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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…
YOU JUST FRACTURE ME – HA, HA, HA!
Fractions – Class 4 – Main Lesson
The family sedan sped along the country road; it was late afternoon and the twins, Lily and Tan, relaxed in the back seat – it had been a long day at school indeed! Their bodies may have been at rest, but their eyes were active.
“That’s four 8th!” exclaimed Tan as they passed yet another kilometer ‘milestone’. “No it’s not, it’s two 4th – I simplfried.” (deliberate misspelling) replied Lily “I can do that too, it’s 1/ – we’re half way home.”
Mother knew what her double-barrel offspring were talking about; after all, she’d just spent half an hour on the parking lot talking to their Class Teacher, Nyyl, about the current Class 4 maths main lesson – Fractions. By the way, that’s ‘simplified’!
“Our understanding” he said patiently “of the 10-year-olds, is that they’re unfolding the Conceptual Aspect of the Etheric Body. This Body of Formative Forces yearns to be informed by the newfound, objective, ‘conceptual’ faculties of the children. It is the ‘9 ½ year split’ – ½ as a fraction of a year is 6 months.”
“These class teachers are pretty single-minded when they’re onto some new main lesson.” Thought Mother as she quartered an apple to give to the children and their two friends “hmmm.”
Nyyl continued “Maths is ¼ of the 4 main lesson streams, the 2nd in fact. It calls on the forces of the astral or ‘sentient’ body. This maths stream is further fragmented – fractioned – into 3 clearly definable strands, appealing to Will, Feeling, and Thinking. One main lesson of each is taught each year; these are nominally Mensuration, Numeration and Calculation.
Fractions falls within the ambit (although one can be liberal here) of Calculation. This ‘thinking’ (conceptual) quality of calculation makes a main lesson on fractions especially important this year.”
“I see,” said Mother, mot quite seeing either the details of these streams/strands, or her peripatetic children “but I understand that thinking is its nature is divisive – fraction-making!”
“Yes, but to be taken through a thinking initiation from Unity to Diversity and back to Unity again is a safe way to begin.”
“Fractions, being of the nature of fragmentation, is a potentially hazardous cerebral journey. After all, heartless thinking is destroying our planetary home faster than anything else, in the form of amoral science – clever but dangerous.
Anyhow, we proceed with the ‘jigsaw’ method, where we first see the whole picture, before scrambling it into its separate elements – then we re-create the whole again. Think of learning as a tree.” Said Nyyl to an increasingly distracted Mother, she had to cook dinner when she got home! “To understand the tree, we must perceive it as a whole being; to know it further, we have to analyze it – study its parts. To simplify – a fraction term ha, ha – we see it as a root, leaf, and seed. The roots of the Tree of Learning represent the past, the thinking indeed. All that we bring from the past embraces the children’s thinking.
The generous leafy crown is the present. Everything for the class which is born out of the moment – humor, stories, poems, drawings, etc. – nourishes the feeling life as the child responds in the spirit of spontaneity.
We ‘see the will’ by referring, in positive terms, to the future; the future of the child, the world, of mathematics even – but most importantly, to this main lesson. That’s why, on day 1, I provided a precis of the major learning activities that will unfold for them in the next 3 weeks. This will-seeding descends into the cauldron of the unconscious, where it bubbles invisibly away, to later re-emerge when that particular aspect of the lesson is presented – it then manifests as enthusiasm.”
“So how did your Fraction main lesson go?” enquired Mother later to the back seat of the wagon.
“The circle!” they chimed “Let me tell!” – “No, it’s my turn! Okay, you say – I’ll do the next bit.” Mother groaned in spirit, what had she unleashed?! The twins were almost as obsessed with each new lesson as the teacher; sometimes more so! Lily began.
“We started with the circle, which means ‘one’, or everything, you know?” – “Yes, Unity, a symbol of the divine.” Muttered Mother “go on.” – “Then we divided the circle like a pie, into all kinds of fractions – halves, 12ths, quarters. We used degrees, like 180° and 30°.”
“All that pie talk made me hungry,” said Tan ruefully “So I was glad when we got on to finding fractions in other shapes, like squares, triangles and rectangles. That was the 2nd bit.”
“WE re-learn lots of stuff from earlier lessons – really quick but – all about square areas and…I thought we’d forgotten all that, but we didn’t. And then we made fractions of all other things, like water, time – you know, two 6ths of a year, and …”
“Hmmm, a 3-fold soul structure beginning with the Will of the circle.” Thought mother “Even my feelings are fractioned sometimes, one quarter exasperations, 3 quarters exhilaration.”
“…and Thinking Mum,” Lily thought Tan had had enough “you can’t even see it; 3 quarters of my thoughts are happy, the…”
“I know, they’re sad – but what about terminology? Are you learning the language of fractions? The difficult maths words always got in the way of the process for me at school.”
“Yeah, Nyyl said to find the picture in the word, like in ‘cancel’. We invented a man in our little play who was afraid of flying, so every time he’d book a ticket, he’d cancel it again. Old Colonel Cancel would be there when we had to , say, cancel 2 naughts in a times sum.” explained Tan, before the inevitable distraction.
“Hey, we’ve covered five eights of the trip – that’s er…”
“You can’t simplify that; there’s no common factor – we’ve covered 5 eights of the trip that’s all.” Lily smiled.
“Anyway, we had a story about two adventurers; one was called The Numerator, he was a famous mountain climber – he always like to be on the top. (Mother smiled whimsically.) And the other was a cave man…a spunker or something?”
“A spelunker perhaps? – a cave explorer.” Mother smiled still.
“That’s him, he was called The Denominator, and he always like to go to the bottom.” – “You know Mum, the numerators’ on the top of the fraction bac, and the denominator underneath, or ‘down below’. We also had a game with The Fraction Twins. Naturally me and Tan played the parts.” – “Tan and I.” – “You? No, this was at school, it was Tan and me!” – “Oh forget it, we’ll worry about the grammar when you have an English main lesson. So what did you do?”
“I had to sit on Tan’s shoulders (he’s really strong you know) while he held his arms straight out to the side – that’s the fraction bar. We were called Num and Nom. Num is the ‘number’, Mom from ‘name’. The top figure is the number of the fraction, the bottom on its name. Gosh it was funny; Num, me on the top, could only answer questions with numbers, and Tan could only give an answer with a name, like quarters.”
“the class would ask me things, like half my age, or how many days in 4 weeks less ¼. If they asked me questions with no number answers, I’d go cross-eyed – ha, ha, ha. Tan could only answer questions which had a name, like quarters, half, and things. Then we had other games to learn more of those awful words; like factor, simplify, equivalent and un, vulgar.”
“Vulgar?” queried Mother “Oh I know, a vulgar fraction is one with a larger numerator than denominator – what an awful word.”
“It didn’t look awful when we drew it, funny more like it. Mine was a great big egg sitting on a flat board held up by a little googy. We had to do the operation beside it, so the vulgar fraction became a whole number plus a proper fraction. Hey, that’s seven 8ths, we’re almost home, one 8th to go.”
Lily seemed to have exhausted her fraction resources, so her brother took up the baton. “I was good at the next part; the 4 processed cheeses we called them – hee, hee.”
“Ah, the 4 processes – division, multiplication, addition and subtraction – all with fractions?” said Mother impressed.
“you bet, I got it because we had a story where the 4 maths people we had a Class 1 and 2 came in. I like Magic Multiplier best. The sums were pretty easy too – well most of the time. We learnt a poem about each cheese – I mean each process, ha, ha. Let’s see, how did the Dividing Fractions one go? (All 4 are published in Magic Mangrove Seed.)
To divide a fraction by a fraction
Change division to a times;
Then invert the sum’s divisor – on the right, on the right,
Multiply the numerators, and denominators too,
Then that fraction is the answer – see the light?”
(Example: Two 8th divided by nine 3rds = two 8ths x three 9ths = ¼ x one 3rd = one 12th)
“That’s it!” cried Lily, assuming that Tan was creatively exhausted “It’s really fun learning them; we even put in actions, like turning on the light and stuff. We say them every morning – when you’re trying to do a fraction sum, all you have to do is say the verse to yourself and it tells you how – mostly anyway. We learn it through rhythm, just like the tables.”
“Yeah, some were harder than others. It’s funny Mum, in normal sums, adding and taking away is the easiest, and multiplying and dividing much harder; with fractions it’s the opposite. When Nyyl told us that, we didn’t believe them; especially when he told us he was going to teach us the hardest, adding fractions, first.
He said, in unit teaching anyway, you tackle the hardest first, then the rest seems easy. It’s the same in life; in climbing a mountain, it is better to climb the steep cliffs first, then stroll up to the summit, than the opposite. To leave the hardest till last means that you’re already tired. Nyyl said that’s a reason many children don’t like maths; they’re taught it the wrong way round.”
“To multiply 2 fractions is easy, you just times the 2 mountain people and the 2 cave people, in that order.”
“What?” said Mother, still not conversant with the imagery.
“Um, the numerators and the denominators, you know – 3 quarters x five 7ths is 15 over 28 – easy! But with adding and subtracting fractions, first you’ve got to find the common denominators, er, two 4ths + five 8ths = …? I know, four 8ths + five 8ths, which is nine 8ths – then you’ve got to simplify to get 1 and one 8th! Sometimes you’ve got to simplify before you start by finding common factors – whew! Thank heaven for the 4 Process Verses – To add a fraction to a fraction, is easy it is true…”
“And what else did you do in your Fractions main lesson?” said Mother as she turned into the drive.
“We played with toy cars.” Said Lily as she jumped out of the car to an ecstatic (but low decibel) welcome form Silence.
Later, after a worrying hour or so after tea, Mother was on the phone to Nyyl After the usual civilities, she got to the point.
“…it was just that the children said they were playing with toy cars in the main lesson. Do they really have time for games? I mean there’s so much to learn to survive in the big world.”
“They were learning.”
“Oh? And what has playing toy car rally games in class got to do with learning? We thought you maintained a high academic…”
“as you know, young Tan is crazy about Matchbox cars; you find them sneaking up onto his desk all the time. Anyway, I needed a tangible teaching aid to both demonstrate fraction facts, and draw my examples and exercises from – both mental and written. So the class drew up and designed a very large rally track!”
“The ‘rally track’ stretched across 2 tables; to make a workable maths situation, it was 60 kilometers long, divided with different colored flags into its major ‘factors’ (12, 10, 30, etc.). there was a square for every kilometer; it was a dice game, and all the traps and so on were fraction oriented. Say a child might throw a 6; they would then move their little car to – oh no! – the Oggy Boggy Swap square (graphically drawn by them I might add).
Here the instruction on the square is to return to er, one 6th of the track – which of course they have to work out (we help with the slow ones, or work in teams) – that’s the 10 kilometers. I use this rally track imagery later for all kinds of fraction problem-solving, including fuel-use, travel time, etc.
By the way, there’s a story that goes along with this each day: the adventures of the good-hearted but not too bright, eponymously named Frank Partsoff. He is matched against a worthy opponent, the villain; that super-sharp but mal-intentioned Percy Pursestrings. After many preliminary heats, where Percy, with his sleek and high-powered limousine, tried to cheat the rattletrap Frank, the big final is on. Naturally they tie; and again to demonstrate the principle of Unity, they join forces with their respective virtues – Good hearted intelligence – to become the most formidable rally team on the circuit!”
Mother returned to the lounge room, a look of foolish relief on her face; the twins had their model cars out again, and had transformed books, cushions, and ornaments into a spontaneous rally track – complete with checkered (tea towel) flag!
“I’m Frank Fartsoff,” yelled Tan happily “Grrrttt…” then he saw his frowning mother and reddened. “Don’t tell Nyyl will you?! It’s just that whenever he draws Frank’s old car on the board, blowing great puffs from the exhaust…” Mother’s eyebrows arched as she turned to continue dinner, from where she could still hear the drifts of playful banter coming from ‘the track’.
“I want to be Percy this time; I’ll just reverse the road sign so you get lost…and I find a short cut through the bush and come out – ah, three quarters further along the course.”
“Okay, but then I blow my big claxon horn under the Snowy Blowy Alps to make a snowslide to block the road – then you’ll only be…seven 8ths? there.”
“Ha, ha – but all the snow drops on you instead, and I drive right over the top, not even knowing you’re underneath!!”
After dinner, Mother popped her head round the corner “It’s twenty 24ths of the day you know – time for bed.”
“No it’s not.”
“Yes it is, a day has 24 hours, and it’s not 8 o’clock. That’s 20 hours – twenty 24ths…”
“But you didn’t simplify; the largest common factor of 20 and 24 is 4 – it’s only, un, five 6ths of the day – hee, hee – giggle”
“Maths is just a big joke for you two isn’t it?!” sighed Mother as she herded them off to clean their teeth.
“What fraction of our teeth is in my bottom jaw?”
“Um, 32 in each child, or is that 24? Oh go to bed?”
Frank Partsoff and Percy Pursestrings
Frank Partsoff and his vintage Fort Fraction,
Favorite to win the 60-kilometer race –
He’s not very bright but he’s very good-hearted;
With luck on his side
He will set a sizzling pace!
Half or tow fourths, it’s all the same to him;
Nige tenths to go, he’s started in a spin.
Eleven nights he’s Past the line –
He’ll go right round again – again – again – again…
Nothing can stop this intrepid rally driver
When his Numerator blows he can fix it up like new.
If he passes a car with a flat Denominator
He will stop and lend a hand –
What a generous thing to do!
Percy Pursestrings and his silver limousine,
In the rally that he thinks
can be only won by him.
He’s mean but he’s smart – his 100-plus machine
Has everything he needs,
From leather seats to tinted screen.
One over one, he’s right round in a flash,
Two over four – it’s only half a dash…
Percy watch that bend aheeeeead –
Slow down or you’ll crash – you’ll crash – you’ll crash – you’ll crash…
He must win the race
His ambition drives him on;
He will stoop to any trick,
Run the drivers off the track.
Pour oil in their path,
Even let their tires down!
But his perfect plans backfire –
BANG! They always pay him back.
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