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Geometry

Dear Friends,

Spring is here in Italy. In my attic room, I can hear the birds singing well-before the dawn. Their songs come through loud and clear, so I’m up with them in the early morning hours. There is still snow on the mountains, although the trees are flowering around these lowers hills. The world seems a beautiful and peaceful place.

This weekend I am finishing Botany Charts. This is the last set of required materials for the course. There are other materials and charts, of course, and these will fill out and complete the basics for an Elementary classroom. In fact, it seems that we could spend a life-time creating materials for the Elementary classroom because any subject area can be presented in a simple and clear way through images. It is left up to the teacher to create the things she will need for her class. We only make the essentials while on the course.

For some fun: the last few days we have begun looking at Montessori materials for the Pythagorean Theorem. Attached are images from the internet (Forgive me: my camera is on the fritz!). These are terribly exciting and wonderful materials and one of the signs, I think, that Montessori done correctly really is an education for life in a deeply intellectual way ~ while integrating the mind and body through sensorial exploration.

The Pythagorean Theorem states that given a right-angled triangle, the sum of the squares formed on the short legs equals the square formed on the hypotenuse. Simple and not too hard at all really! What makes it special, and doubly special for children, are the clear and reasoned steps to arrive at that conclusion.

Pythagoras Material

Montessori devised three materials that help children to explore the proof.  The first is sensorial exploration: the triangles are simply exchanged and viola! it’s obvious that the large square contains the smaller because the triangles fit. The second material is considered and arithmetical proof. The little squares (remember, area is measured by square) fit into the large square. This is also quite easy and rather fun.

Euclid’s proof requires a little more reasoning. We have to know that any triangle is equal to 1/2 the square that has the same base and height. This makes up the ‘middle term’ and hence an essential part of the proof.

EuclidI47a

Next week we will practice with this material a little more and I will show you some more steps. Remember, this work if for children around the age of 9 or 10, or even a little younger!

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We often hear about the insights of Dr. Montessori into the growth and development of the human mind. Obviously, this involves children to the a great degree. She is also well-known for devising beautiful and precise materials for classrooms. ‘Montessori School’ brings to mind lovely airy rooms with well-behaved children and lovely activities for them. All this is well and good and as it should be.

But Maria Montessori did not ‘discover’ her method under these conditions. She did not start out with the lovely things we are able to purchase today. She had almost nothing, just a ‘hunch’ if you will, about how the mind works.

So what is the connection between the materials and the child? I believe that the connection ~ and at a very deep level ~ lies between an understanding of natural world that Dr Montessori had when she devised the materials and her intuition about the person.

In order to be clear, I have to take another step back. When Aristotle set out his ten categories of existence he really didn’t come up with anything new. He simply showed us a way of thinking and acting in accord with reality. He organized what we already know. That doesn’t mean its easy to read him however. In fact, it can be rather difficult and rather fun to figure out what it all means. But down at the bottom of it all, we come out with a clearer mind because we’ve organized, categorized, and compartmentalized reality. St. Thomas refers to Aristotle frequently, if that’s any vote of confidence.

Its sort of like having a very very messy room or house. Its your house and you might know exactly where everything is kept, where it was put, at what time and so on. However, it doesn’t look very tidy and in fact, its rather a hassle for yourself and anyone else to tries to help. We might think of Aristotle as the philosopher who tidied things up a bit. Its nothing new really, just the same house organized.

All this is to say that we can operate in the world with a clear sense of what’s real and with assurance of objective truth that can be known, loved, and understood. That is,  to the degree that our minds are organized enough to receive the truth. (Small ‘t’ truth here.)

Maria Montessori understood all this when she began to devise her materials. She put it in her own words: “The simpler and clearer thing is the origin of things: as I use to say, the child has to have the origin of things because they origin is clearer and more natural for his mind. We simply have to find a material to make the origin accessible.” (Rome Course May, 1931).

How do we do that? How many of us have sat down to wrestle with the concept of how quality is a category of existence that  must must reside in a material substance? How do we begin to convey the postulates of Euclid? They are self-evidently true, but not obviously true. And if we don’t know (and don’t even care because life is busy) why would we ever think to try to teach a seven-year old that all horizontal lines are measured in reference to still water and vertical lines to the pull of gravity? How could we begin to teach the intricacies of music theory to children who won’t even sit still to play piano?

If you have figured these things out for yourself, then great! Most of us haven’t because we can’t organize our knowledge in reference to the origin of things.  We were all born on the same planet, but not all of us have organized our knowledge. The point here is that Dr. Montessori already understood the origin. As she designed materials for children, it was always with the effort to “give the most the simplest way.”

The part that shocks us is that a child can learn these concepts when we find it so difficult. There are many examples that I want to share, but for the moment let me give a (rather poor photograph) of some basic math material:

Here is an example:

numberrods

These are the number rods. Yes, simple rods in red and blue.No, they don’t break apart. Each one is its own rod.  These are used for teaching numbers. Dr. Montessori was nominated for the Nobel Prize for designing these. In fact, if she had done nothing else, she would be known as the person who invented how to teach number. You see, the idea of numbers as lines ~ whole lines, measured by a unit ~ goes back to the ancient Greeks. But this is where formal number teaching begins. We don’t stay here very long (obviously!) but a child gets the idea that 3 or 6 or 7 or 10 is a definite value. And its easy to see that the unit measures all the numbers.

That is a great place to start!

Here is another example, also math but a little farther down the line:

OLYMPUS DIGITAL CAMERA

This is the little tiny unit. On the ten bar there are ten units. It take 10 ten bars to make a 100 and 10 hundreds to build a thousand. This is just the ‘meet the decimal system’ stage. (Please know that there are many more of each of these.)  But with these simple things, its possible to teach the child all the operations of arithmetic. How? Because all that is required is the skill of counting to ten.

But that is not all. Montessori didn’t waste any time teaching concepts in  slow singular-case stages. Can we later bring out these same materials and begin a lesson on geometry? The unit corresponds to a point; in motion we could understand how a line is made up of infinite points. And from a line we can have a surface, and from a surface, we can understand a solid.

We could go on through the materials for a long time. Bu what I wanted to illustrate is simply this: Dr. Montessori was able to design her materials based on a fundamental and objective understanding of the world; her materials bear witness to the origin of things as ancient philosophers describe. When we teach little children with these materials and bring to life, make accessible, the origin of things, it is as though we gaze upon a  beauty ever ancient and ever new.

OLYMPUS DIGITAL CAMERA

 

One advantage of studying in an old Montessori Training Centre is the exposure to old Montessori materials. Really old materials. Materials that are so permeated with oral tradition that all that is necessary to understand the concept is to hold the material.  I jest but slightly.

These are some of the geometric solids (the sphere, the cone, and the cube). Nowadays, these shapes are base 10cm and painted a dark, almost navy, blue. In the ‘old days,’ they were base 5cm in light blue.

It might seem like a small thing, but l recall that Dr. Montessori developed these materials to a particular size according to the children’s response. When Ms. Grazzini took these ancient geometric solids from the shelf, I almost fainted with delight.

See how the light reflects off the them? We can see a delicate shading as the light falls on the faces of the solid; even the sphere seems round! These are solids that are easy to draw, color and copy!

As opposed to these:

450px-GeometricSolids

 

Of course they are still beautiful. And they are useful for teaching.

But I prefer the ancient loveliness of the light blue solids.