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Letter angels? Arithmetic dwarfs?

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A plea for objectivity and creativity in the teaching of writing and arithmetic.

Arithmetic dwarfs for first graders, angels for language education and nocturnal elves? Does it make sense to include elemental beings and angels in school lessons or is this simply esoteric sentimentalism? An article by Thomas Wildgruber.

«Why are you using five angels to teach your 1st grade students about vowels?» I often asked this during my visits to Waldorf schools in Central and South America. «Because I was told that's how you do it.» I get about the same answer when I ask why, of all things, four arithmetic gnomes teach the arithmetic operations to the first graders. Or: «Why do you tell the children that elves drew the chalkboard picture they see every morning?»

I want to know where these conventions come from. My search for clues gets lost in the worldwide web of digital advice for aspiring and searching Waldorf teachers. In Rudolf Steiner's suggestions for learning to write and for the first arithmetic lessons, no trace of letter angels and arithmetic dwarfs can be found!

Waldorf supermarket

I am immersed in another world when I follow some instructions based on tradition. There I can read, for example: «Each vowel has its own angel who brought the soul sound to the people.»1 Or, «Steiner suggested that the vowels sound from a different depth than the consonants. In the first grade of Waldorf school, the vowels are introduced with body gestures and feelings, and the vowel angels bring the sounds over the rainbow bridge to earth.»2Even more colourful and commercial are the results for «math gnomes», or «arithmetic gnomes». I can buy them, make them according to instructions, and also copy the matching stories.

This digital Waldorf supermarket makes teachers very comfortable. Why should I bother to think up something myself? The figures, the pictures, the stories are already there. All I have to do is access them.

Does it make sense to make these elementary beings the carriers of the four arithmetic operations? In the work of Rudolf Steiner, one finds much about the worlds of the spiritual hierarchies. He also characterizes vowels and consonants; for example, in relation to planetary and zodiacal forces. He also speaks about archangel hierarchies, which trained the language tools for man. However, he does not speak about the fact that exactly five angels gave the vowels to the human language. Also, he describes in a lively way the nature of the dwarfs in their relation to man3, but not exactly as «sweet math dwarfs».

So, what is Rudolf Steiner's original indication? How does he characterize the vowels for the first graders, which then coagulate into the letter form?

«When you teach the letter A, for instance, you can say: ‘Think of the sun rising in the morning. Can any of you remember what you did, when you saw the sun rise in the morning? ‘ If the children don’t remember you can help them by describing how they stood there, saying, ‘AH!’ because the sunrise was so amazing. Let them experience that you imitate a feeling when you pronounce a vowel.»4

Proceed from the whole

So, we see his suggestion is that we could start from the immediate real world of children's experiences and then approach the shape of the letters. Likewise, Rudolf Steiner takes his suggestions for the initial teaching of maths from life experience. For example:

«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 – a 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…

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?' Then 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.»5

So, why should we bother angels and gnomes for these initial lessons! It is also possible to do it more vividly, as Steiner shows. He does not demand that the children rise with their imaginations into spiritual worlds. We prepare the children for life on earth. Therefore, I would say: let us free ourselves from such easily offered and little digested traditions. Let's rather take real figures from the children's lives for the arithmetic operations and vowel letters! Let's leave aside the esoteric sentimentalism! Unless someone really has clairvoyant experiences with angels and elemental beings and can responsibly implement this into the teaching.

Here is an example from the teachers' seminar of the Centro de Desarrollo Antroposófico in Mexico, where in the summer of 2021 we tried, according to Rudolf Steiner's suggestions, to create a story from children's experiences, and to translate that into a series of didactic steps and drawings.

«You will always experience a quiet inner joy when you develop a form from some animal or plant and transform it yourself into a letter. And this joy will live in your teaching and be transmitted to your pupils.»6

«What we also aspire to in our teaching is freedom…You will not attain freedom by cramming your head with information…, but by bringing active movement to your own soul life. What exactly you end up doing can differ from one teacher to the next...But you need to be devoted to your teaching. Then you will be free.»7

Thomas Wildgruber


Thomas Wildgruber was a classroom teacher and subject teacher for art, handicrafts and art history in Germany from 1979 to 2011. Since 2011, he has been a trainer and teacher trainer at schools, teacher training colleges and universities, specialising in methodology and art didactics, as well as a school consultant in Europe, Asia and Central and South America. Publications: «Malen und Zeichnen, 1. bis 8. Schuljahr» (German, English, Spanish, Chinese)




2: whywaldorfworks oder waldorfmaterialparaiberomerica

3: Rudolf Steiner: «Man as Symphony of the Creative Word» (GA 230), Lecture from November 3rd, 1923

4: Rudolf Steiner: «Methods of teaching» (GA 311), Lecture 5, August 26th, 1919

5: Rudolf Steiner: «The kingdom of childhood» (GA 311), Lecture 5, August 16th, 1924

6: Rudolf Steiner: «Methods of teaching» (GA 311), Lecture 5, August 26th, 1919

7: ibid

Photo: PädSekPhoto: PädSek