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Alan speaks in a very symbolic and esoteric manner in some parts of his books. Although they can be read anthroposophically, passages speaking of Atlantis, archangels, gods, etc. do not need to be taken literarily to be meaningful. The more you read, the more you will realize he uses many different religions to express ideas in a symbolic manner and not in a religious manner. His writings are not religious. In some places his writings are meant to refer to religious events in a historical way. In some places he is using religious figures (from Christianity, Judaism, Islam, Buddhism, Hinduism, Paganism, Ancient Roman and Greek Religions, etc.) in a symbolic manner. However, at no point is he promoting a specific religion or speaking from a religious point of view.
I have kept the writing as close to one-hundred percent original so you will also find that he speaks of Australia often and some spelling or manners of speaking may be cultural. Any words I have changed are presented like this: <word>.
Also keep in mind that these books are written by a Waldorf teacher with decades of experience who also studied with a Steiner student himself, so he speaks to an audience that is dedicating their lives to the Waldorf method without exception.
Because of this, all of his views are not reflected in the Earthschooling curriculum and not all of them may be ones you want to embrace or are able to use. In all of Alan Whitehead’s writings the opinions are his own and may not align with Earthschooling or Waldorf Books. In some cases, we will be updating some of these chapters in the future with additional and/or updated information.
Ultimately, however, as I read through these passages I find I can distill wisdom from even those paragraphs that do not resonate with me.
We invite you to read with an open mind and heart and with eagerness to learn and discuss…
A SPIRALLING (RABBIT) POPULATION
Signs & Symbols – Class 3 – Main Lesson
“We saw a bunny!” panted the twins as they ran back to the shady spot by the creek where the Sensible family were picnicking. The cloth was spread, and they were all starving.
“And there are lots more there too – we saw their holes.” puffed Lily as her brother Tan was distracted by the lovely lunch.
“I daresay there are,” said Mother wryly “rabbits are noted as being prolific breeders.”
“Rabbits? Breeding?” said Father thoughtfully from a reclining position “That reminds me of the little number puzzle created by a certain Italian gent. It showed without a doubt how the Divine Hand ordains nature through the wisdom of number sequence – let’s see…beginning with 1 pair of rabbits; how many pairs can be produced from the original pair in a year, if each month, each furry couple begets a new pair, which the 2nd month on becomes productive?” Everyone looked blank at this.
“About a millions?” offered Tan, his 9-year-old enthusiasm for a peanut butter and jam sandwich unabated.
“Actually it’s 12 squared, or 144.” Revealed Father, sadly coming to the realization that he was not the sire of a pair of mathematical genii. “It’s not hard to work out on paper…really. But the curious harmony of the number sequence interested this Fibonacci (that was the gent’s name). The numbers were 1 – 2 – 3 – 5 – 8 – 13 – 21 – 34 – 55 and so on. Each number is obtained by adding the previous two.
The 12-ness is the relationship indicates (12X12 in 12 months) the presence of the sun, in relation to the zodiac. Wherever we see 12-foldness in the world, we can fairly safely assume this.”
Actually Father was speaking only to his munching wife at this stage; Lily was picking the sultanas out of a bun (‘I only like the sultanas.’), and her blond-haired brother was feeding crumbs to a disciplined column of green ants. He looked up and spoke.
“That ‘square’ business? We learnt that at school last week – where you put a little number above a big one. Nyyl, our teacher, called them Power Numbers.
He said that the little ones give power to the big ones – that’s weird isn’t it? 103, 10 to the power of 3, is 10×10,10, which is 10,000. You’d think the big ones would be the strongest.”
“Yes,” chimed in Lily, holding a sultana delicately aloft before committing it to eternal darkness “We’re doing a main lesson on Symbols, Terms and Operations. I like the symbols best; we’ve got really big plywood cut-outs of them up on the walls.”
“It’s so easy to see what Nyyl’s talking about when, say, he’s going on about ‘less than’. You know the one, the arrowhead pointing to the left.” Added Tan.
“And we’ve got the cut-outs all in different colors. Let’s see, there’s ‘greater than’ >; the negation slash /; the equivalence <=>. That took me a while to get, it means…”
“I know what it means,” said Father sitting up at last “it means one whole set of operations, or one sum is equal to the sum (with its different operations) on the other side of the symbol. How about the parenthesis, and the bracket – did you learn them? As you’re being introduced – and I hope at this stage it is only an introduction – to power numbers, did your teacher tell you about the root symbol – the inverse operation to powers?”
“You bet; I was good at those.” Chirped Tan proudly.
“Okay, what’s the square root of 144?” An uncomfortable silence was interrupted only by Lily rattling around in the basket.
“It’s 12 isn’t it Tan?” said Mother kindly.
“Yes that’s it, 12 – the bunny, I mean the sun number. But Dad, what did you mean by all that wisdom of nature stuff, you know, with the Fibrearcher numbers?”
“Fibonacci actually; well, his nominal title was Leonardo of Pisa, and he is still regarded as one of the giants of mathematics. He was certainly ahead of anyone else in the Middle Ages. This magical sequence has been found as the key to growth in the plant, animal and human world. The Fibonacci number-sequence is found in the seed-head of the sunflower, you know how the two spirals interpenetrate. It’s also in the sequence of leaf buds of almost all plants as they spiral up the stem – similarly it can be found in pinecones, you name it.”
“But animals don’t have spiral?” said Lily curiously “Hang on, yes they do – a snail shell’s a spiral. Is that a pizza spiral?”
“A pizza spiral?” queried Dad. Leonardo of Pizza – I know another one, animal horns, yes?”
Um, yes and no. Both the snail shell and (some) horns are logarithmic spirals, they conform to the Fibonacci series – and no, it is not Leonardo of Pizza – Pisa – Pisa!”
“It’s true isn’t it,’ intervened Mother, smoothing over her daughter’s pronunciation shortcomings “that the number pairs in the series all have the proportion of the Golden Mean?”
“Oh yes,” replied a calmer Father “5 and 8 is the approximate Golden Section proportion, but a higher pair, 35 and 55…”
“…are even closer I bet.” Said Lily earnestly from beneath her dark fringe. She stared at the twin spirals on the small, sweet pineapple – before she began to lick her lips!
“That’s right Lily; and the numbers get more accurate the bigger the pairs. Actually the Golden Mean proportion is impossible to establish absolutely – the decimal is non-repetitive. This imponderable, infinity principle points to its cosmic, rather than earthly origins.’ At least Mother understood!
“We learnt about the Golden Mean,” continued the irrepressible Lily “Nyyl said that if 1000 people were asked to simply draw a rectangle, then the average size of the drawings would be almost exactly the golden proportion. He did it was us – didn’t he Tan? – they weren’t that close though. He said the sample was too small; or that we were too artistic to do the predictable. That was nice of him wasn’t it? Although later he told us that lots of great artists use it all the time, in stink…stinky…?”
“Instinctively! It’s true though,” replied Dad “the magical proportion is approximately 1 to 1.618. The relationship of say our hand-length to our forearm – and stacks of other body parts.”
“Body parts?!” queried Tan in surprise; he seemed to have missed the context.
“Yes, you measure them.” Said his sister helpfully.
“Yuk! A fine turn this discussion’s taken.”
Father turned to Lily “But with all your symbol work, this must be an introduction lesson to Sets?”
“Um – yeees; we’ve been learning about those. I’m not quite sure what they’re for though, but they’re okay – especially the sums on the ‘number line’; you know, the one with the dots and arches. I even like those Venn diagrams; I make them into faces, moons and beachballs. Nyyl doesn’t mind, he says it makes our main lesson books look pretty. I still get the maths right though (sometimes anyway) – hee, hee.”
“There’s a lot of symbols in set theory isn’t there?” said Dad curiously “I believe the problem many children have with maths is their ignorance of the symbols, and by extension, the operations they represent. Most children can work the numbers okay, the process. Have you got classroom cut-outs of say, the (union); or the (subset) – and how about phi, F or f – the empty set?”
“yes, all of those,” said Tan, his consciousness ascending from Ant Land “I like the universal set like a – and the ‘is a member of’, the epsilon, the funny . Epson Salts I called it – ha, ha – because it…”
“you must be getting use to pronumerals; like x and y if you’re working on set theory?” said Mother quickly, ever-ready to short-circuit a social indiscretion “And what about the correct terms of the elements of the 4 algorithms? The answer is called something different with each one you know. With a division sum it’s the ‘quotient’; multiplication, the ‘product’; of course in an addition sum it’s called the ‘sum’; and in a subtraction they call it the ‘difference’. This might sound confusing I know, but these are internationally recognized mathematical conventions – and the sooner you use them the better. Rudolf Steiner always did.”
“We know – we know!” chorused the 9-summers duet “Hey, lets’ see if we can find some baby rabbits!” yelled Tan to his sister as they ran off. Actually he secretly had in mind the small story he had heard in the main lesson; the one about rabbits and Golden Mean numbers – something about a great winged golden rabbit which visited earth to bequeath harmony onto a chaotic nature!
Father lay on his back in the dappled shade “This 2nd maths main lesson in Class 3 seems to appeal to the children’s feeling life, evidenced by the enthusiasm they muster with an ‘imaginatively’ presented lesson.”
“Yes,” agreed Mother, ignoring the cleaning up and lying down beside him “With all that Golden Mean talk, I was taken back to the esoteric number schools of, say Pythagoras. They held number is such high esteem that their name for this whole science/art was “Archai’. Imagine, calling something so, well, so useful, the name of the highest beings of the 3rd or Astral Hierarchy. Another confirmation of maths being the ‘astral’ subject.”
“Indeed,” Father Sensible chewed amiably on a stalk of grass “although mathematicians have universal recourse to the astral ‘heaven’, there is much inspiration derived from the etheric.”
“Etheric?” queried Mother “Leonardo of Pisa was a ‘lion’ or etheric initiate – hence Leo the Lion was sent to Northern Africa, where he studied in the Islamic esoteric schools which fostered the Ancient Indian number mysteries – secrets of the Lion’ of life. Indeed the trimorph lion is the emblem of India.”
“Yes, the mission of Ancient India was to incarnate the etheric body into civilization – to make of it a cultural entity. Anyway, young Leonardo was led deeply into these number secrets, and on his return to Europe, he gave that world the Hindu number symbols, 1 to 9. The same that we use, more or less, today. These are an invaluable tool with which we can, with economy, construct any number we can conceive of. How would you write 453,839,903,099 in the previously used Roman numerals?
It’s funny tracking the Indian numerals back to source; the different number Lodges, say in Tibet and Kashmir, used the same symbols – but often in different orders! The figure 8 might represent the 4th number in one, the 8th in another – and perhaps the number 2 in yet a third! These numerical systems were really part of the secret semiotic language of, what were then, secret spiritual brotherhoods.
Leonardo’s second valuable gift to the dawning Consciousness Soul of Europe was zero. This is not really a number at all, but a symbol of unimaginable supplication and spiritual implication. He was quite a guy this Leo; his famous ‘series’ is a revelation of growth – of the etheric. These are truly ‘lion’ numbers.
There was even a Fibonacci Society created in 1962 in that mathematical heartland, California. Their mandate was exclusively to study the – and I quote – ‘inexhaustible properties; of the seemingly simple number series Leonardo so dismissively revealed to the world. Hey, you’re not listening!”
Mother was indeed dozing off to the hum of summer-besotted cicadas, but woke up with the thought. “Wasn’t the Holy Roman Emperor at the time Leonardo’s sponsor? Yes, that’s right; around 1200 odd, Frederick the Second was a great patron of science – and the mind. He was a later complement to the more famous Holy Roman Emperor, Charlemagne, patron of letters fand the arts. Frederick was no ordinary emperor; much of the scientific flowering of the Renaissance is attributed to his vision. His epithet, Stupor Mundi, means’ ‘wonder of the world’; that’s how highly his contemporaries thought of him.”
“Good one Ear, I was thinking about teaching, especially maths. At last they’re realizing that the most effective learning process is – Introduction…then forget, while the material is committed to the temple of the subconscious… then meet it again on a higher level, say in a term, or in some cases, a year even! It if found that the children haven’t really forgotten at all; how easily the material emerges when the golden key of a new main lesson unlocks the secret door of memory.” Mother then pushed the discourse just that bit further.
“It not only re-emerges, but something has happened in the soul in the intervening period. The original knowledge, presented creatively as it has been, is enriched and embellished; in short it – like its owner – has grown. This is non-intrusive, organic learning – how this happens – goodness knows?!”
“Uh-huh” responded Dad sleepily “I hope they learn about perfect and prime numbers and …Ouch!! I’ve got an ant up my trousers – get ‘em off – Yikes!”
“Not here Darling; oh dear, look – there’s ants all over the tablecloth – they’re in the honey! Tan – Lily, come and help!”
“Aw Mum, it’s not so bad; there aren’t many ants here.” Said Tan, upon which Lily began counting them to support their case.
“145, 15, 16 – hey, that’s a Square number. 4×4 makes a square. Hello little square ants! Nyyl said that there are 4 kinds of number – Positive integers; Squares…”
“I know,” said Dad wiggling his trouser leg “Perfect and Prime – hmm, these bear a cryptic relationship to the 4 Elements. The squares of course are the mineral; The Prime, with no factors, have the metamorphic quality of the liquid; the Perfects, equal to the sum of their factors, have the sublime form properties of the air element; and the Positives are the underlying structure of the number world, that created in the white hot forge of creation. Negative integers belong, mot to the worldly elements, but to the soul world itself – the Archai!”
“Nyyl didn’t tell us about that?”
The ants had at last been routed, Dad tried to lighten the s=atmosphere “Ants are a bit like rabbits; they breed exponentially.”
“They eat exponentially too.” Mother added “See the waste.”
“It’s not all wasted,” said Lily “They didn’t like the buns either.”
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