Photo: © Charlotte Fischer

The unity of the world as starting point

How do we introduce numbers in Waldorf education? We start with a unity. That corresponds to the child’s experience of the world because the child is one with the world. All other numbers – two, three and so on – arise through dividing one. For one we could take the image of the whole day which divides as two into day and night. Or for four the year which divides into the seasons. For five the image of the rose with its five petals. The opposite of this approach would be to repeat the one: 1+1 is 2, 1+1+1 is 3 etc. At some point the numbers disappear into infinity.

Thus the whole is divided through the numbers in Waldorf school. It is a challenge for class teachers to find examples of such division which arise out of the matter itself, for example that the rainbow has seven colours or the year twelve months. The thought that the whole is divided continuously accompanies arithmetic in lower school. Thus adding is not introduced by saying that three plus five is eight but that eight is equal to three plus five, but also four plus four and so on.

Thus the child unconsciously becomes familiar with the qualities of the different numbers. A twelve can be divided differently to a seven. In this way the child obtains a personal relationship with the numbers. Each number acquires its specific character and this character can always be found again inwardly. In this way the numbers do not slip away into infinity.

We might well use chestnuts as an aid when the children start to calculate with numbers but the three is not a chestnut in triplicate but an independent number. That is the point to which the children have to be brought. That is, to do arithmetic without chestnuts. The child must be given the opportunity to experience the non-sensory side of a number. It is our task when doing arithmetic to satisfy the need of the child for a non-sensory perception of numbers. This need is just as pronounced in the child as the need for sensory visuality. It is a fundamental soul need of the child for non-visuality outside the senses which is met by mathematics and arithmetic.

What conversely does it mean for human beings if we learn about numbers as the product of addition? It sets up a way of thinking which is cumulatively additive, by tendency heads towards infinity and is basically purely quantitative in nature. What does that mean from a social perspective? There is a difference whether we learn about numbers by dividing a whole or whether we learn about them as something that can be continued additively to infinity. After all, social problems only arise because they are seen as the sum of single items and not as parts of a whole.

The task of mathematics lessons in the lower classes is therefore to establish the foundation for a certain form of thinking which does not proceed in an egoistically cumulative way.

Numerical series are also good for the memory

Children learn in series. If we are fortunate enough as mathematicians to go a monthly school gathering, then we will hear the children recite the series there. The class is divided into groups and one of them recites the two times table, another the three times table, etc. That is an immense experience because we notice the different qualities of these numerical series. The five times table for example has something simple about it. When we hear the seven times table, in contrast, it does sound quite a bit more interesting. In other words, the children develop a feeling for the special nature of the numerical series. When we have a panorama of such numerical series, then there are very special numbers in between which no one recites: 37 for example.

Thus the children not only develop a relationship with the individual numbers as non-visual soul content but they also develop a very special qualitative relationship with the numerical series. That stabilises the relationship with numbers immensely. Such a way of doing things which is practiced in the Waldorf schools strengthens the memory. Mathematics lessons in the lower classes primarily serve memory development.

When the aim is to get a feeling for the nature of numbers, it is justified to have the children do calculations without understanding, as it were. Because the child learns not only through understanding but human beings learn through doing. This is taken into account in Waldorf education in that nothing is explained to the lower classes when they do arithmetic but it is done and practiced.

The world of numbers belongs to the inner world of human beings. The task is to let children experience their inner world. This soul world is simultaneously the centre of the imagination. If we do not give children the possibility to become acquainted with this inner world at all, we prevent the development of the imagination. The imagination is ignited by the outside world but it is an inner soul experience. With the world of numbers the inner soul world is thus opened up and structured. Through practice of the times tables the children first of all learn about the quality of the individual numbers. All mathematics teaching builds on that.

Fractures at the Rubicon

In class 4 the relationship between the child and the world of numbers changes. The increasing distancing of the child from the world means that the time has come to become acquainted with a new form of arithmetic. Now we no longer divide but create fractions. In doing arithmetic the child experiences what they experience in their changed relationship with the external world: the unity with the world is broken.

A very simple example: a stick is broken in the middle. Such a breakage must be experienced and learned in doing arithmetic. The laws of fractions are communicated by means of a picture. It is really not that important whether or not a class 5 pupil understands precisely the result of the calculation two thirds time two fifths (he might well not have understood it yet in class 9 either) but the important thing is that they should gain access to this inner gesture of fractions. In calculating fractions the teacher also works very much through the memory.

Negative numbers and deficits in puberty

In class seven the pupils are in the middle of puberty and quite new inner emotions, longings but also deficits appear. It is only at this time that the Waldorf schools, in contrast to the state schools, deal with negative numbers. Concerning ourselves with negative numbers corresponds to this new state of soul.

What is a negative number? Nothing would be more devastating than if we were to attempt to communicate a negative number through an outer perception. The negative number must be experienced. If we have five and give one away then we are left with four. If we give two away we are only left with three. And if we give more away than there is to give away then we have a deficit. And this soul feeling of deficit lies behind the negative number.

We are on a cycle trip on a hot day: let us imagine we cycle the whole day and have too little to drink. We arrive home in the evening and drink three bottles of water. Then we know what minus three is. This experience is the crucial one for understanding a negative number and not if it says minus three degrees on the thermometer. The latter is not in any way connected with such an inner experience. It is determined purely arbitrarily.

It is even more devastating if the pupils are presented with a number line on which the negative numbers carry on to the left beyond the zero and the positive numbers simply continue to the right. It is of course possible to introduce a number line with older pupils. But if we want to communicate a relationship with numbers to the pupils which corresponds to the emotional state of a class seven pupil then the number line is totally out of place here.

So what are we actually doing in mathematics? We are transferring an event which takes place in our soul into our thinking. In other words, mathematics lessons in the Waldorf school support the development of the pupils’ thinking. And we all know that when we put something into thought we become a little bit freer. The prerequisite is that we were able to communicate an inner relationship with numbers. If that basis does not exist, no thoughts can be formed in this respect either.

The ordering power of thinking in chaos

Rudolf Steiner specified combinatorics for the curriculum in classes 8 and 9. Take a simple example. We have four bells and each bell rings once. How many ways of ringing the bells consecutively are there? Even if this example appears straightforward, it cannot be solved off hand. What can the first bell do? It can ring first, second, third and fourth. So the first bell has four possibilities. Now it is first. What possibilities does the second bell have? It has three possibilities, but take heed, it has another three possibilities if the first bell comes later. In other words, in each of these places in which bell one rings there are still two, three other possibilities.

That means ultimately that there are 24 possibilities of creating a tune with four bells. Or with four people sitting in a room there are 24 possibilities to change the seating order. If you try that with the pupils, there is immediate chaos of course – as planned. Although we are only talking about four chairs, it is unlikely that these four pupils will discover all the 24 possibilities. Even if only because it is impossible to remember all the possibilities that have already been tried – at the twenty-fourth possibility that would, after all, be 23 variations. But these so-called permutations are fully comprehensible in the thinking.

Amidst the chaos of this age the study of combinatorics helps to find orientation in the thinking. The thinking takes an ordering role and it does so in an elementary form. Combinatorics is not a gimmick. That is to say, in the mathematics of class 9 we have a non-visual encounter with the world. The key educational factor for the pupils is to experience the ordering power of thought.

The experience of producing objective thoughts

In Waldorf education, geometry remains separate from the number side for a very long time. It is not until class 10 or 11 that geometry is linked with algebra. The reason for this is that the pupils in class 8 and 9 are not yet meant to do mathematics graphically but should still connect with where they stand emotionally. Because if we take a geometrical formula we immediately have a specific geometrical shape before us which does not of course exist in the external world in this pure form. The crucial thing is for the pupils to experience the curve-creating power.

When we deal with differential quotients for example, the main subject in class 12, there are two ways of doing so: the one is very visual and the other is not visual at all and arises from the numbers. Here particular value is placed on first going the non-visual route before we embark on the visual one. The non-visual should therefore not be illustrated. We support the child mathematically on the non-visual side. Proceeding in this way helps the development of the thinking with a lifelong effect. The important thing is to have assimilated the basic mathematical gestures; and they form the human being.

The key thing about mathematics is the experience of producing thoughts through our own activity which are at the same time wholly objective. It must be possible for the pupils to experience in class that the soul produces something which is not present externally in the world. Because the human soul thirsts for more than visual perception.

As mathematics teachers in a Waldorf school we have the task of supporting the development of the children with the corresponding forms of thinking and to give the pupils the experience which they can take with them of becoming familiar with a part of their soul which is not visual. Mathematics is about becoming liberated in our thinking from sensory impressions.

About the author: Dr. Christoph Kühl is a mathematics teacher at the Uhlandshöhe Free Waldorf School in Stuttgart. This article is an edited version of a lecture.