Arithmetic - Waldorf Research Institute

120 Pages·2010·4.26 MB·English

Arithmetic
and Zoological Considerations
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Printed with support from the Waldorf Curriculum Fund
Produced by:
The Research Institute for Waldorf Education
www.waldorfresearchinstitute.org
Title: Arithmetic
Author: Dr. Karl König
Layout and Production: David Mitchell
Proofreader: Ann Erwin
© 2009 by AWSNA
For Study Purposes Only
First printed by Glencraig Printing Department and published
by the Camphill Village Trust at Newton-Dee Bieldside Aberdeen
for circulation in the Camphill Movement. Electronically printed with
permission from the Karl König archives, Scotland.
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Contents
The Arithmetic Curriculum .......................................................... 5
Conference I
Seminar I ............................................................................. 18
Seminar II ............................................................................ 27
Seminar III ........................................................................... 37
Concluding Lecture ............................................................. 45
Discussion I ......................................................................... 54
Discussion II .......................................................................... 62
Discussion III ....................................................................... 73
Conference 2
Seminar I ............................................................................ 82
Discussion I ........................................................................ 92
Seminar II ........................................................................ 95
Discussion II ....................................................................... 102
Seminar III ............................................................................ 106
Discussion III ...................................................................... 115
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THE ARITHMETIC CURRICULUM
Lecture by Hans Christoph Valentien
Dear friends, when we teach arithmetic to mentally handicapped
or disturbed children, our familiarity with the subject covers more or
less the beginnings of number-work only. It is worthwhile to turn to
the Waldorf curriculum in both arithmetic and geometry, especially
in the light of Dr. König’s lecture this afternoon, where he described
mathematics as being rooted in the formative forces working in man’s
sensory organization, particularly in his lower senses. These forces
become conscious in arithmetic and geometry.
Very important, just to us who are working in curative education, are
certain general aspects of the Waldorf curriculum and of mathematics
in particular; for instance, the question of how to introduce arithmetic
and geometry to children in such a way that we reckon with human
nature and life in their totality.
Let us ask, first of all, how arithmetic is placed within the whole of
Waldorf education. As curative teachers we are aware of the curative,
therapeutic aspects of the curriculum. Often we have experienced
its harmonizing and healing value. In trying to adapt the Waldorf
curriculum to the more limited scope of schooling in curative education,
our first objective must be to retain and make use of, and not to lose, its
therapeutic powers.
This objective is very much in keeping with the overall idea of
Waldorf education. Remember that in the introductory courses given
to the first College of Waldorf teachers, “Allgemeine Menschenkunde”
(Study of Man) and “Methodisch-Didaktisches” (Practical Course for
Teachers), Rudolf Steiner points outs: “The aims of Waldorf education
are derived from the spiritual tasks of our age. In order that men can
fulfill those tasks, the method of teaching in the widest sense should
always have the aim of harmonizing the ‘upper’ and the ‘lower’ man,
the spirit-soul and the body (Leibeskörper). In Waldorf education, any
subject, be it botany, history, gardening or shorthand, is to become
a tool in the hands of the teacher with this aim of harmonizing body,
spirit, and soul.
Arithmetic and geometry, too, can work powerfully towards this
harmony. Educationally they are in the first place not knowledge to
be learned, but tools. How does arithmetic work upon the child? One
of the first things we are meant to learn as Waldorf teachers is the
following: In every subject we bring to our classes we must distinguish
between ‘conventional’ knowledge on the one hand, and on the other
hand knowledge which is based on the understanding of man’s being,
spiritual and physical. Conventional knowledge as Rudolf Steiner
broadly defines it, is founded upon human convention, for example,
reading and writing. Our alphabet is a convention. Conventional
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knowledge works only upon the head-nature of man, whereas
knowledge of man, of his spiritual nature, links us to the spiritual world.
Further, we read in ‘Methodisch-Didaktisches’ (all of this is in the first
lecture):

Wir unterrichten im Gebiete des Allerphysischesten, indem wir
Lesen und Schreiben unterrichten.
Teaching reading and writing we teach in a realm that is most
physical. That is teaching conventional knowledge.
Wie unterrichten schon weniger physisch, wenn wir Rechnen
unterrichten.
We teach already less physically when we teach arithmetic.
Und wir unterrichten eigentlich den Seelen-Geist oder die
Geist-Seele, indem wir Musikalisches, Zeichnerisches und
dergleichen dem Kinde beibringen.
And we actually educate the soul-spirit or the spirit-soul when
we bring near to the child that which is of an artistic nature—
music, drawing, etc.
Now follow very significant sentences:
Nun können wir aber im rationell betriebenen Unterricht
diese drei Impulse des Überphysischen im Künstlerischen, des
Halb-Überphysischen im Rechnen, und des Ganz-Physischen
im Lesen und Schreiben miteinander verbinden, und gerade
dadurch werden wir die Harmonisierung des Menschen
herverrufen.
In all education that is handled rationally these three
impulses—the spiritual in the artistic, the half spiritual-half
physical in arithmetic, the physical in reading and writing—can
be combined with each other, and it is just thereby that the
harmonizing of man is called forth.
Can you already see the mediating role arithmetic can play if it is
handled rightly? Everything artistic works upon the will-nature and
involves the whole being of man. The conventional works on his head-
nature only. Therefore:
Wir müssen auf künstlerische Art Konventionelles lehren
in Lesen und Schreiben, wir müssen den ganzen Unterricht
durchdringen mit einem küinstlerischen Element.
We have to teach creatively, artistically, the conventional in
reading and writing; we have to permeate all education with an
artistic element.
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Arithmetic, however, stands midway between art and convention, as
the soul stands midway between spirit and body. Speaking of subjects
that work upon the physical and etheric bodies (such as drawing which
leads to writing, and the study of plants) and of other subjects that work
upon ‘that which leaves the physical body and the etheric body during
sleep’ (e.g. the study of animal and man), Rudolf Steiner goes on to
say:
Rechnen Geometrie spricht zu beiden. Das ist das
Merkwürdige. Und daher ist wirklich in bezug auf den Unterricht
und die Erziehung Rechnen sowohl wie Geometrie, man
möchte sagen, wie ein Chamäleon; sie passen sich durch ihre
eigene Wesenheit dem Gesamtmenschen an. Und während
man bei Pflanzenkunde, Tierkunde, Rücksicht darauf nehmen
muss, dass sie in einer gewissen Ausgestaltung in ein ganz
bestimmtes Lebensalter hineinfallen, hat man bei Rechnen und
Geometrie darauf zu sehen, dass sie durch das ganze kindliche
Lebensalter hindurch getrieben werden, aber entsprechend
geändert werden, je nachdem das Lebensalter seine
charakteristischen Eigenschaften verändert.
Arithmetic and geometry speak to both. This is so
remarkable. And therefore, arithmetic and geometry are, as
regards education, like a chameleon; through their own essence
they adapt themselves to the entire human being. And, whereas
with botany and zoology we have to consider that they are to
be given in a quite definite form at a definite age, with arithmetic
and geometry we have to see to it that they are practiced
throughout childhood, but changing in form according to the
changing charac-teristics of the age, of life.
After having considered the place of arithmetic within education,
and having seen that it actually arises in human nature, we realize that
the Four Rules are not foreign to our human faculties in the way the
letters of the alphabet are foreign to it. We are, however, also aware
that an intellectual element can very easily be given to the child much
too early; and a great deal may be spoiled in the child’s development.
The child should be introduced to mathematics in a way “which can
only be decided upon by one who is able to survey the whole of life
from a spiritual point of view.”
The introduction of the Four Rules, namely addition, subtraction,
multiplication and division, is prescribed for the First Class but in a
rather unusual way, as we know. In contrast to everyday usage, the
process of addition is to be brought near to the child’s understanding
by proceeding from the sum, or total, to the parts—just the opposite of
what is usually being done. Similarly, multiplication is to be introduced
by proceeding from the product. But still more obscure may we find
the indications regarding subtraction (to start with the remainder) and
division (starting with the quotient). Only after this special introduction
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of the four processes, only after having secured a glimpse of analytical
under-standing, should the Four Rules be practiced in the form in which
they are commonly used: synthetical, beginning with the parts, and
finding the total or product, etc.
Now arises the question: What is the meaning and advantage of
the analytical way of introducing the Four Rules? The advantage is not
necessarily greater skill in arithmetic, but is of a moral nature. That is
surprising. Rudolf Steiner often spoke about the relationship between
arithmetic and morality. In the course for teachers given at Oxford
(1922) he says:
In this way (the analytical way) we get the child to enter
into life with the ability to grasp a whole, not always to proceed
from the less to the greater. And this has an extraordinarily
strong influence upon the child’s whole soul and mind. When
a child has acquired the habit of adding things together, we
get a disposition which tends to be desirous and craving.
In proceeding from the whole to the parts, and in treating
multiplication similarly, the child had less tendency to
acquisitiveness, rather it tends to develop that which in the
Platonic sense, the noblest sense of the word, can be called
considerateness, moderation. And one’s moral likes and dislikes
are intimately bound up with the manner in which one has
learned to deal with numbers.
A little further on Rudolf Steiner concludes:
Thus what comes to pass in the child’s soul by working with
numbers will very greatly affect the way ho will meet us when
we want to give him moral examples, deeds and actions for his
liking or disliking, sympathy with the good, antipathy with the
evil. We shall have before us a child susceptible to goodness
when we have dealt with the teaching of numbers in the way
described.
One more statement I would like to quote in this connection,
which can make us aware of the immense importance Rudolf Steiner
attached to the proper use of mathematics in school:
If, then, man had known how to permeate the soul with
mathematics in the right way during these past years, we should
not now have Bolshevism in Eastern Europe.
The first part of this remarkable statement about communism,
namely that men did not know how to permeate the soul with
mathematics in the right way, brings us to another equally important
aspect. It concerns the teacher himself, his obligation to know what
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