Only through mathematical language has it become possible to order phenomena into relation that are comprehensible for understanding, into measurable, precisely ordered relations. The only language that is suitable for all measurable phenomena: a language that makes itself transparent to make the relationship between processes visible. A contribution by Constanza Kaliks, a passionate maths teacher.
In the course that Rudolf Steiner gave in Ilkley in the summer of 1923, he begins the first lecture with three perspectives that were fundamental to his pedagogical impulse, and which are astonishing in their relevance today: the equality of women and men as a condition for school, the fact that the school founded in 1919 was intended for children from working-class families, for proletarian children – a «school of humanity developed from the proletariat»1 – and anthroposophy as a perspective of knowledge. Equality, social responsibility and cognitive orientation are presented as conditions for pedagogical action:
«So, the fact here is that the pedagogical structure has emerged from a social root, which, with regard to the entire teaching spirit, to its entire teaching method, seeks its root in anthroposophy.»2
Among the many educational and anthropological topics addressed by Rudolf Steiner are included the methodology of a living science that seeks to recognize human beings in their development, and education as a moral, free act.
Particularly noteworthy is what was and is being taken up in many schools, namely, the beginning of arithmetic lessons. It may be surprising what Steiner shows, in a straightforward manner, a methodical approach to the four basic arithmetic operations. The approach can be understood as a variation on traditional forms of computing, and certainly as a valuable exercise in rethinking. But if one looks at the paradigmatic character of the course itself, at the attempt to set new anthropological perspectives for a pedagogy of the present day, this exercise in «other arithmetic» takes on an almost shocking dimension. Shocking in the sense that the small child, in the very first beginnings of learning, is made to experience the greatness of this learning.
Modern natural science was made possible, and conditioned as much as anything, by mathematical language: through the language of mathematics, it was possible to order phenomena into relationships that were comprehensible for understanding; into measurable, precisely ordered relationships. The phenomena are mathematically represented in relation to their quantifiability and expressed in equations. This makes it possible to calculate what the change in one variable means for the other variable. In describing planetary orbits and predicting the positions of the planets, mathematics is also fundamental to the astronomical revolution – a paradigmatic field of knowledge at the beginning of modern times. A single language that is suitable for all measurable phenomena: a language that makes itself transparent to make the relationship between processes visible. And it is slowly conquering all areas of knowledge: measurability is sought not only for physics, but for varied processes in almost every discipline. The mathematical expression is considered a guarantee of comprehensibility. Whether an equation is an expression of the relationship between wealth and education, or of gasoline consumption per kilometre driven, or even of psychological states and hormonally determined values - it cannot be determined primarily from the equation itself. The phenomenon described must be explained in addition. Since modern times, mathematics has been decisive for most areas of knowledge: a universal language. Major developments followed and continue to follow this achievement. At the same time, the question arises as to what is not captured by this measurability.
What cannot be captured in the equation are the properties of the phenomena described. They disappear - the language of clear attribution does not express them as such. In this sense, mathematics presents processes as abstractions: many of the properties that are specific to them, and that distinguish them are not expressed in what can be conveyed in the measurable.
On the other hand, the equation represents the knowability of the world like hardly anything else: knowing is a matter of putting oneself in relation to something perceived, something visualized. Through recognition, a relationship is perceived and understood. In the transition from the Middle Ages to modern times, Nicolaus Cusanus (1401-1463) writes impressively about the importance of numbers for the cognitive grasp of relationships:
«All research consists of establishing relationships and drawing comparisons, although this may be easier at one time and more difficult at another. The infinite as infinite is therefore unknowable because it defies all comparison. Relationship means agreement on one point, and in the same instance, difference. It cannot therefore be imagined without a numerical ratio. The number therefore encompasses all phenomena that can be brought into a proportional relationship to one another.»3
Rudolf Steiner called Nicolaus Cusanus a «marker» of his time4 and attributed to him a crucial role in what happened at the beginning of modern times and what was to develop from it.5
Steiner now suggests that teachers start from unity, from the whole. Counting should not be experienced as a cumulative addition, but rather as a structure of unity. This articulation of unity continues to resonate in the life body, in the etheric body of the child.6
When adding, it is also important to start from the whole and then go to the parts; in other words, from the sum to the addends:
«One starts from whatever the whole is and moves on to the parts. Then you will find your way back to normal addition.»7
It is suggested that arithmetical operations be practised at the beginning of arithmetic in such a way that the focus is on the relationship that can be experienced and not on the result. First graders are unfamiliar with the techniques for solving simple equations. The empathetic, searching insertion into the comprehensible, given relationship is practised by not asking the question: How much is seven plus eleven? but by trying to discover, to trace: for eighteen to be eighteen, what must be added to seven?
A different kind of alertness arises – as an adult you have to mentally and contemplatively immerse yourself in the process to feel the different quality. In this way «the living thing comes into the matter»,8 one can experience how the tension of the relationship is present in the equation, how a fine feeling for the connection becomes active. This tension brings liveliness to the experience, enlivening the lessons – yes, the different types of calculations can be «set up again» from the head.9
«Let us help the children in such ways that their physical and etheric bodies continue to work in a healthy manner. But we can only do this if we really bring excitement, interest and life, especially into calculation and geometry lessons.»10
«One addresses the whole human being when one considers the unity, and from there moves on to the numbers; when one considers the sum, the difference, the quotient, and the product, and from there moves on to the terms.»11
Rudolf Steiner's suggestion is also important if one considers the foundations that René Descartes laid in his discours de la méthode which, in many respects has become a foundational orientation in science and also, although often unspoken, in didactics.
Among the four rules that Descartes recommends following, without exception, to achieve certainty in the sciences, he mentions the principle of division, the analytical procedure and also the principle of always moving from the simplest and easiest of the things to be known to the most complex.12 These principles are fundamental to the scientific process – but for the learning experience, which includes a moral dimension in knowing the world, it requires a connection to the whole, and also the courage to go from the complexity of reality into the details, and the richness of the real – not the addition of individual parts that may be replaceable or even superfluous – can be experienced as a starting point for learning.
This horizon, and thus the moral dimension, is addressed in Steiner's proposal for the beginning of arithmetic lessons. Especially when it comes to arithmetic and mathematics, attention should be drawn from the outset to morality. It is as if we wanted to experience, right at the beginning of learning, that cognition is a living in relationships, in relationships that live in an infinitely differentiated context that opens to learning.
Constanza Kaliks
This text is a summary of a contribution at the meeting of the Hague Circle - International Council for Steiner Waldorf Education from November 16 to 19, 20
References
1: Rudolf Steiner. Gegenwärtiges Geistesleben und Erziehung. GA 307. Dornach5 1986, S. 14.
2: Ebd., S. 15
3: Nikolaus von Kues. De docta ignorantia. Die belehrte Unwissenheit. Buch I, Kapitel 1, 3, S. 7f. Hamburg: Felix Meiner Verlag, 2002.
4: Rudolf Steiner. Anthroposophische Leitsätze. GA 26. Basel: Rudolf Stzeiner Verlag, 2020, S. 144.
5: Vgl. Constanza Kaliks. Erkenntnis als Menschwerdung. Nikolaus von Kues und Rudolf Steiner. In: Peter Selg / Marc Desaules (Hg.). Anthroposophie: Erfahrende Wissenschaft des Geistes. Arlesheim: Verlag des Ita-Wegman-Instituts, 2023.
6: Vgl. Rudolf Steiner. Gegenwärtiges Geistesleben und Erziehung. GA 307. Dornach5 1986, S. 183f.
7: Ebd., S. 185.
8: Ebd., S. 183.
9: Ebd., S. 184.
10: Ebd., S. 181.
11: Ebd., S. 187.
12: René Descartes. Discours de la méthode. Bericht über die Methode, die Vernunft richtig zu führen und die Wahrheit in den Wissenschaften zu erforschen. Stuttgart: Reclam, 2012, S. 39ff.