LECTURE 5
16th August,
1924
It will be
essential for you to have some understanding of the real essence of
every subject that you teach, so that you do not make use of things
in your teaching that are remote from life itself. Everything which
is intimately connected with life can be understood. One could even
say that whatever one really understands has this intimate connection
with the life of man. This is not the case with abstractions.
What we
find today is that the ideas a teacher has are largely abstractions,
so that in many respects he is himself remote from life. This brings
very great difficulties into education and teaching. Just consider
the following: Imagine that you want to think over how you first came
to count things and what really happens when you count. You will
probably find that the thread of your recollections breaks somewhere,
and that you did once learn to count, but actually you do not really
know what you do when you count.
Now all
kinds of theories are thought out for the teaching of number and
counting, and it is customary to act upon such theories. But even
when external results can be obtained, one does not touch the whole
being of the child with this kind of counting or with similar things
that have no connection with real life. The modern age has proved
that it lives in abstractions, by inventing such things as the abacus
or beadframe for teaching. In a business office people can use
calculating machines as much as they like — that does not
concern us at the moment, but in teaching, this calculating machine,
which is exclusively concerned with the activities of the head,
prevents one from the very start from dealing with number in a way
which is in accordance with the child nature.
Counting
however should be derived from life itself, and here it is supremely
important to know from the beginning that we should not ever expect
the child to understand every single thing we teach him. He must take
a great deal on authority, but he must take it in a natural,
practical way.
Perhaps
you may find that what I am now going to say will be rather difficult
for the child. But that does not matter. It is of great significance
that there should be moments in a man's life when in his thirtieth or
fortieth year he can say to himself: Now I understand what in my
eighth or ninth year, or even earlier, I took on authority. This
awakens new life in a man. But if you look at all the object lessons
that are introduced into the teaching of today, you may well be in
despair over the way things are made trivial, in order, as one says,
to bring them nearer to the child's understanding.
Now
imagine that you have quite a small child in front of you, one of the
youngest, who is still quite clumsy in his movements, and you say to
him: “You are standing there before me. Here I take a piece of
wood and a knife, and I cut the wood into pieces. Can I do that to
you?” The child will see for himself that I cannot do it to
him. And now I can say to him: “Look, if I can cut the piece of
wood in two, the wood is not like you, and you are not like the wood,
for I cannot cut you in two like that. So there is a difference
between you and the wood. The difference lies in the fact that you
are a unit, a ‘one,’ and the wood is not a
‘one.’ You are a unit and I cannot cut you in two, and
therefore I call you ‘one,’ a unit.”
You can
now gradually proceed to show the child a sign for this
“one.” You make a stroke: I, so that you show him it is a
unit and you make this stroke for it.
Now you
can leave this comparison between the wood and the child and you can
say: “Look, here is your right hand but you have another hand
too, your left hand. If you only had this one hand it could certainly
move about everywhere as you do, but if your hand were only to follow
the movement of your body you could never touch yourself in the way
in which your two hands can touch each other. For when this hand
moves and the other hand moves too at the same time, then they can
take hold of each other, they can come together. That is different
from when you simply move alone. In that you walk alone you are a
unit. But the one hand can touch the other hand. This is no longer a
unit, this is a duality, a ‘two.’ See, you are one, but
you have two hands.” This you then show like this: II.
In this
way you can work out a conception of the “one” and the
“two” from the child's own form.
Now you
call out another child and say: “When you two walk towards each
other you can also meet and touch each other; there are two of you,
but a third can join you. This is impossible with your hands.”
Thus you can proceed to the three: III.
In this
manner you can derive number out of what man is himself. You can lead
over to number from the human being, for man is not an abstraction
but a living being.
Then you
can pass on and say: “Look, you can find the number two
somewhere else in yourself.” The child will think finally of
his two legs and feet. Now you say: “You have seen your
neighbour's dog, haven't you? Does he only go on two feet
also?” Then he will come to realise that the four strokes IIII
are a picture of the neighbour's dog propped up on four legs, and
thus he will gradually learn to build up number out of life.
The
teacher must have his eyes open wherever he goes and look at
everything with understanding. Now we naturally begin to write
numbers with the Roman figures, because these of course will be
immediately understood by the child, and when you have got to the
four you will easily be able, with the hand, to pass over to five
— V. You will soon see that if you keep back your thumb you can
use this four as the dog does!: I I I I. Now you add the thumb and
make five — V.
I was
once with a teacher who had got up to this point (in explaining the
Roman figures) and could not see why it occurred to the Romans not to
make five strokes next to one another but to make this sign V for the
five. He got on quite well up to I I I I. Then I said: “Now let
us do it like this: Let us spread out our fingers and our thumb so
that they go in two groups, and there we have it, V. Here we have the
whole hand in the Roman five and this is how it actually originated.
The whole hand is there within it.”
In a
short lecture course of this kind it is only possible to explain the
general principle, but in this way we can derive the idea of number
from real life, and only when number has thus been worked out
straight from life should you try to introduce counting by letting
the numbers follow each other. But the children should take an active
part in it. Before you come to the point of saying: Now tell me the
numbers in order, 1, 2, 3, 4, 5, 6, 7, 8, 9 and so on, you should
start with a rhythm; let us say we are going from 1 to 2, then it
will be: 1, 2; 1, 2; 1, 2; let the child stamp on 2 and then on to 3
also in rhythm: 1, 2, 3; 1, 2, 3. In this way we bring rhythm into
the series of numbers, and thereby too we foster the child's faculty
of comprehending the thing as a whole. This is the natural way of
teaching the children numbers, out of the reality of what numbers
are. For people generally think that numbers were thought out by
adding one to the other. This is quite untrue, for the head does not
do the counting at all. In ordinary life people have no idea what a
peculiar organ this human head really is, and how useless for our
earthly life. It is there for beauty's sake, it is true, because our
faces please each other. It has many other virtues too, but as far as
spiritual activities are concerned it is really not nearly so much in
evidence, for the spiritual qualities of the head always lead back to
man's former earthlife. The head is a metamorphosis of the former
life on earth, and the fact of having a head only begins to have a
real meaning for man when he knows something of his former earth
lives. All other activities come from somewhere else, not from the
head at all. The truth is that we count subconsciously on our
fingers. In reality we count from 11 o on our ten fingers, then
eleven (adding the toes), twelve, thirteen, fourteen (counting on the
toes). You cannot see what you are doing, but you go on up to twenty.
And what you do in this manner with your fingers and toes only throws
its reflection into the head. The head only looks on at all that
occurs. The head in man is really only an apparatus for reflecting
what the body does. The body thinks, the body counts. The head is
only a spectator.
We can
find a remarkable analogy for this human head. If you have a car and
are sitting comfortably inside it, you are doing nothing yourself; it
is the chauffeur in front who has to exert himself. You sit inside
and are driven through the world. So it is with the head; it does not
toil and moil, it simply sits on the top of your body and lets itself
be carried quietly through the world as a spectator. All that is done
in spiritual life isdone from the body. Mathematics are done by the
body, thinking is also done by the body, and feeling too is done with
the body. The beadframe has arisen from the mistaken idea that man
reckons with his head. Sums are then taught to the child with the
beadframe, that is to say, the child's head is made to work and then
the head passes on the work to the body, for it is the body which
must do the reckoning. This fact, that the body must do the
reckoning, is not taken into account, but it is important. So it is
right to let the child count with his fingers and also with his toes,
for indeed it is good to call forth the greatest possible skill in
the children. In fact there is nothing better in life than making the
human being skilful in every way. This cannot be done through sport,
for sport does not really make people skilled. What does make a man
skilled is to give him a pencil, for instance, and let him hold it
between his big toe and the next toe and learn to write with his
foot, to write figures with his foot. This can be of real
significance, for in truth man is permeated with soul and spirit in
his whole body. The head is the traveller that sits back restfully
inside and does nothing, whilst the body, every part of it, is the
chauffeur who has to do everything.
Thus from
the most varied sides you must try to build up what the child has to
learn as counting. And when you have worked in this way for a time it
is important to pass on and not merely take counting by adding one
thing to another; indeed this is the least important aspect of
counting and you should now teach the child as follows: “This
is something which is ONE. Now you divide it like this, and you have
something which is TWO. It is not two ONEs put together but the two
come out of the ONE.” And so on with three and four. Thus you
can awaken the thought that the ONE is really the comprehensive thing
that contains within itself the TWO, the THREE, the FOUR, and if you
learn to count in the way indicated in the diagram, 1, 2, 3, 4 and so
on, then the child will have concepts that are living. He thereby
comes to experience something of what it is to be inwardly permeated
with the element of number.
In bygone
ages our present conceptions of counting by placing one bean beside
another or one bead beside another in the frame were quite unknown;
in those days it was said that the unit was the largest, every two is
only the half of it, and so on. So you come to understand the nature
of counting
by actually
looking at external objects. You should develop the child's thinking
by means of external things which he can see, and keep him as far
away as possible from abstract ideas.
The
children can then gradually learn the numbers up to a certain point,
first, let us say, up to twenty, then up to a hundred and so on. If
you proceed on these lines you will be teaching the child to count in
a living way. I should like to emphasise that this method of
counting, real counting, should be presented to the child before he
learns to do sums. The child ought to be familiar with this kind of
counting before you go on to Arithmetic.
Arithmetic too must be approached out of life. The living thing is
always a whole and must be presented as a whole first of all. You are
doing wrong to a child if you always make him put together a whole
out of its parts, and do not teach him to look first at the whole and
then divide this whole into its parts; get him first to look at the
whole and then divide it and split it up; this is the right path to a
living conception.
Many
things that the materialistic epoch has done with regard to the
general culture of mankind pass unnoticed. Nowadays no one is
scandalised but regards it rather as a matter of course to let
children play with boxes of bricks, and build things out of the
single blocks. This of itself leads them away from what is living.
The child out of his very nature has no impulse to put together a
whole out of parts. He has many other needs and impulses which are,
admittedly, much less convenient. If you give him a watch for
instance, he immediately has the desire to pull it to pieces, to
break up the whole into its parts, which is actually far more in
accordance with the nature of man — to see how the whole arises
out of its component parts.
This is
what must now be taken into account in our Arithmetic teaching. It
has an influence on the whole of culture, as you will see from the
following example.
In the
conception of human thought up to the thirteenth and fourteenth
centuries very little emphasis was laid upon putting together a whole
out of its parts; this arose later. The masterbuilder built much
more from the idea of the whole (which he then split up into its
parts) rather than starting with the single parts and making a
building out of these. The latter procedure was really only
introduced into civilisation later on. This conception then led to
people thinking of every single thing as being put together out of
the very smallest parts. Out of this arose the atomic theory in
Physics, which really only comes from education. For atoms are really
tiny little caricatures of demons, and our learned scholars would not
speak about them as they do unless people had grown accustomed, in
education, to putting everything together out of its parts. Thus it
is that atomism has arisen.
We
criticise atomism today, but criticism is really more or less
superfluous because men cannot get free from what they have been used
to thinking of wrongly for the last four or five centuries; they have
become accustomed to go from the parts to the whole instead of
letting their thoughts pass from the whole to the parts.
This is
something we should particularly bear in mind in the teaching of
Arithmetic. If you are walking towards a distant wood you first see
the wood as a whole, and only when you come near it do you perceive
that it is made up of single trees. This is just how you must proceed
in Arithmetic. You never have in your purse, let us say, 1, 2, 3, 4,
5 coins, but you have a heap of coins. You have all five together,
which is a whole. This is what you have first of all. And when you
cook pea soup you do not have 1, 2, 3, 4, 5 or up to 3o or 4o peas,
but you have one heap of peas, or with a basket of apples, for
instance, there are not 1, 2, 3, 4, 5, 6, 7 apples, but one heap of
apples in your basket. You have a whole. What does it matter to us to
begin with, how many we have? We simply have a heap of apples which
we are now bringing home (see diagram). There are, let us say, three
children. We
will not now
divide them so that each gets the same, for perhaps one child is
small, another big. We put our hand into the basket and give the
bigger child a bigger handful, the. smaller child a smaller handful;
we divide our heap of apples into three parts.
Dividing
or sharing out is in any case such a queer business! There was once a
mother who had a large piece of bread. She said to her little boy,
Henry: “Divide the bread, but you must divide it in a Christian
way.” Then Henry said: “What does that mean, divide it in
a Christian way?” “Well,” said his mother,
“You must cut the bread into two pieces, one larger and one
smaller; then you must give the larger piece to your sister Anna and
keep the smaller one for yourself.” Whereupon Henry said,
“Oh well, in that case let Anna divide it in a Christian
way!”
Other
conceptions must come to your aid here. We will do it like this, that
we give this to one child, let us say {see lines in the drawing), and
this heap to the second child, and this to the third. They have
already learnt to count, and so that we get a clear idea of the whole
thing we will first count the whole heap. There are eighteen apples.
Now I have to count up what they each have. How many does the first
child get? Five. How many does the second child get? Four. And the
third? Nine. Thus I have started from the whole, from the heap of
apples, and have divided it up into three parts.
Arithmetic is often taught by saying: “You have five, and here
is five again and eight; count them together and you have
eighteen.” Here you are going from the single thing to the
whole, but this will give the child dead concepts. He will not gain
living concepts by this method. Proceed from the whole, from the
eighteen, and divide it up into the addenda; that is how to teach
addition.
Thus in
your teaching you must not start with the single addenda, but start
with the sum, which is the whole, and divide it up into the single
addenda. Then you can go on to show that it can be divided up
differently, with different addenda, but the whole always remains the
same. By taking addition in this way, not as is very often done by
having first the addenda and then the sum, but by taking the sum
first and then the addenda, you will arrive at conceptions that are
living and mobile. You will also come to see that when it is only a
question of a pure number the whole remains the same, but the single
addenda can change. This peculiarity of number, that you can think of
the addenda grouped in different ways, is very clearly brought out by
this method.
From this
you can then proceed to show the children that when you have
something that is not itself a pure number but that contains number
within it, as the human being for example, then you cannot divide it
up in all these different ways. Take the human trunk for instance and
what is attached to it, head, two arms and hands, two feet; you
cannot now divide up the whole as you please; you cannot say: now I
will cut out one foot like this, or the hand like this, and so on,
for it has already been membered by nature in a definite way. When
this is not the case, and it is simply a question of pure counting,
then I can divide things up in different ways.
Such
methods as these will make it possible for you to bring life and a
kind of living mobility into your work. All pedantry will disappear
and you will see that something comes into your teaching that the
child badly needs: humour comes into the teaching, not in a childish
but in a healthy sense. And humour must find its place in teaching.
[At this. point Dr. Steiner turned to the translator
and said: “Please be sure you translate the word
‘humour’ properly, for it is always misunderstood in
connection with teaching!”]
This then
must be your method: always proceed from the whole. Supposing you had
such an example as the following, taken from real life. A mother sent
Mary to fetch some apples. Mary got twentyfive apples. The
applewoman wrote it down on a piece of paper. Mary comes home and
brings only ten apples. The fact is before us, an actual fact of
life, that Mary got twentyfive apples and only brought home ten.
Mary is an honest little girl, and she really didn't eat a single
apple on the way, and yet she only brought home ten. And now someone
comes running in, an honest person, bringing all the apples that Mary
dropped on the way. Now there arises the question: How many does he
bring? We see him coming from a distance, but we want to know
beforehand how many he is going to bring. Mary has come home with ten
apples, and she got twentyfive, for there it is on the paper written
down by the applewoman, and now we want to know how many this person
ought to be bringing, for we do not yet know if he is honest or not.
What Mary brought was ten apples, and she got twentyfive, so she
lost fifteen apples..
Now, as
you see, the sum is done. The usual method is that something is given
and you have to take away something else, and something is left. But
in real life — you may easily convince yourselves of this
— it happens much more often that you know what you originally
had and you know what is left over, and you have to find out what was
lost. Starting with the minuend and the subtrahend and working out
the remainder is a dead process. But if you start with the minuend
and the remainder and have to find the subtrahend, you will be doing
subtraction in a living way. This is how you may bring life into your
teaching.
You will
see this if you think of the story of Mary and her mother and the
person who brought the subtrahend; you will see that Mary lost the
subtrahend from the minuend and that has to be justified by knowing
how many apples the person you see coming along will have to bring.
Here life, real life, comes into your subtraction. If you say, so
much is left over, this only brings something dead into the child's
soul. You must always be thinking of how you can bring life, not
death, to the child in every detail of your teaching.
You can
continue in this method. You can do multiplication by saying:
“Here we have the whole, the product. How can we find out how
many times something is contained in this product?” This
thought has life in it. Just think how dead it is when you say: I
will divide up this whole group of people, here are three, here are
three more and so on, and now I ask: how many times three have we
here? That is dead, there is no life in it.
If I
proceed the other way round and take the whole and ask how often one
group is contained within it, then I bring life into it. I can say to
the children for instance: “Look, there is a certain number of
you here in the class. Let us count. There are fortyfive of you in
the class. Now I am going to choose out five, 1, 2, 3, 4, 5 and put
them over here.” Then I let them count up; how many times are
these five contained within the fortyfive? You see that here again I
consider the whole and not the part. How many more of these groups of
five can I make? Then I find out that there are eight more groups of
five. Thus I do the thing the other way round and start with the
whole, the product, and find out how often one factor is contained in
it. Thereby I bring life into my arithmetical methods and above all I
begin with something that the child can see before him. The chief
point is that we must never, never separate thinking from visual
experience, from what the child can see, for otherwise we shall bring
intellectualism and abstractions to the child in early life and
thereby ruin his whole being. He will become dried up and this will
not only affect the soul life but the physical body also, causing
desiccation and sclerosis. (We shall later have to speak of the
education of spirit, soul and body as a unity.)
Here
again much depends on our teaching Arithmetic in the way we have
considered, so that in old age the human being is still mobile and
skilful. You should teach the children to count from their own bodies
as I have described, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, first with the
fingers and then with the toes — yes indeed, it would be
good to accustom the children actually to count up to twenty with
their fingers and toes, not on a beadframe. If you teach them thus
then you will see that through this childlike kind of
“meditation” you are bringing life into the body; for
when you count on your fingers or toes you have to think about these
fingers and toes, and this is then a meditation, a healthy kind of
meditating on one's own body. If this is done a man will still be
able to use his limbs skilfully in old age; the limbs can still
function fully because they have learnt to count by way of the whole
body. If a man only thinks with his head, rather than with his limbs
and the rest of his organism, then later on the limbs lose their
function and gout sets in.
This
principle, that everything in teaching and education must be worked
out from what can be seen (but not from what are often called
“object lessons” today) — this principle I should
like to illustrate for you with an example, something which can
actually play a very important part in teaching. I am referring to
the Theorem of Pythagoras which as wouldbe teachers you must all be
well acquainted with, and which you may even have already come to
understand in a similar way; but we will speak of it again today. Now
the Theorem of Pythagoras can be taken as a kind of goal in the
teaching of Geometry. You can build up your Geometry lessons to reach
their climax, their summit, in the Theorem of Pythagoras, which
states that the square on the hypotenuse of a rightangled triangle
is equal to the sum of the squares on the other two sides. It is a
marvellous thing if we see it in the right light.
I once
had to teach Geometry to an elderly lady because she loved it so
much; she may have forgotten everything, I do not know, but she had
probably not learnt much at her school, one of those schools for the
“Education of Young Ladies.” At all events she knew no
Geometry at all, so I began and made everything lead up to the
Theorem of Pythagoras which the old lady found very striking. We are
so used to it that it no longer strikes us so forcibly, but what we
have to understand is simply that if I have a rightangled triangle
here (see diagram) the area of the square on the hypotenuse is equal
to the sum of the other two areas, the two squares on the other two
sides. So that if I am planting potatoes and put them at the same
distance from each other everywhere, I shall plant the same number of
potatoes in the two smaller fields together as in the larger one.
This is something very remarkable, very striking, and when you look
at it like this you cannot really see how it comes about.
It is
just this fact of the wonder of it, that you cannot see how it comes
about, that you must make use of to bring life into the more inward,
soul quality of your teaching; you must build on the fact that here
you have something that is not easily discernible; this must
constantly be acknowledged. One might even say with regard to the
Theorem of Pythagoras that you can believe it, but you always have to
lose your belief in it again. You have to believe afresh every time
that this square is equal to the sum of the other two squares.
Now of
course all kinds of proofs can be found for this, but the proof ought
to be given in a clear visual way. (Dr. Steiner then built up a proof
for the Theorem of Pythagoras in detail based on the superposition of
areas; he gave it in the conversational style used in this Lecture
Course, and with the help of the blackboard and coloured chalks. For
those who are interested a verbatim report of this proof, with
diagrams, can be found in the Appendix on page 106.)
If you
use this method of proof, i.e. laying one area over the other, you
will discover something. If you cut it out instead of drawing it you
will see that it is quite easy to understand. Nevertheless, if you
think it over afterwards you will have forgotten it again. You must
work it out afresh every time. You cannot easily hold it in your
memory, and therefore you must rediscover it every time. That is a
good thing, a very good thing. It is in keeping with the nature of
the Theorem of Pythagoras. One must arrive at it afresh every time.
One should always forget that one has understood it. This belongs to
the remarkable quality of the Theorem of Pythagoras itself, and
thereby you can bring life into it. You will soon see that if you
make your pupils do it again and again, they have to ferret it out by
degrees. They do not get it at once, they have to think it out each
time. But this is in accordance with the inner living quality of the
Theorem of Pythagoras. It is not good to give a proof that can be
understood in a flat, dry kind of way; it is much better to forget it
again constantly and work it out every time afresh. This is inherent
in the very wonder of it, that the square on the hypotenuse is equal
to the squares on the other two sides.
With
children of eleven or twelve you can quite well take Geometry up to
the point of explaining the Theorem of Pythagoras by this comparison
of areas, and the children will enjoy it immensely when they have
understood it. They will be enthusiastic about it, and will always be
wanting to do it again, especially if you let them cut it out. There
will perhaps be a few intellectual goodfornothings who remember it
quite well and can always do it again. But most of the children,
being more reasonable, will cut it out wrong again and again and have
to puzzle it out till they discover how it has to go. That is just
the wonderful thing about the Theorem of Pythagoras, and we should
not forsake this realm of wonder but should remain within it.
