“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 Gnomes; Free the Math Gnomes, Part 2)

Free the Math Gnomes; Free 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