on to letter calculation. You can safely say: “We have learned that a sum of 25 was equal to 8 plus 7 plus 5 plus 5, \(25 = 8 + 7 + 5 + 5\).” Once the child has understood this, you can tell them: “However, instead of 25, there could also be a different sum, and instead of 8, 7, 5, 5, there could be other numbers.” Now that you have explained this to them, you can say: "But instead of 25, there could also be another sum, and instead of 8, 7, 5, 5, there could be other numbers, so that we could also say that there is ‘any’ number. So, for example, there would be: \(S\), a sum. And there would be: \(a + b + c + c\). But if there were \(c\) instead of the first 5, it must also be there instead of the second 5. Just as I use any capital letter \(K\), I use the letter \(c\) in this place."
Once you have demonstrated the transition from numbers to letters in a specific case, you can now develop the concept of multiplication, and from this specific \(9 \cdot 9\) you can develop \(a \cdot a\). Or you can develop \(a \cdot b\) from \(a \cdot 2\), and so on. So that would be the way to move from these numerical calculations to letter calculations. And from this to area calculation, \(a \cdot a = a^2.\)
Task for tomorrow: Develop interest calculation in a way that is sufficiently ingenious and clear for children in the eleventh and twelfth years, with everything that goes with it, including the reverse: percentage, time, and capital calculation. — Then, from there, develop how to explain discount calculations. Then how to teach children discount and packaging calculations, and how to teach them the concept and calculation of a bill of exchange. This belongs in the twelfth and thirteenth years, so that it remains for life; otherwise, it will be forgotten again and again later on. It can be taken in a simple way, but it belongs there. If someone can do this properly, then they can do the methodology of all arithmetic. Compound interest calculation does not belong in these years.
So move organically into letter calculation up to multiplication and from there into area calculation.
Now I would ask that we address the other questions from yesterday. Because there, too, it is important that you stimulate the children's presence of mind through arithmetic.
G. suggests setting up a small sales stand with fruit, vegetables, potatoes, and so on, where the children have to shop, sell, pay, give change, and calculate everything independently.
Rudolf Steiner: This merchant principle is quite good for the second grade. And it is good to insist that the person who has been given a bill actually solves it himself, and that no one else is allowed to do it for him. Always keep everyone's interest alive!
There is talk of mental arithmetic; of calculating without writing.
Rudolf Steiner recounts that Gauss, as a six-year-old boy, once came up with the following solution: The task was to add the numbers from 1 to 100. Gauss thought that it would be more advantageous and easier to get the result quickly by taking the same numbers again, but arranging them in the first row from 1 to 100 in such a way that the first row could be written as usual from left to right as 1, 2, 3, 4, 5... 100, but then, in reverse order, the second series 100, 99, 98, 97, 96... 1, so that 100 comes under 1, 99 under 2, and 98 under 3. Then each time, the two numbers standing below each other would add up to 101. The sum must be multiplied by 100, resulting in 10100, and then only needs to be halved—because the numbers from 1 to 100 have been added twice, once forwards and once backwards—resulting in 5050. This is how Gauss solved this problem in his head, much to the astonishment of his teacher at the time.
T. cites two types of problems, among others: 1. Calculating time and distance when locomotives with different wheel circumferences are given; 2. Problems involving filling and emptying vessels with spouts of varying widths.
Rudolf Steiner: You can use your imagination when thinking up arithmetic problems. You can generate presence of mind through movement tasks. You can put yesterday's example into practice by saying: I have sent a courier with a letter. The letter has become irrelevant. I have to send another courier. How fast does he have to go to arrive before the letter has caused any harm? The child should be able to calculate this at least approximately, which is quite good.
A participant points out error calculations.
Rudolf Steiner: Such error calculations are very common. It is common to include the errors in the calculation. Well, in one respect, such an error calculation is made today and will have to be corrected at some point. When Copernicus established his “Copernican system,” he formulated three theorems. If all three were used to sketch the path of the Earth through space, the result would be a completely different movement than that currently assumed by our astronomers and taught in our schools. This elliptical motion is only possible if the third theorem is disregarded. When the astronomer points his telescope, the facts do not add up. For this purpose, errors are also taken into account; Bessel's equations are used to insert errors every year for what is not correct in reality. Bessel's error equations contain Copernicus' third theorem.
Methodologically, one must proceed in such a way that the child is not merely occupied with imaginary examples, but that one comes up with practical examples from life. Everything must be allowed to flow into practical application. In doing so, one can always let the preceding enrich the following and vice versa.
How would you apply all these motion calculations, the slow flow of liquids through small holes and the fast flow through large holes, the circular motion tasks on machines with wheels of different sizes?
The best way would be to explain to the children the different types of clocks, such as pendulum clocks, pocket watches, and so on.
Tasks for tomorrow:
First: to deal with a historical topic according to the model example given earlier, cultural history. Second: to deal with something from general nature, the rising and setting of the sun, the seasons or the like, something that is close to you, something from the structure of the world. It is important to apply the teaching method.
Third: the principles of music in the very first year of school.
Fourth: how should poetry be taught in English and French? How can children be taught to appreciate poetry in English and French?
Fifth: How is it possible to teach children the concepts of the ellipse, the hyperbola, the circle, the lemniscate, and the concept of geometric location? — All of this must be taught to children immediately before they leave school.