When the spherical cap smiles. Mathematics as experience and art

By Stephan Sigler, October 2014

Mathematics lessons in middle and upper school are unfortunately only rarely suspected of being places where imagination, an artistic and aesthetic sense and real independent judgement can develop. Rather it is thought to be necessary to prepare the historically grown pearls of wisdom of this subject so that the pupils can understand them. A good mathematics teachers is one who can “explain well” – that at least is the received wisdom. In this sense mathematics is a despot: pupils have to accept it as it is and adapt. If this works, mathematics lessons are deemed to have been a success.

Rudolf Steiner in a lecture on 5 September 1919 in his general education course (The Study of Man) called for something quite different: it was quite possible to make greater demands of children in the early years at school as they learn to read and write with more of an emphasis on the development of the intellect. But then on no account should there be a failure in middle school to make the imagination part of the growing ability to make judgements.

A remarkable statement: better to develop the intellect more strongly in the first years of school than to omit introducing the imagination into the ability to make judgements of pupils in middle school!

How can we deal with such a statement in practice? And how do we do so at an age in which the power of the intellect begins to stir mightly, in which the cleverness it contains is used with calculated precision, but in which the arguments are also used with incredible flexibility and in which everything is presented in such a clever way that what is said is undoubtedly correct. Even if five minutes earlier precisely the opposite was argued, which does not however lead to the consequence that what has just been said is withdrawn. Our thoughts follow our wishes and we legitimise what we happen to want at the time. Logical thinking is an incredible help in this respect but does not, of course, guarantee that such thoughts bear any relationship with reality! Logic serves as the agent of our own wishes.

Mathematics supports the imagination and judgement

What support does the thinking need to be able to liberate itself from the life of our wishes? So that judgement is sparked by the facts of the world and in turn leads back into the world? So that it becomes our very own thinking and judgement?

And what role does the imagination play? Here is an example from a mathematics lesson in middle school which starts with the following thought exercise:


In the lesson, the pupils have to create these shapes in their imagination based on a description given by the teacher. The teacher’s words have to be chosen in such a way that the intended shapes and their movements can develop inside us – and with great clarity. Such an exercise of course has to appeal to the imagination in order for pictures to arise in the first place. But the pupil is no longer free in the composition of the pictures but guided into something that has to be precisely like this or like that. But it is my imagination, and thus it is I, who produces the phenomenon. It only appears because I produce it. I enter fully into the described context and the processes and live in them. I completely get away from myself, identify with the matter in hand. There is an unconditional fusion between the object and myself. I live in the visualising consciousness – but  more on the productive side of the imagination, that is, on the side of the will and not on the side of reflective reason.

Then a drawing of these various stages of the process is made in the lesson, the length of the sides of the rectangles is determined as they are drawn and the circumference and area calculated and noted in a table.

The first stage: g and h are the respective sides of the rectangle; U is the circumference, A is the respective area.

Summary of the different stages: side length, circumference and area of the rectangles

We note with surprise that the two rectangles always have the same area no matter where we put the intersection of the horizontal and vertical lines. The summary in the table makes this shape accessible to the life of the imagination. But the thought exercise does not yet bring the kind of clarity which extends down as far as the numbers for the measurement of the lengths and areas. For that we have to break down the process into individual steps, each of which in turn has to be looked at and analysed to be pictured. But that enables us to understand the correlations shown by the shape: now we know. At the same time we get an overview of the progression and can understand the qualities of the changes which take place in the course of the movement more precisely. The main lesson can be concluded with that.

The first step in the next day’s lesson takes a completely different form in each class. The activity originates wholly from the pupils who now discover the laws inherent in the shape which, on the one hand, lie in the number sequence in the table and, on the other, in the equal area of each of the rectangles.

The trained eye of the mathmatician sees the proof immediately:

The diagonal line divides the rectangle into two right-angled triangles with the same area. Each of these triangles contains a small dark grey and a slightly bigger light grey triangle. If we remove these two smaller triangles from the larger triangles, we are left with each of the yellow rectangles. They must have the same area because something identical was removed from each of the equally-sized starting triangles. If identical amounts are removed from two things that are identical, the area of what remains must also be identical.

We can see that all the elements which were used for the proof already had to be produced in the thought exercise. It is a matter of how we look at things for this proof to be “seen”. The task of the teacher thus is to guide the eye to all these elements through an open question. Such a question might be: “How many triangles can you see in this shape?” The correct answer, “six”, creates an awareness of the simultanous presence of all the geometrical elements required for the line of thought. The thought turns into perception at the moment that the evidence is experienced. It is not the pupil who creates the evidence but it arises from outside; the parts of the puzzle, which initially seem to have no context, fall into place as if by themselves. The eye of the imagination has consolidated to the point where it has turned into knowledge.


The pattern of rectangles which are equal in area to the green rectangle illustrates another law: the corner points lie on a hyperbola.

A way into the world

If we take another look at the course of the lessons, the pupils are immersed completely in the sphere of the will at the beginning in this thought exercise in order then to penetrate what has happened with their consciousness. The key thing is that nothing is added conceptually which did not exist (potentially) in the phenomenon beforehand. The conceptual context is already contained in the mathematical phenomena, which always have to be seen simultaneously as a process and event, and it only needs to be discovered. To this extent we can say that the life of the will of the pupils themselves is illuminated in the consciousness. The will processes become clear and transparent. The thoughts are born out of the life of the will which is ignited by the world. In this respect mathmatics lessons are also first and foremost an education of the will! As the pupils grasp reality, the educational process is realised which in truth is self-education: the pupils develop the life of their will from the inside out. They come to a standpoint of their own in and not in parallel with the world.

Do we still believe in “normal” lessons?

Fundamental questions follow from this example of a lesson: do we as Waldorf teachers still believe in the unconditional educational power of lessons framed by a subject in middle school? Or have we become resigned and actually only any longer place our trust in events, Alpine walking projects, cycle tours and other de-schooling measures, projects such as year projects and class plays which make “normal” lessons bearable, if not redundant; which actually, alongside eurythmy, make Waldorf schools into Waldorf schools?

Rudolf Steiner’s call for imagination in forming a judgement, for elegance and style to be introduced to mathematics and science lessons, which should not be “deadly serious” and “unpleasant” but which should have a punchline so that the spherical cap smiles when its area has to be calculated, justifies such trust. In artistic lessons, sensory matter should lead to the idea through the pupils’ imagination and power of thinking. What an incredible challenge for class teachers who as a rule rarely have the time to work their way into the individual subjects to such a depth. Yet it is clear that there is no way round such a deep study of the subject because that is the only way to access the material which is then meant to appear in the classroom in the form of the idea. Without the subject matter, without the corresponding field of the world as material, lessons at best remain Platonic in that great tableaus are “talked at” the pupils, and at worst only amount to intellectual instruction – the superficial froth on the surface of an infinitely deep ocean.

The conclusion to be drawn from all of this is that in the debate about the length of time that a class teacher should be in charge of a class we should not always sweepingly play off the “so-called experts” against those who always have to “reinvent the wheel”. We should at last focus once again on the lessons and thus particularly also the subject matter in the sense described above. And we should therefore focus on the conditions which it is necessary to create to make it possible for the teacher of a (middle school) class to come to grips with the aspect of the phenomena and the aspect of mind with regard to the lesson content in such a way that the spherical cap does indeed smile occasionally and the process in the lesson can become an artistic one.

About the author: Stephan Sigler is a lecturer at the Kassel teacher training seminar and a middle and upper school mathematics and geography teacher at the Kassel Free Waldorf School.