“Al-Jabr means union of the parts.”
Algebra – Class 7 – Main Lesson
“Al-jabr means ‘union’ – of the parts.” Said old Al Jabr cheerily to the frowning Class 7 teacher. The two were new friends, having just met in the small park outside the local mosque. Taur had that same frown even then, which prompted this perfect stranger to inquire what troubled him. It transpired that, although he loved teaching, Taur always dreaded the approach of a maths lesson.
In the past this anxiety had evaporated when he began working on the content, and infusing the children with its creative potential. But this one was different; this was the Class 7 Algebra main lesson! Taur had loathed the subject in school, possibly because it was introduced too early – or too late.
The 13th year is that in which the particular Soul Lotus is opening to receive the shift from Arithmetic, dealing mainly with positive, real numbers – to the limitless application of Algebra. The latter is a kind of conjurer’s art, where pronumerals (letters, etc.) are introduced into the equation of life.
“this illimitable potential so excited Isaac Newton that he called Algebra ‘Universal Arithmetic’ – arithmetic merely being Greek for ‘number science’.” Said the old man with a smile “I’m glad the term, although having substantially the same meaning, went back to the Arabic. Al-Jabr means ‘union’ – of the parts.”
“I know, you already told me.” Grumbled Taur; the two were actually sitting inside the domed mosque by now. Al Jabr had invited the young teacher in – probably to help remedy his poor attitude to maths in general, and Algebra in particular. The building, in an inner-city suburb, looked a little squalid from the outside, even its tiny park seemed in a state of siege from the indigent humans and dogs hanging around – but within there was beauty and peace – an oasis in a desert of concrete and cars!
The deep blue dome with the tiny sliver crescent moons and stars painted right across it elevated the worshipper within to a mood of spiritual optimism. This stirred in the of Taur as he adjusted his glasses and asked another question. “But how did Algebra begin? Being an Arabic word, I presume it was a Moslem contribution to human culture.” (If you could call it that! He thought).”
Old Al-Jabr’s eyes twinkled as he stroked his silken white beard “I’d like to take credit for my countrypersons for this ‘key to the door of world mastery’, as Algebra has been referred to; but I can’t! No, the idea to replace a numeral with a symbol, thus expressing the unknown, first occurred in Greece. But their level of number skills was so limited, due to a cumbersome numeral system, that they couldn’t exploit it.
In the Greek Empire expansion eastwards, they exported the idea to mystic India – which did have a sensible system of numerals; these were essentially the ones we use today. If the Greeks were great geometricians, which they were, the Indias were Masters of Arithmetic.
Anyway, as you know, we Moslems conquered India, or a large part of it at least, and brought back the precious numerals (1 to 9) – as well as this seminal algebraic concept – back to the halls of learning of Islam. Here Algebra was developed as a kind of abstract science, one which challenged our best scholars, who saw its reflection in the cosmos – or astral world as some call it.”
Al-Jabr looked meaningfully dome ward before continuing “Even though this was essentially secret knowledge (in those days, as today, computative skills equated with worldly power!), the Jews, those traders supreme, with whom we Arabs had such geographical and racial mutual interests, soon realized Algebra’s ‘key to the gate of world mastery’.
This aphorism would seem to be true; from that time, with the remarkable scientific/commercial tool that Algebra is, European culture has girded the earth as no other before it.”
The old scholar leaned slowly back into the over-plush sofa in a gesture of intellectual over-exertion. The Class 7 teacher was still frowning, but his countenance has lightened a little “I suppose a short historical exposition of Algebra would put some imaginative flesh on the conceptual bones of the subject. It is interesting to consider the timing of teaching in this context.
The Arabs incarnated, or formalized at least, this magical maths. The Europeans, those masters of materialism, applied it. In the epochal unfolding of the child, 13-year-olds are actually re-living that particular historical period, the so-called Dark Ages. What better time to formally teach Algebra?!”
“The Ages weren’t ‘dark’ in Islam; that period from 600AD to the 15th Century were our halcyon days!” interjected Al-Jabr firmly.
“Oh yes, the children have already learned about the dark-light dichotomy of the Age of Heraldry. European Christians saw themselves as an inner light in a dark, dark world; the Moslems as the opposite, a dark matter-bound body surrounded by golden spiritual luminance.”
Class 7 teacher looked curiously round the beautifully-proportioned internal space of the mosque; perhaps there was a kind of golden glow – a spiritual ambience – around them? “Dark Age, Light Age? It seems to depend on your point of view, or your religious persuasion; anyway, it is this period of history that we teach in Class 7. The Age of Chivalry is a main lesson which could be enriched by a sub-strand of Algebra. Most certainly the Algebra main lesson has much to gain by a historical embellishment, one of knights and 1stMillennium mores.”
It seems that Al-Jabr had regained some of his strength as he sat upright and said brightly “Algebra means ‘union’ – of the parts.”
“Yes I know, you already told…”
“I understand that this principle of Arabic ‘union’ and Newtonian ‘universality’ corresponds very well with the Steiner approach to mathematics – and life for that matter – ‘from unity to diversity to unity’? Very holistic!”
“What do you know about Steiner? He wasn’t Islamic!”
“He was everything. Some think his movement is merely Christian, but it’s more than that, a path of pure ‘universality’ itself. By tying the Christian tag to Spiritual Science, one risks reducing it to a religion, instead of the global spiritual revelation it truly is, free of doctrine, ritual, formal obligation and creed.
Rudolf Steiner’s revelations of the Revelation that is Islam are among the most enlightening ever uttered. It is our responsibility to recognize who the great teachers of humankind are – or were-no matter what religion or philosophy – Anthroposophy is of course a philosophy rather than a religion.
The Doctor would certainly have agreed with the axiom ‘If maths is the language of science, Algebra is the grammar of maths’. And grammar it is, being part ‘word’, with its prolific use of letters – abc, xyz. These incidentally usually follow the convention that the early part of the alphabet is used for known quantities (4a, 16b); while the end with its X, Y, etc. are for the unknown.
It took us Arabs to see the power of symbolism, which Algebra essentially is, which underlies the algebraic breakthrough in human numerical consciousness. You see, we (as have our brothers the Jews) have had a proscription – albeit poorly understood – on the representation of image – ‘Thou shalt worship no graven image;, that sort of thing. So symbology became a real art form in Islam, second nature to us really. We had to make some kind of image to satisfy our souls, being a moon/imagination culture as we are. So we expressed it in decorative motifs and complex semiotics.”
“You can see” continued Al-Jabr “this tradition even in the lavish embellishments of this very building, but you won’t find a single ‘picture’ in here. Even the crescent moon on the top of the dome outside bespeaks this love of crypticism. If folk ask, we say that the moon up there is merely decorative; but it really is a most timeless and profound image for we Moon people. So numbers are symbols, of things – 3 apples whatever; Algebra provides a second layer of symbolism with its letters (pronumerals) for numbers – 3 apples is 3a.”
“That’s interesting, about the Moon I mean; in the unfolding of the 7 Planets in the human being’s 7-year evolutionary cycles, 13-year-olds, in the final year of their 7-14 primary school run, are imbuing their souls with the forces of the Moon (beginning in Class 1 with Saturn of course). We see this most evidently in the pubescent stirrings in the Moon ‘region’ itself, the genitalia.
But we know of a higher aspect of Moon development, that which dwells in the sublime Moon ‘organ’ (rather than the ‘region’), the Brain. The spiritual center of the brain, the Pineal Gland, is completing its ethericdevelopment in 13-year-olds; one known as the Shamballic Aspect of the Etheric Body!”
“That doesn’t sound Islamic?!”
“Well no it’s not; this esoteric knowledge actually originates from the Hindu Mysteries (‘Shamballa’ is Sanskrit) – other cultures can be repositories of wisdom too you know! Anyhow, it is the awakening of the pineal/shambolic factor which allows the 7-year development to conclude in a healthy way, providing a foundation for the beginning of adolescence in the 14th year, their new Saturn year.
Algebra, with its symbol-image nature, wedded to a refined conceptual idiom, is an excellent aid in formalizing this vital development, placing emphasis on the higher – brain – Moon impulses rather than merely those of the nether realms!”
Al-Jabr rose and strolled around the hallowed space of the Mosque; he pondered on his own early education, and of his introduction to Algebra in particular. His esteemed teacher had begun at Class 1 really, proceeding slowly in clearly defined steps until reaching the vestibule (main lesson!) of the new Class 7 content.
Firstly there was ‘tree plus stone equals stone plus tree – a+b=b+a. This was an Associative Law, and was introduced with its two brothers, the Commutative and Distributive. These fundamental concepts, called so beautifully the Ting of Unity by modern number lovers, were necessary to underwrite, in a simple static form, the whole range of the naturally mobility which Algebra introduces into computation.”
“Then followed a series of 3-steop equations demonstrating the 3 Laws, using first simple drawings of trees, stars and so on.” Al-Jabr swept his arm grandly to include the spangled universe of his own lofty dome “Then one merely translates these into algebraic form – then arithmetic. Tree is not equal to stone…
The old Number Master actually enjoyed this process of clarification; this confirmation of the holy Ring of Unity (or the Annul of Allah as he called it!). But what really got his pineal gland pulsing was the demonstrations of how Algebra could solve complex equations by reducing the elements to symbols; he fondly recalled the one about little Farouk: He had 16 oranges on this tree; his friend Fez picked 12 and ran away, putting them in his basket with 4 apples. Fez then divided the lot between 4 children. Farouk took all the fruit from 3 of the children and ate one third. How much fruit did Farouk have left?!
16a – 12a = 4a (12a + 4b) divided by 4 = 3a + b (3a +b)3 = 9a + 3b 9a + 3b divided by 3 = 3a + b 3a + b + 4a 7a + b. Farouk was left with 7 oranges and 1 apple (I think!?). Draw it first with oranges and apples rather than a and b. After this carefully stepped presentation, the sums should gradually shed the drawings and contain only pronumerals.
Naturally as the 3-week main lesson progresses, there are lots of simple exercises. These are the daily workouts to keep the number muscles toned; little equations like x times x = x squared.
“I think that the spirit of review of our number work over the years requires the creation of exercises based on say, linear measurement; the circle, the times tables, set theory, graphs, money – and many other maths lessons enjoyed since Class 1 – did I say ‘enjoyed’?”
Taur brushed his fringe back off his highbrow in puzzlement “Well the children certainly enjoy the maths lessons, presented as they are in an imaginative, fun way. But I didn’t realize that I did as well. In fact I can’t wait to start this one. Maybe I’ll write a poem incorporating a few basic axioms of algebra. Poems, with their rhythmic-rhyming life find a special access to the hearts of children, especially those who find the gateway to clinical concepts (temporarily we hope) firmly closed.
If they learn the verse by heart, a word which conjures the mystical element of Love in learning, then a wide range of allusions from it can be made throughout the whole unit to the principles being taught…oh, oh, I’m sorry Al-Jabr, but the light of Inspiration is streaming into my soul right now – it must be this building! Excuse me while I jot it down.” Al-Jabr just smiled and waited.
WHAT IS X?
X can stand for whatever we like,
Or for something not known at all.
It could be an axe, or maybe an ox?
Or the long or the short or the tall.
So X is really only a name,
And everything has its own name
When X plays along in the Algebra game –
As long as they’re in the same sum!
If Xs are axes, then 4x plus 2x,
Is 6x (or axes) – it’s done.
But in the next sum we see Xs again,
They then stand for oxes (oh, oxen for shame!);
Or for anything under the sun.
Now X may not be things at all,
But unknown numbers of things
If 4axes plus 3axe are 6X it’s true,
Just take 4 from 6 – and X equals 2!
Xaxes will add to Xoxen for sure,
Axe and Ox in the same sum you see.
Xaxe plus Xox plus 3axe plus 3ox;
Now it’s (Xaxe plus Xox) plus 3.
But maybe we know what X is at first,
Say X equals 2 – now the sum,
14 plus X equals 14 plus 2 equals 16 –
Already it’s done!
Al-Jabr returned from his ruminations as Taur finished scribbling down the first rough draft of his Algebra verse. “I understand that you’ve maintained a connection with the original 4 Operations characters you introduced in Class 1?” said the old man curiously “Hmm, Algebra is, to define it, ‘A generalization of arbitrary arithmetic qualities.’”
A smile widened under Al-Jabr’s white beard as he continued “I see those four much-loved Number Beings as still active in creating ‘sympathy by association’ in this algebra unit. From first principles, Algebra is merely a higher form of the Operations, it is a 7-year reflection of that particular Class 1 main lesson really. You’ll find this principle of ‘Shamballa’, or 7-year overview, common in this final year of primary. It exemplifies the – again 4-fold – curriculum canon in Steiner education – Growth, Life, Movement and Creativity…”
“Steiner education – I thought you didn’t know who he was?!”
“What? Er, I just remembered that’s all – anyway these four were given variously and often by The Doctor so that his educational impulse wouldn’t paralyze after his departure. Well, perhaps not paralyzed, but in the 70 years since his death, in many cases it’s certainly been set in amber – beautiful but fossilized. This is due to the four antonyms of the golden canons insinuating themselves into the work – decay, torpor, stasis, and convention.
The four canons relate, in the same order, to the first four ‘Aspects’ of human development, the Spatial, Timely, Pictorial, and Conceptual – all specific function of the four ‘bodies’, physical, ether, astral, and ego – again in the same order. Well the 4 Operations won’t remain in amber with this revisiting of their fundamental, metaphysical even, natures.
Of course the use of pronumerals – Xs, empty boxes and the like – have been usefully employed in the maths program right through, but it is only formalized as a singular study, given its very own main lesson, in Class 7. It’s a bit like the Fractions main lesson in Class 4; of course children of a younger age – even infants – know about halves, quarters and so on, but the rendering of the higher skills, like dividing and multiplying fractions, is the stuff of the Fractions main lesson.
Indeed Algebra forms one of the 5 fundamental Number systems, each representing a higher threshold of mathematical consciousness – Algebra being the highest of the 5. The first is the system of Natural Numbers, related as they are to the physical body. Even so-called primitive peoples had simple positive number counting – of things.
When number progressed to etheric awareness, we see the emergence of the second system, the Integer. Here we see the concept of zero, from which the positive numbers march off to infinity to the right – and the negative integers, the minus numbers, disappear down a numerical mine shaft – an etheric vacuum – to the left! The ether body always manifests through this polarity principle of lift-right, light-dark, good-bad, Yin-Yang, major-minor…! Indeed the fact that there are 5 Number Systems so far revealed to humankind suggests a connection to Atlantis!”
“What? Algebra – Atlantis? How so?!”
“Not Algebra as such, but the 5 Number Systems, of which Algebra is one; these relate to the 5 Post-Atlantean Civilizations, from Ancient India to today. Number is such an intrinsic part of our cultural heritage. For instance the 3rd Civilization, that of the Egypto-Chaldean, discovered the 3rd system, that of Rational Numbers. This encompassed the idea of parts of numbers, Fractions!
This is called the ‘architects’ system, due to its use in measuring and dividing measurements in buildings – something Egyptians did a lot of! The Greeks, with their obsession with columns, arrived at the 4th, the Real Number system, that of decimals – or the ‘banker’s’ system as it’s known. The Romans of course took advantage of its monetary potential with their love of the base 10 system.
But Algebra is our very own, the 5th, system, nominally known as the Complex. Algebra, as a global legacy, came to earth with the beginning of the 5th Post-Atlantean Civilization, the co-called Anglo-Teutonic. This was in about the year 1413. There seems to be a new stage in the evolution of number consciousness, that the 5 systems surely are, with each new civilization every 2160 years.
Wo knows how number will be perceived in the 6th culture, the Caucasian, in some 1500 years’ time? It’s hard to imagine anything transcending the number magic of Algebra – although it’s safe to say that the seeds of this new number epiphany are already sown…?”
“Hmmm, it must go beyond the chain reasoning, which is the core of today’s computation – to comprehensive and spontaneous revelation of the spirit behind the number systems perhaps? A Union with the Spirit of Numeracy itself, the mighty Archios as the Greeks called it. What do you think Al-Jabr?”
By this time the old man had settled down on the sofa again, his rheumy eyes glazing over with weariness “Al-Jaber means union of the parts.” Was all he could think of in reply.
Leave a Reply