Sunday, May 25, 2014

Montessori and "thinging"

The Montessori method, originated by pioneer educator Maria Montessori (1870 - 1952), has a very special place in my heart.  I went through Montessori education from preschool to third grade, my children went through Montessori, my sister teaches at a Montessori school, and my mother has run Montessori schools in California for over thirty years.  Of course, with so much exposure to Montessori, many of our family's close friends and the fantastic individuals I was around while I was growing up also come from the Montessori community. Some of the most influential teachers I ever had were my early Montessori teachers, and I am tremendously grateful to each one of them to this day.

Not only is Montessori a wonderful approach to education, but it is also centered on respect for the child as an individual and a person, and respect for the child's own initiative and ability to learn by himself or herself.  Montessori also inculcates in the child a respect for other children and the ability to work with and help others.  All of these wonderful aspects of Montessori are evident in the above video, entitled "A Montessori Morning," and which shows in about four minutes a series of photographs taken during the course of three hours in the morning of a four-year-old named Jackson, along with his friends at the Dundas Valley Montessori School in Ontario.

In addition to all these outstanding characteristics (and there are many more I have not mentioned), Montessori also provides an excellent example of the esoteric method of enabling the human mind to grasp big or profound concepts (previous discussions of the esoteric include "Wax on, wax off" and "Like a finger, pointing a way to the moon . . .").

Montessori uses ingenious physical materials to represent abstract concepts.  In doing so, it echoes the method employed by the sages responsible for the mythologies which make up the world's ancient sacred traditions, according to thinkers such as Gerald Massey (1828 - 1907) and Alvin Boyd Kuhn (1880 - 1963) -- although most conventional historians and academics erroneously approach ancient sacred texts and traditions as if they were intended to be understood literally.

Gerald Massey vigorously refutes the conventional view that the world's ancient myths were intended or anciently understood to be literal in the second and third sections of his essay "Luniolatry, Ancient and Modern," in which he explains:
They [meaning the conventional historians and professors of mythology, several of whom he cites in the essay] have misrepresented primitive or archaic man as having been idiotically misled from the first by an active but untutored imagination into believing all sorts of fallacies, which were directly and contradicted by his own daily experience; a fool of fancy in the midst of those grim realities that were grinding his experience into him, like the grinding icebergs making their imprints upon the rocks submerged beneath the sea.  It remains to be said, and will one day be acknowledged, that these accepted teachers have been no nearer to the beginnings of mythology and language than Burn's poet Willie had been near to Pegasus.  My reply is, 'Tis but a dream of the metaphysical theorist that mythology was a disease of language, or anything else except his own brain.  The origin and meaning of mythology have been missed altogether by these solarites and weather-mongers!  Mythology was a primitive mode of thinging the early thought.  It was founded on natural facts, and is still verifiable in phenomena. [. . .]
In modern phraseology a statement is sometimes said to be mythical in proportion to its being untrue; but the ancient mythology was not a system or mode of falsifying in that sense.  Its fables were the means of conveying facts; they were neither forgeries nor fictions.  Nor did mythology originate in any intentional double-dealing whatever, although it did assume an aspect of duality when direct expression in words had succeeded the primitive mode of representation by means of things as signs and symbols.
Alvin Boyd Kuhn picks up on the importance of Massey's concept of "thinging" and says:
As Gerald Massey says, thinking is in essence a process of "thinging," since thoughts must rest on the nature of things.  And things are themselves God's thoughts in material form. Lost Light, 42.
This "thinging" that Massey and Kuhn are talking about is perhaps best illustrated by the Montessori materials, some of which can be seen in the beautiful little video above.  For example, in the video above, Jackson works with the Montessori "sensorial material" project known as the trinomial cube beginning at about 2:30 into the video, through about 2:44 in the video (the video moves fast -- you can see the trinomial cube segment following immediately after Jackson and his friends have a snack, at about 2:25 -- immediately following the "window squeegee" scene -- just after Jackson finishes cleaning his dishes from the snack and puts them into a drying rack to air-dry).

The trinomial cube is an example of "thinging" the somewhat abstract algebraic concept of cubing a trinomial (a trinomial is a mathematical expression containing three variables, with the vairables commonly designated as a, b, and c).   If we have a trinomial (a + b + c), and we wish to cube it, we must multiply the trinomial by itself three times (this is the definition of cubing something).  In other words, we must multiply (a + b + c) (a + b + c) (a + b + c).

If you remember your algebra, you will remember that the way to tackle this particular operation is to begin with the first term in the first trinomial, and multiply it by each of the terms in the next two instances of the trinomial, and continue this process all the way through the operation.  The outcome of that process is illustrated rather well on this discussion of the Montessori trinomial cube, on the website of Montessori World Educational Institute.

After multiplying it all the way out, and adding it all together, one finds that the cube of (a + b + c) can be written:

 a+ 3a2b + 3a2c  + 3ab+ 6abc + 3ac+ b3 + 3b2c  +3bc2 + c3

The trinomial cube used in Montessori classrooms makes this rather intimidating-looking formula into a thing, into a model which children such as Jackson can manipulate and explore at a very early age (remember that Jackson is four years old, and he can be seen assembling the trinomial correctly in the video).

The way the model cube "things" the expression of the cubed trinomial shown above is ingenious.  You can see that in the full formula, the cube of each variable appears one time each, a-cubed, b-cubed, and c-cubed appear at the beginning, the "middle," and the end of the formula, respectively.  The  variable a is represented by the largest dimension of the blocks in the cube -- when a is cubed it is represented by the largest cube in the model, painted red on all surfaces, of length a on each side of the cube.  The variable b is the next-largest dimension of the blocks in the cube: when it is cubed it is painted blue on all sides and appears as a cube with sides of length b (a shorter distance than length a).  Finally, the variable c is the shortest of the dimensions represented in the cube; when c appears as a cube (which it does one time in the above formula for a cubed trinomial), it is depicted as a cube in which all faces are painted yellow, and the sides are a length c (shorter than b, which in turn was shorter than a).   

Note that in the solution formula above, the term following  a3  is 3a2b.  The term a2b is "thinged" in the Montessori trinomial cube as a solid with a face that is length a on each side (that is, it is a physical representation of a2 and it is painted red on the square face), but which is only a depth of b (these sides, b in length, are painted black).  Thus, the Montessori trinomial cube represents a2b as a solid with a height and width of a and a depth of b, and it contains three such solids, to match the a2b in the solution to the cubed trinomial.  

The model of the trinomial has solids to represent each of the terms in the full formula above.  It has three that are again a height and width of a but this time only a depth of c, to represent the next term which is 3a2c (and again, the face representing a-squared is painted red).  It has three solids which are b in height and width and a in depth, to represent the 3ab2 (and this time, of course, the face representing b-squared is painted blue, while the depth representing a is painted black -- colors are only used when a term is either squared or cubed, otherwise the side is black).  And it has six solids which have a height of a, a width of b, and a depth of c, which are black on all their sides, and represent the term 6abc.  To help visualize all of this, follow this link to the excellent schematic on the trinomial cube page of Wikisori, which lays it all out visually.

Now, the interesting thing about all of this is that the child learning how to work with the trinomial cube (and its slightly less-complicated cousin, the binomial cube, which represents the binomial a + b multiplied by itself three times) is not taught anything at all about the way that the cube is an ingenious physical representation of a rather advanced and very abstract algebraic concept.  That would not really be helpful to a four-year-old child.  

However, when the child is old enough, and is being introduced to binomials or trinomials in algebra, then the teacher can explain the connection to the old, beloved, familiar binomial cube and trinomial cube, and show the "esoteric" connection between the physical model and the formula they are learning.  What a flash of recognition will go off in the young person's mind!  It is exactly akin to the sudden dawning of recognition experienced by Daniel-san when Mr. Miyagi showed him what "wax on, wax off" was really all about!

You can, in fact, see for yourself that the webpage for the binomial cube on the Montessori World Educational Center website expressly states: "Do not explain to the child why you are setting the cube out in this order, or talk about the mathematics of the cube."  Is this because the Montessori teachers do not want children to know the "esoteric secrets" of the binomial cube?  Of course not!  The whole point is to eventually help the child to learn about binomials, in a way more profound than the child might ever be able to understand otherwise.  But trying to explain it in a "left-brained" way first would just invite confusion and questions as the analytical "left-brain" tries to absorb the abstract and complicated concepts involved, likely causing the brain to "choke" on it (and possibly never feel comfortable around binomials or trinomials ever again).  Instead, the webpage advises: "The math is presented to the children when they are older and are ready for it."

This example from Montessori (and there are many others that could be used, such as the bead-chains which you can see Jackson and his friend working with after the trinomial cube segment, beginning at around 2:47 and going to about 3:00) really illustrates Massey's point about the value of "thinging" an abstract concept (a point Alvin Boyd Kuhn also underscores as being of supreme importance).  It is easy to see the source of Massey's frustration with conventional academics who insist that the myths were simply a bunch of "fallacies" which ancient men and women believed literally.  

Kuhn disagreed with Massey, however, in saying that these exquisite mythical metaphors, which so wonderfully "thinged" profound spiritual concepts, could not have originated as a "primitive mode" of early thought.  He argues that these incredible metaphors betray the handiwork of sages who already understood completely the deepest spiritual truths, saying:
Primitive simplicity could not have concocted what the age-long study of an intelligent world could not fathom.  Not aboriginal naiveté, but exalted spiritual and intellectual acumen, formulated the myths.  Reflection of the realities of a higher world in the phenomena of a lower world could not be detected when only the one world, the lower, was known.  You can not see that nature reflects spiritual truth unless you know the form of spiritual truth.  Lost Light, 71-72.
In other words, no one could start with the physical model and come up with the spiritual truths -- the makers of the model had to know the spiritual truths already.  We can immediately agree that the designer of the trinomial cube had to understand the full formula of

a+ 3a2b + 3a2c  + 3ab+ 6abc + 3ac+ b3 + 3b2c  +3bc2 + c3

before designing the wooden model.  By this analogy, it stands to reason that the designers of the exquisite esoteric myths of the world understood the profound spiritual truth they wished to convey before they ever created the myths -- the myths were not the product of "an active but untutored imagination," as Massey thought.

Furthermore, it is also evident that one could learn the trinomial cube as a child (as a four-year-old, for example), and never fathom the connection to the trinomial expression shown above -- even if they later became quite advanced at mathematics and algebra and learned all about trinomials!  To make the leap from the model with the solid forms painted red, blue, yellow and black on their various sides, to the formula shown above, is not necessarily intuitive until the connection is shown to the student.  This concept is expressed in the New Testament book of Acts, in which a man is depicted reading an Old Testament scroll (Isaiah), and is asked if he understands what he reads.  He replies: "How can I, unless someone guides me?" (Acts 8:31-32).

What a tragedy it would be if the stories in the ancient scriptures were really intended to act as a sort of "trinomial cube" pointing to profound spiritual truths, but those who were able to teach the connection were prohibited from doing so!  It would be as if children were prevented from being shown the true purpose of the Montessori materials, such as how the bead chains teach multiplication and squaring and cubing of the various digits from 1 to 10.

What a tragedy if all those who knew the esoteric connections were, at some point in ancient history, marginalized and suppressed by people who wanted to teach that these stories should all be understood literally first and foremost, and if these literalists did their best to destroy or cast out all the texts which opposed that literal interpretation or said that the scriptures were not really literal but rather esoteric.  

Fortunately for the human race, the finely-crafted "Montessori materials" which are the ancient metaphors of the myths of all the world's cultures (including those which were preserved in the scriptures of the Old and New Testaments) are still with us today, and can be turned over in our minds as we might turn over a finely-crafted trinomial cube.  The connections to the spiritual concepts that these stories were intended to teach (via the method of "thinging") were not entirely eradicated by the literalists, but survived in various channels over the long centuries, and have been elucidated by various teachers in various texts.  The connections can be made again.