Tuesday, December 8, 2015

Teaching Arithmetic Without Gnomes

“Ok... I don't understand... I began division with the first grade, sharing seashells with them that I had found at the beach. A colleague said that he had always started with addition. I directed him to where ol' Rudy said to start with division. All is well. Then he asked me about my stories for math, because he uses these ‘amazing’ stories about gnomes and their adventures, and ‘here is a book you should check out and use for your classes,’ Math Lessons for Elementary Grades, by Dorothy Harrer. Alas, I take the book, pull up your blog post about math gnomes, and promptly put down the book. But then I thought… ‘Why does this make me any different than him? Dorothy Harrer is telling him to do something, and he believes it is true. Steve Sagarin is telling me to do something, and I believe it is true.’ So I picked up Dorothy’s book again. There are SO MANY direct contradictions to Steiner’s recommendations that it’s mind blowing. It’s even funnier that Dorothy quotes Steiner in the beginning of her book from not one, not two, but three different lectures about teaching math. Then, just a few pages later, she creates games and stories that are so confusing for even adults to understand. Why, why, WHY are teachers not doing the research on stuff like this?!”
(From a former teacher education student, now a teacher at a Waldorf school; some details changed to preserve confidentiality.)

If you are interested in teaching math without using gnomes, animals, royalty, or peasantry to name mathematical operations, you may wish for guidance on what actually to do.

So, for your ease of use, I have compiled quotations on arithmetic and math teaching from about a dozen key Rudolf Steiner lecture series on education, including Steiner’s initial course for teachers (Foundations of Human Experience, Practical Advice for Teachers, Discussions with Teachers), his next two courses at the first Waldorf school (Education for Adolescents, Balance in Teaching), his two courses given in England (A Modern Art of Education, The Kingdom of Childhood), and a few others. While these are not comprehensive, they certainly cover the territory. I have focused on math teaching approximately through third grade because, in my experience, this is where the gnomes live and from where they need to be freed. These references, beyond the few included in the paragraphs immediately below, are pasted at the end of this article.

Most of the quotations found here are referenced, often in somewhat briefer form, in E.A.K. Stockmeyer’s Curriculum, a guide I heartily recommend.

One of the challenges is that many or most of those who use math gnomes and their ilk have studied Steiner—maybe even currently study him—and somehow don’t notice that what they read is at odds with the use of math gnomes. So the rest of this article will focus on an argument in this direction based on Steiner’s words.

Let’s consider the objections or counter-arguments. These, as I understand them, are two:

First, we are asked to teach imaginatively and artistically, and imagining mathematical operations as little beings is imaginative and artistic—or can be. For instance, here is Steiner on this topic, ostensibly: “…all the instruction given for geometry, and even for arithmetic, must not fail to appeal to imagination. We appeal to imagination if we always make an effort to have the children use their imagination, even in geometry and arithmetic.” (Foundations of Human Experience, 209) Or, “we must permeate all of our teaching with an element of art.” (Practical Advice, 4) Or, “It is a fact that all mathematical material taught throughout the school years must be presented in a thoroughly artistic and imaginative way. Using all kinds of means teachers must contrive to introduce arithmetic and geometry artistically, and here, too, between the ninth and tenth years teachers must go to a descriptive method.” (Soul Economy, 207)

So, we need to ask, is this what Steiner means? Is picturing addition—the plus sign and what it stands for—as a (greedy, acquisitive) gnome imaginative and artistic?

Perhaps we need to back up and try to say something about what imagination and artistry are. Without digressing into Steiner’s aesthetic philosophy—like much of his work, based on Goethe’s—we may say three things about art: 1.) Art may be seen as the resolution of the polarity between dry meaning on the one hand and sheer, sensuous beauty on the other. Art is that which is both beautiful and meaningful. And, 2.) Art is the human attempt to reconnect to the oneness of creation; art always symbolizes both our separation from and our desire to reconnect to the all, even if in creating art we are not conscious of this motive. Finally, 3.) Creativity, in art as in all human endeavors, aims at moments of insight or inspiration, so-called “ah-hah” or “eureka” moments.

I would say that gnomes fail to meet any one of these considerations. They have the gloss of imagination, of imaginative teaching, but they introduce something false into the process. They do not introduce meaningful beauty into our work (as beautifully felted as the gnome idols may be); they do not further our connection to creation; and they are more likely to distract from rather than lead to the true moments of insight to which we wish to lead our students.

We may also consider what Steiner means by “artistic teaching,” or “artistry” in teaching. He is referring primarily to our work with our students. Instead of clay or paint or a musical instrument, the material of the teacher’s art is a student. Artistry here does not refer to the teaching of art or to fantastical gnomies. It refers to the insight and practice with which we pursue our teaching. And, in math, Steiner is clear that this regards a few things, none of them gnomes.
1.)   Begin from the whole
2.)   Introduce division first, then subtraction, then addition, and, finally, multiplication
3.)   Use examples from real life  (This seems like a gnome-freeing injunction right here.)
4.)   Address these operations to students according to their temperaments
 
Here is Steiner on the introduction of arithmetic. No gnomes.

Let’s start with addition, and first see what our view of addition should be. Let’s suppose I have some beans or a heap of elderberries. For our present task I will assume that the children can count, which indeed they must learn to do first. A child counts them and finds there are 27. “Yes,” I say. “27—that is the sum.” We proceed from the sum, not from the addenda. You can follow the psychological significance of this in my theory of knowledge. We must now divide the whole into the addenda, into parts or into little heaps. We will have one heap of, let’s say, 12 elderberries, another heap of 7, still another of say 3, and one more, let’s say 5; this will represent the whole number of our elderberries: 27 = 12 + 7 + 3 + 5. We work out our arithmetical process from the sum total 27. I would allow this process to be done by several children with a pronounced phlegmatic temperament. You will gradually come to realize that this kind of addition is particularly suited to the phlegmatics. Then, since the process can be reversed, I would call on some choleric children, and gather the elderberries together again, this time arranging them so that 5 + 3 + 7 + 12 = 27. In this way the choleric children do the reverse process. But addition in itself is the arithmetical rule particularly suited to phlegmatic children.
Now I choose one of the melancholic children and say, “Here is a little pile of elderberries. Count them for me.” The child discovers that there are, let’s say, 8. Now, I say, “I don’t want 8, I only want 3. How many elderberries must you take away to leave me only 3?” The child will discover that 5 must be removed. Subtraction in this form is the one of the four rules especially suited to melancholic children.

Then I call on a sanguine child to do the reverse process. I ask what has been taken away, and I have this child tell me that if I take 5 from 8, I’ll have 3 left. Thus, the sanguine child does the reverse arithmetical process. I would only like to add that the melancholic children generally have a special connection with subtraction when done as I have described.

Now I take a child from the sanguine group. Again I put down a pile of elderberries, but I must be sure the numbers fit. I must arrange it beforehand, otherwise we find ourselves involved in fractions. I have the child count out 56 elderberries. “Now look; here I have 8 elderberries, so now tell me how many times you find 8 elderberries contained in 56.” So you see that multiplication leads to a dividing up. The child finds that the answer is 7. Now I let the sum be done in reverse by a melancholic child and say, “This time I do not want to know how often 8 is contained in 56, but what number is contained 7 times in 56." I always allow the reverse process to be done by the opposite temperament.

Next I introduce the choleric to division, from the smaller number to the greater, by saying, “Look, here you have a little pile of 8; I want you to tell me what number contains 8 seven times.” Now the child must find the answer: 56, in a pile of 56.

Then I have the phlegmatic children work out the opposite process: ordinary division. The former is the way I use division for the choleric child, because the rule of arithmetic for the choleric children is mainly in this form division.

By continuing in this way I find it possible to use the four rules of arithmetic to arouse interest among the four temperaments.

Adding is related to the phlegmatic temperament, subtracting to the melancholic, multiplying to the sanguine, and dividing—working back to the dividend—to the choleric. I ask you to consider this, following what N. has been telling us.

It is very important not to continue working in a singular way, doing nothing but addition for six months, then subtraction, and so on; but whenever possible, take all four arithmetical rules fairly quickly, one after another, and then practice them all—but at first only up to around the number 40. So we shall not teach arithmetic as it is done in an ordinary curriculum.

By practicing these four rules, however, they can be assimilated almost simultaneously. You will find that this saves a great deal of time, and in this way the children can work one rule in with another. Division is connected with subtraction, and multiplication is really only a repetition of addition, so you can even change things around and give subtraction, for example, to the choleric child.
Discussions with Teachers, 48-50

Rather than fanciful gnomes in, what, a cave of gemstones?, Steiner asks us to base our teaching on “real life,” “practical life.” Here are some supporting quotations:

“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.” (Discussions with Teachers, 156. Emphasis added.)

“Suppose you had such an example as the following, taken from real life. A mother sent Mary to fetch some apples. Mary got twenty-five apples. The apple-woman 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 twenty-five 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 this person 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 twenty-five, for there it is on the paper written down by the apple-woman, 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 twenty-five, 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.
(Kingdom of Childhood, c. 79)

At first one should endeavor to keep entirely to the concrete in arithmetic, and above all avoid abstractions before the child comes to the turning point of the ninth and tenth years. Up to this time keep to the concrete as far as possible, by connecting everything directly with life. (Kingdom of Childhood, 126. Emphasis added.)

“In this way, you can extend to all of arithmetic as an art the method of always going from the whole to its parts.” (Practical Advice, 9)

The second argument in favor of using math gnomes is this: I, or a respected colleague, have used math gnomes and the students liked them and learned math successfully.

This is, unfortunately, no argument at all. Human beings learn math using all kinds of methods, even bad ones. Steiner is concerned that we begin from the whole so that we do not lead students into a materialistic (acquisitive, parts-to-whole, atomistic, reductive) view of the world. He is concerned that we use examples from real life to connect the otherwise abstract world of math with its proper place in the world. We awaken genuine interest and enthusiasm (and awakening these requires artistic teaching) when we don’t sensationalize our work by adding gnomes, but link our students’ activity to what they know of the activities in the world and lead them to a conceptual understanding of mathematical truth and beauty.

(And here’s a note on my method, for those wishing to compile other such collections of quotations on various topics, including especially in education. Almost all of Steiner’s educational lectures and publications are available as free downloads, usually as pdfs, from two sites: rsarchive.org, and the SteinerBooks spiritual research archive. Once you have downloaded these, it is a simple and enlightening matter, if somewhat time consuming, to search on key terms (“arithmetic,” “math”), then read, copy, and paste.)

(If you're puzzled about the whole math gnome thing, here's what I know:
Free the Math GnomesFree the Math Gnomes, Part 2)


A "Square-Root Gnome" that I made from beeswax in an idle moment... 
It's just a shirt he's wearing to humor you, and he'll get back to his real gnome work shortly...


OTHER QUOTATIONS FROM RUDOLF STEINER:
(These focus on arithmetic operations. Steiner also spoke about counting (prior to the introduction of operations), geometry, including especially the Pythagorean Theorem, and several other topics in math. Because these haven’t traditionally included gnomes, I haven’t included them here.)


FOUNDATIONS OF HUMAN EXPERIENCE
In the same way, all the instruction given for geometry, and
even for arithmetic, must not fail to appeal to imagination.
We appeal to imagination if we always make an effort to have
the children use their imagination, even in geometry and
arithmetic. We attempted this in the practical part of this
seminar when considering how to make surfaces comprehensible
not only to the child’s intellect, in such a way that the
child would truly comprehend the nature of a surface.
As I
said yesterday, I am surprised that no one ever thought of
explaining the Pythagorean theorem in the following way:
Suppose there were three children. One child blows enough
powder onto a square to cover it. The second child does the
same with a second square, the third with a small square. We
can encourage the children’s imagination by showing them
that the amount of powder that covers the large area equals
the combined amounts of powder on the small and middlesized
surfaces. Then we would draw the children’s comprehension
into the powder blown onto the squares, not through
mathematical exactness, but with imagination. The children
would follow the surface in their imagination. They would
understand the Pythagorean theorem through their imagination
and through the flying powder blown onto a square. We
cannot, of course, perform this in reality, but we can engage
the children’s imagination.
In these years we must always take care that, as teachers, we
create what goes from us to the children in an exciting way so
that it gives rise to imagination. Teachers must inwardly and
livingly preserve the subject material; they must fill it with
imagination.
209-210


PRACTICAL ADVICE TO TEACHERS
The reading and writing
you teach children is based on convention; it came about
within the realm of physical life itself.
Teaching children arithmetic is a very different matter. You
get the sense that the most important thing in arithmetic is not
the shapes of the numbers but the reality living in them. This
living reality has much more meaning for the spiritual world
than what lives in reading and writing. Finally, if we begin to
teach children various activities that we may call artistic, we
enter an area that has a definite, eternal meaning—something
that reaches up into the activity of the human spirit and soul. In
teaching children reading and writing, we work in the most
exclusively physical domain; in arithmetic our teaching becomes
Lecture One  3
less physical; and in music or drawing, or in related fields, we
really teach the children’s soul and spirit.
In a rationally conducted lesson we can combine these three
impulses of the supraphysical in artistic activity, the partially
supraphysical in arithmetic, and the completely physical in
reading and writing. In this way, we harmonize the human
being.
2-3

We must
not allow ourselves to think only in abstractions. Instead, we
must teach art in drawing and so on, teach soul substance in
arithmetic, and teach reading and use art to teach the conventional
in writing. In other words, we must permeate all of our
teaching with an element of art.
4

This sequence of starting with the whole and proceeding to
its parts must, in fact, be present in all that we teach. In
another situation, we could take a piece of paper and cut it into
a number of pieces. We might count the pieces—let’s say there
are twenty-four—and say to the child, “Look, I describe these
pieces of paper I cut up by what I wrote here: twenty-four
pieces of paper.” It could just as easily be beans or whatever.
“Now watch carefully. I take some pieces of paper away and
make another little pile. Then I make a third and fourth pile. I
have made four little piles from the twenty-four pieces of paper.
Now I will count the pieces. You are still unable to do that, but
I can. The pieces in the first pile I will call ‘nine,’ those in the
second ‘five,’ those in the third ‘seven,’ and those in the fourth
‘three.’ You see, at first I had only one pile of twenty-four pieces
of paper. Now I have four piles of nine, five, seven, and three
pieces. It is all the same paper. If I gather it all together, I call it
‘twenty-four’; if I have it in four little piles, I call it ‘nine,’ ‘five,’
‘seven,’ and ‘three’ pieces. Now twenty-four pieces of paper are
nine, five, seven, and three pieces together.”
This is how I have taught the children to add. I did not start
with the separate pieces from which a sum would be derived.
This would, in fact, be out of keeping with the original nature
of the human being.1 It is actually this reversed procedure that
is appropriate to human nature—first the sum is considered,
which is then divided into the separate parts. We teach children
addition by reversing the usual procedure; we begin with the
sum and then proceed to the addenda. Children will understand
the concept of “together” much better this way than if we
take the parts separately first and then bring them together in
the usual way. Our teaching methods will have to differ from
the ordinary; we will teach children the reversed way, so to
speak, about what a total is as opposed to its separate parts.
Then we can also expect a very different comprehension from
the children than we would if we used the opposite procedure.
You will discover what is most important about this method
only with practical experience. You will notice how children
immerse themselves in the subject in a very different way and
how they will have a different capacity to absorb what is taught
when you begin in this way.
You can apply the opposite process for the next step in arithmetic.
You say, “Now I will put all the pieces of paper together
again. I take some away, making two piles, and call the pile I
took away ‘three.’ How did I arrive at three? By taking it away
from the others. But when they were together I called the pile
‘twenty-four.’ Now I have taken three away and call the
remainder ‘twenty-one.’” This is how you proceed to the concept
of subtraction. Once again, do not begin with the whole
and what is to be subtracted; instead, begin with the remainder
that is left over and lead from that to the whole from which it
came. Here you go by the reverse path.
In this way, you can extend to all of arithmetic as an art the
method of always going from the whole to its parts. You will see
this later when we come to the methods for particular subjects.
We must simply accustom ourselves to a teaching process that is
very different from what we are used to. We proceed in a way
that not only nurtures the subject we impart (which cannot, of
course, be ignored, though a rather disproportionate amount of
attention is given to it today) but also, at the same time, fosters
the children’s sense for authority. We say continually, “I call this
‘twenty-four’” or “I call this ‘nine.’” When I stress in lectures on
spiritual science that a “sense for authority” must be nurtured
between the ages of seven and fourteen, I do not mean that we
must drill children into a feeling for authority. The element that
is needed flows from the very technique of teaching, which
reigns as an undertone. For example, a child listens and says,
“Oh, he calls that ‘nine,’ and he calls that ‘twenty-four.’” A
spontaneous obedience arises by listening to a person teaching
in this way, and children are thus permeated with what should
emerge as the sense for authority. This is the secret. Any unnatural
drilling of the sense for authority should not be included
because of the very nature of the method.
7-9

No child ought to reach
age fifteen without having been guided in arithmetic lessons to
an understanding of the rules of at least the simplest forms of
bookkeeping.
160

DISCUSSIONS WITH TEACHERS

RUDOLF STEINER: We will now continue the work we have set
out to do, and we will pass on to what will be said about how
to deal with arithmetic from the perspective of the temperaments.
We must primarily consider what procedure we should
follow in teaching arithmetic.
Someone showed how to explain a fraction by breaking a piece of
chalk.
RUDOLF STEINER: First, I have just one thing to say: I would
not use chalk, because it is a great pity to break chalk. I would
choose something less valuable. A bit of wood or something
like that would do, wouldn’t it? It is not good to accustom
young children to destroy useful things.
47


You can get
the smart children like this to do something with their special
powers that will help the others, so that they do not work just
for themselves, but for the other children as well. If they are
better at arithmetic, have them do the problem first, and let the
others learn from them. Their greater ability is channeled properly
when they hear from the teacher the consequence of a line
of thought that could be expressed in this way: John is a good
boy. Look how much he can do. Such people are a great help to
others, and I’m very pleased with all of you that you learned so
much from John.”
79

Someone spoke about the children who are not good at arithmetic.
Rudolf Steiner: When you discover a special weakness in arithmetic,
it would be good to do this: generally, the other children
will have two gymnastics lessons during the week, or one
eurythmy lesson and one gymnastics lesson; you can take a
group of the children who are not good at arithmetic, and
allow them an extra hour or half-hour of eurythmy or gymnastics.
This doesn’t have to mean a lot of extra work for you: you
can take them with others who are doing the same kind of
exercises, but you must try to improve these children’s capacities
through gymnastics and eurythmy. First give them rod
exercises. Say to them, “Hold the rod in your hand, first in
front counting 1, 2, 3, and then behind 1, 2, 3, 4." Each time
the child must change the position of the rod, moving it from
front to back. A great effort will be made in some way to get
the rod around behind at the count of 3. Then add walking:
say, 3 steps forward, 5 steps back; 3 steps forward, 4 steps back;
5 steps forward, 3 steps back, and so on. In gymnastics, and
also perhaps in eurythmy, try to combine numbers with the
children’s movements, so they are required to count while moving.
You will find this effective. I have frequently done this
with pupils.
But now tell me, why does it have an effect? From what you
have already learned, you should be able to form some ideas on
this subject.

A teacher commented: Eurythmy movements must be a great help
in teaching geometry.
RUDOLF STEINER: But I did not mean geometry. What I said
applied to arithmetic, because at the root of arithmetic is consciously
willed movement, the sense of movement. When you
activate the sense of movement in this way, you quicken a
child’s arithmetical powers. You bring something up out of the
subconscious that, in such a child, is unwilling to be brought
up. Generally speaking, when a child is bad both at arithmetic
and geometry, this should be remedied by movement exercises.
You can do a great deal for a child’s progress in geometry with
varied and inventive eurythmy exercises, and also through rod
exercises.
105-106

Second, figure out for tomorrow how you would give the
children arithmetical problems to solve without writing down
any figures—in other words, what we could call mental arithmetic.
You could, for example, give the children this problem
to do: A messenger starts from a certain place and walks so
many miles per hour; another messenger begins much later; the
second messenger does not walk but rides a bicycle at a certain
number of miles per hour. When did the cyclist pass the messenger
on foot?
The object of these problems is to develop in children a certain
presence of mind in comprehending a situation and evaluating
it as a whole.
150

Mental arithmetic was discussed.
RUDOLF STEINER related how Gauss 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.
154-155

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.
156

In arithmetic, for example, if we do not stress what the child
cannot do, but instead work with the student so that in the end
the child can do it—following the opposite of the principle
used until now—then “being unable” to do something will not
play the large role it now does. Thus in our whole teaching, the
passion for passing judgment that teachers acquire by marking
Discussion Fifteen  179
grades for the children every day in a notebook should be
transformed into an effort to help the children over and over,
every moment. Do away with all your grades and placements.
If there is something that the student cannot do, the teachers
should give themselves the bad mark as well as the pupil,
because they have not yet succeeded in teaching the student
how to do it.
178-179

We must still apportion everything related to arithmetic,
mathematics, and geometry and distribute it among the eight
grades.
You know that standard superficial methodology dictates
that in the first grade we should deal primarily with numbers
up to 100. We can also go along with this, because the range of
numbers doesn’t really matter in the first grade, where we stick
with simpler numbers. The main issue, regardless of what
range of numbers you use, is to teach the arithmetical operations
in a way that does justice to what I said before: Develop
addition out of the sum, subtraction out of the remainder,
multiplication out of the product and division out of the quotient—
that is, exactly the opposite of how it’s usually done.
Only after you have demonstrated that 5 is 3 plus 2, do you
demonstrate the reverse—that adding 2 and 3 yields 5. You
must arouse in the children the powerful idea that 5 equals 3
plus 2, but that it also equals 4 plus 1, and so on. Thus, addition
is the second step after separating the sum into parts, and
subtraction is the second step after asking “What must I take
away from a minuend to leave a specific difference?” and so on.
As I said before, it goes without saying that you do this with
simpler numbers in the first grade, but whether you chose a
range of up to 95 or 100 or 105 is basically beside the point.
After that, however, when the second dentition is over, we
can immediately begin to teach the children the times tables—
even addition, as far as I’m concerned. The point is that children
should memorize their times tables and addition facts as
soon as possible after you have explained to them in principle
what these actually mean—after you have explained this in
principle using simple multiplication that you approach in the
way we have discussed. That is, as soon as you’ve managed to
teach the children the concept of multiplication, you can also
expect them to learn the times tables by heart.
Then in the second grade you continue with the arithmetical
operations using a greater range of numbers. You try to get the
students to solve simple problems orally, in their heads, without
any writing. You attempt to introduce unknown numbers
by using concrete objects—I told you how you could approach
unknown numbers using beans or whatever else is available.
However, you should also not lose sight of doing arithmetic
with known quantities.
In the third grade everything is continued with more complicated
numbers, and the four arithmetical operations practiced
during the second grade are applied to certain simple
things in everyday life.
196-197 (etc. for older grades)

BALANCE IN TEACHING
Everything in teaching that requires one to form mental images
of number and space, like geometry and arithmetic, helps the
ego to settle itself well into the organism when the child forms
such images and works on them.
47

EDUCATION FOR ADOLESCENCE
Essentially our lessons consist of two interacting parts. We
instruct, we exhort the children to participate, to use their
skills, to be physically active. Be it in eurythmy, music, physical
education, even writing or the mechanical processes in arithmetic—
we try to engender activity. The other part of our lessons
is concerned with contemplation. Here we ask the
children to think about, to consider the things we tell them.
20

When we now teach the various subjects expected of a
school—reading, the thought processes in arithmetic, those in
the natural sciences, everything that is of a cognitive nature—
we give the children ideas and mental pictures.
60

RENEWAL OF EDUCATION
We will see what the situation is with
arithmetic. What damages people so much is often the result of
their instruction in arithmetic. The way that we learn to do arithmetic
generally goes against human nature. Everything that occurs
in many people today as a tendency toward materialism is essentially
the result of improper instruction in arithmetic around the
age of nine. Another thing that is so destructive for the later development
of the soul in many people is that they begin to reason too
early, and we present the material there to learn in a way for which
they are not yet mature enough. They take in a large number of
predetermined judgments that then continue to affect them.
People often speak about the fact that in human beings one concept
or idea associates itself with others. There is nothing more
unfortunate than this talk about the association of ideas. When
ideas associate with one another in us, when they clump and we
run after them, then in our thinking, we are under their control
and no longer have power over ourselves. Through education, we
must protect people from allowing these associations to gain the
upper hand over the will. I will speak more of that tomorrow.
165-166

We often do not sufficiently consider the
relationship of arithmetic to the child’s soul life. First of all we
must differentiate between arithmetic and simple counting. Many
people think counting represents a kind of addition, but that is
not so. Counting is simply naming differing quantities. Of
course, counting needs to precede arithmetic, at least counting up
to a certain number. We certainly need to teach children how to
count. But we must also use arithmetic to properly value those
analytical forces that want to be developed in the child’s soul. In
the beginning, we need to attempt, for instance, to begin with the
number ten and then divide it in various ways. We need to show
the children how ten can be separated into five and five, or into
three and three and three and one. We can achieve an enormous
amount in supporting what human nature actually strives for out
of its inner forces when we do not teach addition by saying that
the addends are on the left and the sum is on the right, but by saying
that we have the sum on the left and the addends on the right.
We should begin with analyzing the sum and then work backwards
toward addition.
If you wish, you can take this presentation as a daring statement.
Nevertheless those who have achieved an unprejudiced
view of the forces within human nature will recognize that when
we place the sum on the left and the addends on the right, and
then teach the child how to separate the sum in any number of
ways, we support the child’s desire to analyze. Only afterward do
we work with those desires that actually do not play a role within
the soul, but instead are important with interactions of people
within the external world. What a child analyzes out of a unity
exists essentially only for herself. What is synthesized exists always
for external human nature.
172

Similarly, in multiplication we should not attempt to begin with
the factors and proceed to the product. Instead we should begin
with the product and form the factors in many various ways. Only
afterwards should we turn to the synthesizing activity. This way
through arithmetic people may be able to develop the rhythmic
activity within the life of the soul that consists of analyzing and
synthesizing. In the way we teach arithmetic today, we often
emphasize one side too strongly. For the soul, such overemphasis
has the same effect as if we were to heap breath upon breath upon
the body and not allow it to exhale in the proper way. It is important
to take the individuality of the human being into account in
the proper way. This is what I mean when I speak of the fructification
that education can experience through spiritual science.
We need to become aware of what actually wants to develop
out of the child’s individuality. First we need to know what can
be drawn out of the child. At the outset children have a desire to
be satisfied analytically; then they want to bring that analysis
together through synthesis. We must take these things into
account by looking at human nature. Otherwise even the best
pedagogical principles—although they may be satisfying to use
and we believe they are fulfilling all that is required—will never
be genuinely useful because we do not actually try to look at the
results of education in life.
173-174

We certainly cannot object to the fact that in the nineteenth
century computing machines were introduced into schools.
Nevertheless computing machines should not lead to an overly
materialistic valuation of illustrative materials. While we should be
clear about the value of examples, what is important is that human
capacities be developed through teaching. The primary task of the
period from the change of teeth until maturity is to develop memory.
We should avoid underestimating the value of examples as a
basis for forming memory as well as the value of memory when
viewing examples. We should begin in a simple way—and here for
those who are capable of teaching in a living way, the ten fingers
on our hands are sufficient—by presenting the number ten in all
kinds of ways that show the various arithmetic operations. In
doing so, however, we should present arithmetic in a way that is
appropriate to life, to the life of the soul in a human being.
230

SOUL ECONOMY
When children enter class one, they are certainly ready to
learn how to calculate with simple numbers. And when we
introduce arithmetic, here, too, we must carefully meet the
inner needs of children. These needs spring from the same
realm of rhythm and measure and from a sensitive apprehension
of the harmony inherent in the world of number. However,
if we begin with what I would call the “additive
approach,” teaching children to count, again we fail to understand
the nature of children. Of course, they must learn to
count, but additive counting as such is not in harmony with
the inner needs of children.
It is only because of our civilization that we gradually began
to approach numbers through synthesis, by combining them.
Today we have the concept of a unit, or oneness. Then we have
a second unit, a third, and so on, and when we count, we mentally
place one unit next to the other and add them up. But, by
nature, children do not experience numbers this way; human
Children from the Seventh to the Tenth Year  149
evolution did not develop according to this principle. True, all
counting began with a unit, the number one. But, originally,
the second unit, number two, was not an outer repetition of
the first unit but was felt to be contained within the first unit.
Number one was the origin of number two, the two units of
which were concealed within the original number. The same
number one, when divided into three parts, gave number three,
three units that were felt to be part of the one. Translated into
contemporary terms, when reaching the concept of two, one
did not leave the limits of number one but experienced an
inner progression within number one. Twoness was inherent in
oneness. Also three, four, and all other numbers were felt to be
part of the all-comprising first unit, and all numbers were experienced
as organic members arising from it.
Because of its musical, rhythmic nature, children experience
the world of number in a similar way. Therefore, instead of
beginning with addition in a rather pedantic way, it would be
better to call on a child and offer some apples or any other suitable
objects. Instead of offering, say, three apples, then four
more, and finally another two, and asking the child to add
them all together, we begin by offering a whole pile of apples,
or whatever is convenient. This would begin the whole operation.
Then one calls on two more children and says to the first,
“Here you have a pile of apples. Give some to the other two
children and keep some for yourself, but each of you must end
up with the same number of apples.” In this way you help children
comprehend the idea of sharing by three. We begin with
the total amount and lead to the principle of division. Following
this method, children will respond and comprehend this
process naturally. According to our picture of the human being,
and in order to attune ourselves to the children’s nature, we do
not begin by adding but by dividing and subtracting. Then,
retracing our steps and reversing the first two processes, we are
led to multiplication and addition. Moving from the whole to
the part, we follow the original experience of number, which
was one of analyzing, or division, and not the contemporary
method of synthesizing, or putting things together by adding.
148-150

Now, everything that teachers do with the children, until the
turning point around nine, should have a formative effect, but
in a way that stimulates them to participate freely and actively
in this inner shaping. I indicated this with my strong appeal for
an artistic approach during the introductions to reading, writing,
and arithmetic. The artistic element is particularly important
at this age.
All teaching during the early school years must begin with
the child’s will sphere, and only gradually should it lead over
toward the intellect. Those who recognize this will pay special
attention to educating the child’s will. They will know that
children must learn to drive out the will forces from their
organism, but in the right way. To do this, their will activities
must be tinged with the element of feeling. It is not enough for
teachers to do different things with the children; they must also
develop sympathy and antipathy according to what they are
doing.
201


THE SPIRITUAL GROUND OF EDUCATION
Children are able to take in the elements of arithmetic at a
very early age. But in arithmetic we can see how easy it is to
give an intellectual element to children too early. Mathematics
per se is not alien to anyone at any age. It arises from human
nature; the principles of mathematics are not foreign to human
nature as letters are foreign in a succeeding civilization. It is
exceedingly important, however, that children be introduced
to arithmetic and mathematics in the right way. And this can
really be determined only by those enabled to oversee the
whole of human life from a certain spiritual standpoint.
Arithmetic and moral principles are two things that, in terms
of logic, seem very removed from each other. It is not common
to connect arithmetic with moral principles, because the logical
connection is not obvious. It is obvious, however, to those who
do not view the matter in terms of logic but in a living way that
children who are introduced to arithmetic correctly will have a
very different feeling of moral responsibility than those who
were not. Now, what I am about to say may seem like an
extreme paradox to you, but since I am speaking of realities
and not of the illusions of our age, I will not fear the seemingly
paradoxical, for these days truth often seems paradoxical. If
people had known how to permeate the soul with mathematics
in the right way during these past years, we would not now
have Bolshevism in Eastern Europe. One perceives the forces
that connect the faculty used in arithmetic with the springs of
morality in humankind.
Perhaps you will understand this better if I give you a little
illustration of the principles behind teaching arithmetic. Today
it is common to begin arithmetic by adding one thing to
another. But just consider how alien it is to the human mind
when we add one pea to another, and with each addition specify
a new name. The transition from one to two and then to
three; such counting is an arbitrary activity for the human
being. It is possible, however, to count in a different way. We
find this when we go back a little in human history. Originally,
people did not count by adding one pea to another and deriving
a new thing that, for the soul anyway, had little to do with
what came before. Rather, people counted more or less as follows.
It was their view that everything in life can always be
grasped as a whole, and the most diverse things might constitute
a unity. If I have a number of people in front of me, that
can be a unity at first. Or if I have an individual before me, that
person is a unity. A unity, in reality, is purely relative.
Now, I keep this in mind as I count in this way:
In other words, I take an organic whole—a whole consisting of
members. I begin with unity, and in that unity, viewed as a
multiplicity, I seek the parts. This, in fact, was the original view
of number. Unity was always a totality, and one looked for the
parts in the whole. One did not think of numbers as arising
from the addition of one and one and one; rather, one conceived
of the numbers as belonging to a whole and proceeding
organically from that whole.
When we apply this to teaching arithmetic, we get the following:
instead of placing one bean after another for children
to see, we give them a whole pile of beans. The bean pile constitutes
a whole, and we begin with this. Next, with our pile of
beans (or a pile of apples if you think it might appeal more to
their imagination), we explain that there are three children of
various ages who need different amounts to eat. We want to do
something that might apply to real life. So, what shall we do?
We could, for example, divide the pile of apples so that we
have one pile, on the one hand, and, on the other, two portions
that together equal to the first pile. The whole pile represents
the sum. We have a pile of apples, and we say, “Here are three
parts.” And we get the children to see that the sum equals the
three parts. In other words, in addition we do not go from the
parts and arrive at the sum; we begin with the sum and proceed
to the parts. Thus, to get a living understanding of addition,
we begin with the whole and proceed to the addenda, or parts.
Addition is concerned essentially with the sum and its parts—
the members contained, one way or another, in the sum.
This is how we get children to enter life with the ability to
grasp the whole, rather than always proceeding from less to
more. This has a very powerful influence on a child’s whole
soul and mind. Once children have acquired the habit of adding
things together, a disposition arises that tends to be desirous
and craving. By proceeding from the whole to the parts, on
the other hand, and by treating multiplication in a similar way,
children are less prone to acquisitiveness. Instead, they tend to
develop, in the noblest Platonic sense, consideration and moderation.
Our moral likes and dislikes are intimately connected
with the way we learned to deal with numbers. At first sight,
it’s difficult to see any logical connection between one’s
approach to numbers and moral ideas. Indeed, those who look
at things only from the intellectual point of view may laugh at
the idea of a connection; it may seem absurd to them. It is also
easy to understand that people may laugh at the idea of going
from the sum in addition, instead of from the parts. When one
sees the true connections in life, however, the things that seem
most remote logically become exceedingly close.
Consequently, the processes that take place in the souls of
children through working with numbers have a tremendous
effect on the way they meet us when we wish to present them
with moral examples and actions intended to evoke sympathy
for good and antipathy toward evil. We will see children who
are open to goodness if we have taught them numbers as
described.
70-73


A MODERN ART OF EDUCATION
But we should strive to help that etheric activity continue
during sleep, and we do this when we begin by
communicating a concrete representation of space,
instead of beginning geometry with triangles and the
like, in which the intellect is already in evidence. In arithmetic,
too, we must proceed in this way.
A pamphlet on physics and mathematics by Dr. von
Baravalle (a teacher at the Waldorf school) gives a good
idea of how to bring concrete reality into math and geometry.
The pamphlet also extends this whole way of thinking
into the realm of physics, although it deals primarily
with higher mathematics. If we go into its underlying
Arithmetic, Geometry, History 145
spirit, it is a wonderful guide for teaching math in a way
that corresponds to the natural needs of a child’s being. It
created a starting point for stimulating reform in the
method of teaching mathematics and physics, from early
childhood to the highest levels of instruction. We must be
able to take what the pamphlet says about concrete concepts
of space and extend it to arithmetic.
The whole point is that everything arithmetic conveys
externally to children, even counting, destroys something
in the human organism. To begin with a unit and add to
it, piece by piece, merely destroys the human organism.
But the organism is made more alive when we begin by
awakening an awareness of the whole, then awakening
an awareness of the members of that whole; we begin
with the whole and proceed to its parts. This must be
kept in mind, even when children are learning to count.
Usually, we learn to count by observing purely physical,
external things. We begin with 1, a unity, then add 2, 3, 4,
and so on, unit by unit—and we have absolutely no idea
why one follows the other, or what happens in the end.
We are taught to count through an arbitrary juxtaposition
of units. I am well aware that there are many methods for
teaching children to count, but very little attention is paid
to the principle of starting with the whole and proceeding
to the parts. Children should first see the unit as a whole.
Everything is a unity, no matter what it is.
Here we have to illustrate this with a drawing. We
must therefore draw a line; but we could use an apple
just as well to do what I
am doing with a line.
This, then, is 1. Now we
go from the whole to the
parts, or members. Thus,
we have made a 1 into a 2, but the 1 still remains. The unit
has been divided into two. Thus we come to the 2. And
we continue, and another partition brings the 3 into
being. Unity always remains the all-embracing whole.
And so we go, through 4, 5, and so on. Moreover, at the
same time, using another means, we can give an idea of
how well we are able to hold in the mind everything that
relates to number, and we discover just how limited we
are in our power of mental presentation when it comes to
numbers.
In some countries today, the concept of number that is
clearly held in the mind’s eye goes only up to ten. Here in
England, money is counted up to twelve. But that really
represents the maximum of what is mentally visualized,
because in reality we then begin again and repeat the
numbers. For example, we count up to ten, then we begin
counting the tens, 2 x 10 = 20, and 3 x 10 = 30. Here we no
longer consider the things themselves but begin to calculate
the number itself, whereas the more elementary concept
requires things themselves to be clearly present in
the mind.
People are proud of the fact that they use very
advanced methods to count, compared to primitive people,
who depend on their ten fingers. But there is little
basis for such pride. We count to ten because we feel the
members of our hands. We feel our two hands and ten
fingers symmetrically. Children also experience this feeling,
and we must evoke the sense of number through
transition from the whole to the parts. Then we easily find
another transition, which leads us to counting, in which
we add to another. Eventually, of course, we can move on
to the ordinary 1, 2, 3, and so on. But merely adding one
or more units together must not be introduced until the
second level, for it has significance only in physical space,
whereas dividing a unity into members has an inner
meaning that continues to vibrate in a child’s ether body,
even when we are no longer present. It is important that
we know such things.
After teaching children to count in
this way, something else becomes
important. We must not proceed to
addition in a dead, mechanical way,
by merely adding one item to
another in series. Life arises when
we begin not with parts of addition,
but with the whole total. We begin
with a number of objects; for example, you throw down a
number of little balls. We’ve gone far enough in counting
to say there are fourteen balls. You divide them up,
extending the concept of parts even further. You have
five here, four there, and five again. You have separated
the total into 5 + 4 + 5. We go from the total to the units
that comprise it, from the whole to the parts. The method
we should use with children is to set up the total for them
first, and then let the children perceive how the given
total can be divided up. This is very important. One does
not harness a horse with its tail to the front; likewise,
when teaching arithmetic, we must go the right way. We
start from a whole actually present in the total—a reality—
and then separate it into parts; later, we find our
way back to the ordinary sum total.
Continuing in this way, from a living whole to separate
parts, one touches the reality behind all mathematical calculations:
the vibration of the ether body of formative
forces. This body needs a living stimulus for its formative,
perfecting activity, which it continues with no need
for the presence of the astral body and I-being and their
disturbing elements.
Your teaching will be essentially enhanced and vivified
if, in a similar way, you reverse the other simple forms of
calculation. Today, one might say, children are upside
down and must righted. For example, try to get a child to
think in this way: “If I have seven, how much must I take
away to get three?” instead of “What is left after I take
four away from seven?” Having seven is the real thing,
and what I have left is equally real; how much must we
take away from seven to get three? Beginning with this
kind of thinking, we stand in the middle of life, whereas
with the opposite form we face an abstraction. Proceeding
this way, we can easily revert to the other eventually.
Thus, again, in multiplication and division, we should
not ask what will result when we divide ten into two
parts, but how must we divide ten to get five. The actual
aspect is given; in life, eventually we want to get to something
with real significance. Here are two children; ten
apples will be divided among them. Each is supposed to
get five. These are realities. What we must first contribute
is the abstract part in the middle.
When we do things this way, things are directly
adapted to life, and, if we are successful, the usual, purely
external way of adding by counting one thing after
another with a deadening effect on the arithmetic lessons
will instead become a vivifying force of particular importance
in this area of educational work. We must really
consider the subconscious aspect of human beings—that
is, the part that not only continues to work during sleep,
but also works subconsciously during the waking hours.
We do not always think of everything. We are aware of
only a small fragment of our soul’s experience, but the
rest is always active. Let’s create the possibility for children’s
physical and ether bodies to work in a healthy
way, recognizing that we can do so only when we bring
atmosphere, interest, and life into our lessons in arithmetic
and geometry.
144-149

There are three golden rules for the developing memory:
concepts load the memory; the perceptible arts build
it up; activities of will strengthen it and make it firm. We
are given wonderful opportunities to apply these three
golden rules when we teach nature and history as suggested
in these lectures. Arithmetic, too, may be used for
this, because even here we should always begin with an
artistic feeling, as I tried to show. When children thoroughly
understand the simpler operations—say, counting
to ten or twenty—there is no need to worry about
allowing them to memorize the rest. It is incorrect to
overload children with too many concrete pictures, just
as it is incorrect to strain their powers of memory,
because when concepts become too complex, they have
the same effect. We must carefully observe how memory
develops in each individual child.
172

THE KINGDOM OF CHILDHOOD
Arithmetic too must be drawn out of life. The living thing is
always a whole and must be presented as a whole first of all. It
is wrong for children to have to put together a whole out of its
parts, when they should be taught to look first at the whole and
then divide this whole into its parts; get them first to look at
the whole and then divide it and split it up; this is the right
path to a living conception.
78

We criticize atomism today, but criticism is really more or
less superfluous because people cannot get free from what they
have been used to thinking 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 you should particularly bear in mind
when teaching arithmetic. If you are walking toward 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 30 or 40 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 begin with, how
many you have? You simply have a heap of apples that you are
now bringing home (see diagram). There are, let us say, three
children. You will not now divide them so that each gets the
same, for perhaps one child is small, another big. You put your
hand into the basket and give the bigger child a bigger handful,
the smaller child a smaller handful; you divide your heap of
apples into three parts.
Dividing or sharing out is in any case such a strange 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 learned 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. The child 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 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: humor comes into the
teaching, not in a childish but in a healthy sense. And humor
must find its place in teaching.1
This then must be your method: always proceed from the
whole. Suppose you had such an example as the following,
taken from real life. A mother sent Mary to fetch some apples.
Mary got twenty-five apples. The apple-woman 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 twenty-five 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 this person 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 twenty-five, for there it is on the paper written down by the
apple-woman, 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
twenty-five, 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 way. 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: We will divide up this whole group of people, here are three,
here are three more, and so on, and then you ask: how many
times three have we here? That is dead, there is no life in it.
If you proceed the other way round and take the whole and
ask how often one group is contained within it, then you bring
life into it. You can say to the children, for instance: “Look,
there is a certain number of you here.” Then let them count
up; how many times are these five contained within the fortyfive?
Here again you consider the whole and not the part. How
many more of these groups of five can be made? Then it is
found out that there are eight more groups of five. Thus, when
you 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 you bring life into your arithmetical methods
and above all you begin with something that the children can
see before them. The chief point is that thinking must never,
never be separated from visual experience, from what the children
can see, for otherwise intellectualism and abstractions are
brought to the children in early life and thereby ruin their
whole being. The children will become dried up and this will
affect not only the soul life but the physical body also, causing
desiccation and sclerosis. (I 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 skillful. 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 bead-frame. 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. Doing this will allow the
grown person to remain skillful of limb in old age; the limbs can
still function fully because they have learned to count by using
the whole body. If a person only thinks with the head, rather
than with the limbs and the rest of the organism, then later on
the limbs lose their function and gout sets in.
79-85

At what age and in what manner should we make the transition
from the concrete to the abstract in arithmetic?
At first one should endeavor to keep entirely to the concrete
in arithmetic, and above all avoid abstractions before the child
comes to the turning point of the ninth and tenth years. Up to
this time keep to the concrete as far as possible, by connecting
everything directly with life.
When we have done that for two or two-and-one-half years
and have really seen to it that calculations are not made with
abstract numbers, but with concrete facts presented in the form
of sums, then we shall see that the transition from the concrete
to the abstract in arithmetic is extraordinarily easy. For in this
method of dealing with numbers they become so alive in the
child that one can easily pass on to the abstract treatment of
addition, subtraction, and so on.
It will be a question, then, of postponing the transition from
the concrete to the abstract, as far as possible, until the time
between the ninth and tenth years of which I have spoken.
One thing that can help you in this transition from the
abstract to the concrete is just that kind of arithmetic that one
uses most in real life, namely the spending of money; and here
you are more favorably placed than we are on the Continent, for
there we have the decimal system for everything. Here, with
your money, you still have a more pleasing system than this. I
hope you find it so, because then you have a right and healthy
feeling for it. The soundest, most healthy basis for a money system
is that it should be as concrete as possible. Here you still
count according to the twelve and twenty system which we have
already “outgrown,” as they say, on the Continent. I expect you
already have the decimal system for measurement? (The answer
was given that we do not use it for everyday purposes, but only
in science.) Well, here too, you have the more pleasant system of
measures! These are things that really keep everything to the
concrete. Only in notation do you have the decimal system.
What is the basis of this decimal system? It is based on the
fact that originally we had a natural measurement. I have told
you that number is not formed by the head, but by the whole
body. The head only reflects number, and it is natural that we
should actually have ten, or twenty at the highest, as numbers.
Now we have the number ten in particular, because we have
ten fingers. The only numbers we write are from 1 to 10: after
that we begin once more to treat the numbers themselves as
concrete things.
Let us just write, for example: 2 donkeys. Here the donkey is
the concrete thing, and the 2 is the number. I might just as well
say: 2 dogs. But if you write 20, that is nothing more than 2
times 10. Here the 10 is treated as a concrete thing. And so our
system of numeration rests upon the fact that when the thing
becomes too involved, and we no longer see it clearly, then we
begin to treat the number itself as something concrete, and
then make it abstract again. We should make no progress in
calculation unless we treated the number itself, no matter what
it is, as a concrete thing, and afterwards made it abstract. 100 is
really only 10 times 10. Now, whether I have 10 times 10, and
treat it as 100, or whether I have 10 times 10 dogs, it is really
the same. In one case the dogs, and in the other the 10 is the
concrete thing. The real secret of calculation is that the number
itself is treated as something concrete. And if you think this out
you will find that a transition also takes place in life itself. We
speak of 2 twelves—2 dozen—in exactly the same way as we
speak of 2 tens, only we have no alternative like “dozen” for the
ten because the decimal system has been conceived under the
influence of abstraction. All other systems still have much more
concrete conceptions of a quantity: a dozen: a shilling. How
much is a shilling? Here, in England, a shilling is 12 pennies.
But in my childhood we had a “shilling” that was divided into
30 units, but not monetary units. In the village where I lived
for a long time, there were houses along the village street on
both sides of the way. There were walnut trees everywhere in
front of the houses, and in the autumn the boys knocked down
the nuts and stored them for the winter. And when they came
to school they would boast about it. One would say: “I’ve got
five shillings already,” and another: “I have ten shillings of
nuts.” They were speaking of concrete things. A shilling always
meant 30 nuts. The farmers’ only concern was to gather the
nuts early, before all the trees were already stripped! “A nut-shilling”
we used to say: that was a unit. To sell these nuts was a
right: it was done quite openly.
And so, by using these numbers with concrete things—one
dozen, two dozen, one pair, two pair, and so on., the transition
from the concrete to the abstract can be made. We do not say:
“four gloves,” but: “Two pairs of gloves;” not: “Four shoes,” but
“two pairs of shoes.” Using this method we can make the transition
from concrete to abstract as a gradual preparation for the
time between the ninth and tenth years when abstract number
as such can be presented.2
126-129

THE ROOTS OF EDUCATION
It is most important
that mathematics, for example, should not be intellectualized;
even in mathematics, we should begin with what is real.
Now imagine that I have ten beans here in front of me. This
pile of 10 beans is the reality—it is a whole—but I can divide it
into smaller groups. If I began by saying, “3+3+4 beans = 10
beans,” then I am starting with a thought instead of an actuality.
Let’s do it the other way around and say, “Here are 10
beans. I move them around, and now they are divided into
groups—3 here, 3 again here, and another group of 4 that,
together, make up the whole.”
When I begin this way with the total actually in front of me,
and then go on to the numbers to be added together, I am
sticking with reality; I proceed from the whole, which is constant,
to its parts. The parts can be grouped in various ways—
for example, 10 = 2+2+3+3—but the whole is constant and
invariable, and this is the greater reality. Thus, I must teach
children to add by proceeding from the whole to the parts.
Genuine knowledge of the human being shows us that, at this
age, a child will have nothing to do with abstractions, such as
addenda, but wants everything concrete; and this requires a
reversal of the usual method of teaching mathematics. In teaching
addition, we have to proceed from the whole to the parts,
showing that it can be divided in various ways. This is the best
method to help us awaken forces of observation in children,
and it is truly in keeping with their nature. This applies also to
the other rules of mathematics. If you say, “What must we take
away from 5 in order to leave 2?” you will arouse much more
interest in children than if you say, “Take 3 from 5.” And the
first question is also much closer to real life. These things happen
in real life, and in your teaching methods you can awaken
a sense of reality in children at this age.

58

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