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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…
LABYRINTH OF NUMERACY
Number Systems – Class 4 – Middle Lesson
“Sir – you buy genuine tomb artifact, miniature bronze sphinx – I mek you special deal!” It was the hustlers and the heat which at last made Archie abandon his guided tour of The Sphinx. Instead, he spent the afternoon in his seedy hotel room, lying on his bed and staring moodily at the slow-turning fan above.
Between dozing, he thought of his strange pilgrimage to Egypt; of how, since a child, he has been captivated by stories and images of The Sphinx. These were from many lands: The Levant; Asia Minor; Greece; Crete; or further east even. It was the omniscient quality, that contained in all the tales, like that of Oedipus, that fascinated Archie. This great, divine entity knew everything; and he a mere part of seething humanity, knew nothing at all – about spiritual things at least.
The word sphinx, in a literal translation, means ‘closed gate’ – as in the ‘sphincter’ muscle on the esophagus, and other orifii! The inference of gate however is that it can open; but, like Oedipus, one had to pass an initiation of some kind, usually mental, or thought-based. Though, it esoteric terms at least, is synonymous with Spirit.
The most comprehensive sphinx form is an expression of the Tetramorph, the 4-creature heavenly power described in the Vision of Ezekiel. Or as seen as the emblems of the 4 Gospels – or the cardinal points of the Zodiac!
This collective sphinx had the face of a human being, or ‘archetypal Man’, a symbol of the Physical Body, expressed through Aquarius and St. Matthew. The ‘animal circle; animal had the chest and forepaws of a lion – Leo/St. Mark/Etheric Body; and the hind quarters and abdomen of a bull – Taurus/ST. Luke/Astral Body. Finally it was winged, the eagle of Scorpio (formally ‘Aquila the Eagle0/St. John/Ego.
Even though Archie loved the consistent symbolism of The Sphinx, it was the conundrum aspect he was entranced by. There were times when he would love to meet this awesome creature along some lonely byway, and be posed a cryptic question.
Archie tended to underplay the ‘fail and you die’ factor; focusing more on outwitting his lofty opponent, and ‘squeezing’ (another interpretation of ‘sphinx’) his way into higher worlds. Indeed it was this seed of longing that prompted him, as he looked out of his window over a sea of stained domes and air-con ducts, to slowly get dressed.
A friendly buttermilk moon was rising, inviting him to share its hours of nocturnal coolness. Archie was going to put some questions to The Sphinx all right – he hadn’t come all the way from Australia for nothing! – but not in front of milling tourists.
He strolled through narrow streets of bazaars and food stalls, all smelling and sounding as they had smelt and sounded for thousands of years. Soon the clamor faded, and he trudged alone across a silver sea of sand. His pace slowed as he approached the colossal but impassive recumbent figure.
The layered stone threw ink-black shadow lines along its length; even with its defaced features, The Sphinx emanated a primordial dignity – Napoleons’ vandals notwithstanding! The young Australian caught the ‘You have arrived.’ Look, and involuntarily stopped. He sat on a soft ridge of dune to contemplate the object of his spiritual fantasy life.
“So who are you?” he said, laughing at his own folly – luckily nobody was around to hear him (or so he thought).
“I’m supposed to ask the questions!” the ‘voice’ didn’t actually come from The Sphinx, but rather all around, from the smiling moon perhaps. Or inside his own head! Wherever, it spoke with a commanding tone, one not to be dismissed; so, in spite of feeling foolish, he answered, his voice tremulous.
“I…er, I’m Archie form, I mean, from, Australia – I’m …um, I’m a schoolteacher, Class 4 actually. I’m on holiday and decided to come to Egypt…”
“I don’t want your life history; besides, I know all that (I know everything remember). That was merely a test question to see if you could tune into my consciousness. And so you have – reasonably anyway. But try to free up – be a little less sphincteric – ha, ha. Think of…think of stars. After all, the great Egyptian Civilization was one of ‘Astrology’. One of the Spirit’s earthly emissaries, Rudolf Steiner, would have told you that. Heknew how to find the answers to the difficult questions all right. That’s why we gave him access to our wisdom mansions, Now, Who am I?”
After giving a learned run-down on the history of The Sphinx, in which, after a time, his interrogator seemed to grow distracted – bored even – Archie added “…but the most important element of your Being is again 4-fold, that of the Cherubim, they of the ‘flaming swords’ as Genesis tells us. Some Greek sphinxes were depicted with flames rising out of their heads.
The 4 cardinal points of the Zodiac – Aquarius, Leo, Taurus, Scorpio – is a collective image of the Cherubim. Your little clue, ‘wisdom mansions’ put me onto it. The Cherubim are of course the Spirits of Wisdom. This wisdom path has been one of higher initiation right through the ages – or four paths more like it. Is that enough?”
“I ask the questions?” Archie was taken aback by the razored tone, and recalled the story of being terminated if he answered wrongly, He didn’t believe it, not in this enlightened age; but shivered slightly anyway – The Sphinx went on.
“Of the 4 creatures in the Tetramorph, which is most relevant to Egypt? – careful here, a lot of seekers have failed this one.”
“Let’s see…I’d say Taurus, the Bull. Yes, the Northern Hemisphere sun was rising in the vernal equinox in Taurus during the great ‘Egyptian’ age. I take this into account with my Class 4; they’re little Egyptians in a sense – and by extension, Taurians. Egypt was the 10th age (starting from Ancient Saturn), so 10-year-olds are re-living the cultural awakening that occurred right here. Indeed it is the Taurus sense, that of Thought, which I especially cultivate with them.”
Again The Sphinx seemed to be looking over Archie’s head to the horizon. (If he dislikes long answers – why does he ask a Steiner teacher!?). He asked another question.
“So how was this Taurus/Thought duality conceived of in the next ‘Civilization’, that of Greece? (This should catch him!)”
“Mmmm, let’s see … I don’t think this is fair, I come all the way from Australia to ask you questions! Why can’t I?”
“That’s another question – answer the question??”
“Um, Greece? I know! In the Greek Zodiac, Taurus was represented by the Minotaur. It might seem an uncomplimentary image, but one has to take it in context. Greece was an Aries Age; therefore Taurus was the sign of the immediate past, that which held them back. In a sense the Bull (astral) Mysteries were inimical to their progress; so they created a virgin-eating monster!”
The Minotaur lived in a cave (of the head – thought!); his Labyrinth was the convolutions of the brain itself, both physical and mental. Theseus, the quintessential Greek hero, used intelligence – of Taurus – to achieve his goal of slaying these retarding forces – and to find his way out again. It was great; you see, he used a ball of string…”
“I know the story – indeed I inspired it. The Greeks weren’t all that original; our pyramids were a kind of Labyrinth of Initiation. Bewildered neophants would be led in by the priest, through seemingly endless, lightless passages, and left there – alone!
After agonizing days without light, food water or comfort, desperately trying to find their way out, they would be ‘rescued’. ‘And so the first trial of your Thought Initiation is to realize that you can’t master the Labyrinth of the mind without a guide – follow me, we’ll be out in a few minutes!’ saith the priest.
And neither can you schoolteacher. Now, in terms of both Steiner’s ‘Rationalism’, one of the 12 Philosophical Standpoints, and Mathematics, one of the 12-Subject Zodiac lessons, what sign do these live under – in what heavenly home do they dwell?”
“Oh that’s easy, Taurus of course. Mathematics is always regarded as the highest province of thought. If you ask anyone who the smartest person in the world is, they will say Einstein – a mathematician. The power of thought – the sheer discipline – required to master higher mathematics is beyond most mere mortals. A complex equation is a ‘labyrinth’ indeed; but why are we talking about maths?”
“Hold it!!”
“Sorry, you’re suppose to ask the questions – go ahead.”
“Thank you – the Greek sphinx would have decapitated you – removed your thought organ – for that slip! One of the keys to many – indeed most – of the ‘closed gates’ of the Labyrinth of Number are contained in the so-=called Number Systems. (Dear Reader: Sorry it’s taken so long to get down to business; I thought I’d indulge myself on this last article in the book.) These have been painfully ‘discovered’ over the generations – with a little help from me. In what context do you teach these to Class 4?”
“Ah, another easy one; Number Systems are presented in the middle or ‘heart’ lesson. This mitigates the potential desiccation of this heady subject. It is taught in the Numeracy stream; and of the 3 strands therein, Problem Solving – Naturally!
One might call this problem-solving/thinking unit the ‘Sphinx Strand’, concerned with finding answers to enigmatic questions – problems – as it does. There we teach – introduce I should say – the function of specific groups of numbers.
Firstly we make mention of that Aries number system, that which preceded our own, the cumbersome Roman Numerals. The largest one known, on that temple of conservatism, the former Darlinghurst Court in Sydney, is MDCCCLXXXVIII – 1888! These ‘letter numbers’ might be seen to be numerical Spirits of Retardation, belonging as they do to our former civilization – conservatives love ‘em!”
“Don’t get political with me; what do you do next?”
“Well we draw the distinction between a Prime and a Rectangle Number. As such I tell the class how to find the factors of any non-prime number. There is a heaven/earth dichotomy hidden there; the rectangles inherently bear the 4-square earth form, all being the product of smaller numbers as they are. The primes are more metaphysical n their nature; the pair might be thought of as the Yin (prime) Yang (non-prime) of numeracy. The largest prime known as discovered after a 2-month search by a supercomputer (a mechanical labyrinth); is the 244497.
A 3rd number system is that of the Triangle, found easily by adding each consecutive natural number. If these are arranged under each other as counters, they create a triangle form – from the top: 1, 3, 6, 10, 15 – (1+2=3, 1+2+3=6, 1,2,3,4=10, etc.)
The square Numbers form a system in their own right, even though they are all rectangles as well. These are found by multiplying any number by itself; like 7×7=49 = 72 – the little number is called the exponent (‘put out’)…”
“Hey, I’ve got another question: if you put an amoeba in a jar, and after 1 second it splits in tow, making two amoebas, they did likewise and on and on for an hours, When was the jar half full?”
“What!?”
“Don’t answer a question with a question!”
“Oh, er…maybe in half an hour? No – I know, one second before the hour was up!?
“Gooood – I hope your 10-year-olds don’t take as long to get this square number problem. So then what?”
“Well I try to show how square and odd number (odds and evens are another number system) are related.”
Additions of consecutive odd numbers always make a square number – 1=1, 1+3=4 (22), 1+3+5=32 (9) – and so on forever. This is easy for the child to see if drawn in a square umber box. What do you…forget it!”
“Question time again: of the 3 answers to the following sums, which is not a square number? 81+144=225, 64+225=289, 144+255=369. (Varieties of these can be give to the class in this problem-solving lesson.)”
“Ahhh, for each answer I have to do a guestimate sum; let’s see, 225 sounds like about 14×14 …196? No, 15 then – ah, that’s it, 152 is 225 (extended period of calculation followed). There! The last sum of 369 is nota square umber – 182=324!
A charming diversion is to look at 162, (which is 256, the frequency of middle C) and other serendipitous square numbers.”
“Really? Ah, of course – anyway you passed the last question; only just though, there is a time limit on these things you know. Do you introduce Cube Numbers at this stage, expressing the 3rd dimension as they do?”
“Oh no, this matter-heavy consciousness is better left to Class 5. From here I might present the concept of the radix, or Base Number Systems. Our number system, which we inherited from you ‘Egyptians’, the civilization of the 10, is the Decimal, one based on repeating cycles of 10 – 1, 10, 100, 1000, etc.
But it’s good for children to know, and be given exercises on, different base systems. For instance, they can write a sequence of numbers in base 4: 0, 1, 2, 3 -10, 11, 12, 13 -20, 21, 22, 23 -,30,31, 32, 33-100, 101, 102, 103… ad infinitum. The 4 refers to numbers in the system, it doesn’t actually appear at all.
If the base system one chooses is bigger than 10, one has to invent new words – and symbols – for the extra numbers in the sequence – that can be both fun and empowering for the children. Most of these arcane number systems don’t get much purchase in our age – except one, the Base 2, that which is the foundation of all computer technology.”
“I must admit I don’t know much about computers; do they really do sums quicker than human thought?”
“Quicker than most humans yes, but that’s only because our generation is conceptually lazy. If we used the potential thought capacity we all have – if we trod every pathway of our ‘labyrinth’ – the computer would be recycled into ploughshares!”
The Binary System works on 2 numbers only, 0 and 1; the electronic impulses in the computer or calculator switching form one to the other with the speed of light.”
“Lightning more like it – the computer is a sub-earth, Ahrimanic device!”
“Do you want to know or not!?”
“Impertinence and a question – you’d lose 2 heads for that!”
“Ha, ha – you’re just joking yes? Er, I mean…these two numbers are derived from properties of odd and even. If you divide an odd number – any odd number – by 2, you always have a remainder of 1. When dividing even numbers by 2, the remainder is always 0. O stands for even, 1 for odd.
The number system forms very small multiplication tables: 0x0=0, 1×0=0, 1×1=1! As usual, the simplest keys lead to the most complex rooms, seen in even the basic even + even = even, odd + odd = even, odd + even = odd – always. This is only part of this mysterious even/odd world.
Perfect numbers; those rare specimens which are equal to the sum of their divisors, form a ‘toy system’, as many of these small number systems are called. The lowest perfect number is 6 (1+2+3=6). The highest – again ferreted out by a toiling computer – is (244497 – 1) x 244496! A far cry indeed from the least numerate people on earth, A Brazilian tribe in the Matto Grosso, who have no named numbers at all!”
The leprous-white moon disc shone down on a camel with rider, who appeared silently over a dune, quite close. The Bedouin, garbed in blue/black, merged with the velvet sky. The whites of eyes and teeth gleamed in moonglow. He stopped, seeming to tower over the somewhat unnerved Australian, who stammered.
“Er, me no buy…me no money!”
The glint of a scimitar in the wide belt further eroded his composure. A grunt commanded the camel to kneel, and the man dismounted with a single, sweeping movement.
“Relax councilor,” the Arab replied in perfect English “You’ve been chatting away with The Sphinx haven’t you? This wisdom source used to be very popular. As part of the timeless stream of pilgrims, we Moslems had academies of excellence all over our empire – much of the knowledge derived from this very spot.
Not too many come these days though; you’re from the Great South land I see (Your ‘I’m an Aussie’ T-shirt!).
“I hope you don’t mind sharing, but I was gong to confer with The Sphinx on a few aspects of Number Systems.” Well you could have knocked Archie down with a harem veil!
“That’s what I’m asking…I mean conferring on!!” he gasped in disbelief “But.. but how…?”
“Stop! I ask them remember.” Yes, it was The Sphinx, trying to prevent the Class 4 teacher making a fool of himself again.
“I want to know how you understand the 3 forces of Will, Feeling and Thinking in the major number systems – well?”
Archie was flummoxed, and turned to the Bedouin, Kismet by name (meaning ‘destiny’). He sat down on the sand and proceeded, slowly to begin. “The first number system is that of Natural Numbers; 1, 2, 3, 4 – forever.
It is a Will exercise of some value to count, not only with children, but, say, when trying to go to sleep. Counting sheep stills the mind, drawing Will forces up out of the unconscious on their relentless, coma-inducing march. The ‘natural’ is the oldest number system known, and stretches from a given point, to infinity.
After all, if you add or multiply any 2 natural numbers, you get a higher natural number. This system is used not only by defacto pillow shepherds, but by gatekeepers and the like. They do just fine counting the heads as they pass, but they’re stuck if you ask them to do, say, 2 minus 5.
The gatekeeper can’t, but the banker can. He uses the second number system, that which calls on the major/minor, happy/sad elements of Feeling – those of the positive and negative numbers, the Integer System.
Here, from the same point we started with the natural numbers, zero, we can go backwards as well as forwards – backwards into minor…I mean, minus numbers. As any camel trader knows, you can have less than no money in the bank, you can be in debt! So the integer system adds another possibility, you can subtract any number, as well as adding and multiplying (as with the natural numbers).
A couple of examples might be” (-2)+(=5) = -3: (with this system, the signs should be included in the answer) (-2)+(-3) = -5; (+2)-(+5) = -3. In multiplying, you always look at the sign on the multiplier (the 1st figure) and attach it to the answer: (-2)x(-3) = -6. And just when you thought you could do it all, you get a sum like 5 divided by 2!
“This sum is impossible for both the gatekeeper and the banker, but not the carpenter. S/he works daily with the 3rd number system, that of Rational Numbers – the Thinking system! Put simply, these are the fractions, which our Class 4 were introduced to so imaginatively this year.” Kismet turned and faced his friend, his dark eyes sparkling.
“How did you know about my…?” sputtered Archie.
“Not another question thanks, you might spoil it for me. The Sphinx does seem to be getting more irascible in his old age! The rational numbers system means you can divide any 2 natural numbers, as well as the other 3 operations. This calls upon the Thinking faculty more than the other 2 systems. Contrary to popular belief, carpenters spend quite a lot of time thinking about things like: ‘1050 centimeters divided by 17 to five wall stud centers.; (61 thirteen 17ths!? Well it was just an example!)”
“Can I interrupt? Don’t answer that!” said The Sphinx, a tiny blush of dawn rose reflecting from his sandstone profile “When, schoolteacher, do you intend teaching the Real Number System – decimals; and algebra, the Complex System?”
“Later, in Class 5 and 7 respectively – isn’t all we’ve talked about enough for one 23-hour middle lesson?”
“That’s it – I’ve had it! You heard him Kismet, he asked yet another question!” With The Sphinx’s outburst, the Bedouin sadly shook his head, mounted his protesting camel, and sauntered off among the dunes, muttering something like ‘He couldn’t find his way out of a phone box, let alone a labyrinth!”
No amount of apologies from a contrite Archie would elicit another word from his stone inquisitor. As the sleepy moon slid out of sight, and the golden eye of Ra rose above the desert, the first party of clamorous tourists and hucksters arrived.
“Sir – you buy genuine tomb artifact, miniature bronze sphinx – I mek you special deal!” Archie looked up appealingly to The Sphinx, but he just gazed with igneous indifference to the horizon.
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