benedetto scoppola: from euclid to montessori, the construction of rational geometry

Praha, March 16, 2013.

From Euclid to Montessori, the construction of rational

geometry

Benedetto Scoppola, Universita’ di Roma “Tor Vergata”

Summary

- Two quotes

- ….

First quote (Maria Montessori Rome course May 5, 1931)

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 the origin is clearer and more natural for his mind. We simply have to find a material to make the origin accessible.

What about the origin of mathematics?

This turns out to be a quite subtle question…

Second quote (A. Einstein)

Everything Should Be Made as Simple as Possible, But Not Simpler (1948)

Summary

- Thanks

- Two quotes

- The “origin of things”: Platonic and Euclidean attitude

- Montessori and Greek science

- Pedagogical outcomes

The origin of things

The origin of math, as we know it today, dates back to the ancient Greeks. We have very interesting math from prior populations, but such math, as far as we know, is less structured from a logical point of view. So let us start from the Greeks.

Suggestions

Two books:The Forgotten Revolution: How Science Was Born in 300 BC and Why it Had to Be Rebornby Lucio Russo

Euclid’s Elements, online versionby D.E. Joyce

The Greek science: a long story

We are used to think to “the ancient Greeks” as a whole.

The Greek science ranges from the VI century B.C. to the V century A.D.

A thousand years.

Was it a linear development?

The first centuries

From Pythagoras, who introduced for political reasons a mixture of math and magic, to Archimedes, in III century B.C., the Greek science had a roaring evolution.

Then Rome destroyed the original tradition, and the subsequent science was just a comment of the old one.

Plato

We really know too little about Pythagoras to talk about him. So let us start with Plato. He lived in Athens, at the beginning of the crisis of its power (first half of the IV century B.C.). He stated a complex philosophical theory, in which math had a great role.

You certainly know that who manages math, arithmetic and other similar things, takes for granted the even and the odd, the shapes and the three kinds of angles, and other similar things, depending on the science he studies, and he assumes them as hypotheses, and then it is not necessary to discuss them, taking them as evident principles. Starting from that principles he discusses the other questions, deducing the conclusions that he wanted to prove. (From “Republic”)

Evident principles = truth

In platonic geometry what is evident needs not to be proved, because it is intrinsically true. This saves a lot of work on the fundament of the theory.However, the proofs of some of the “conclusions” mentioned by Plato are not trivial. Here’s an example, presented with “montessorian style”.

Platonic solids

We want to classify all the possible regular solids. They have to have regular faces, and the same number of edges incident to all the vertices.

Let us start from triangular faces.

The basic element (in cardboard)

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The basic structures to construct regular solids:

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We can’t construct a solid starting from an hexagonal shape, because it is flat.

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Starting from a triangular shape we have a tetrahedron

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Starting from a square shape we have a octahedron

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Starting from a pentagonal shape we have a icosahedron

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With the same principle we can construct the esahedron (a.k.a. cube) and the dodecahedron, with pentagonal faces. And that’s it.

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We have used many times the fact that the things are “evidently true”.

However, we can say that the result was not at all evident, and we had to think a bit to obtain it.

SkepticismOne hundred years later the confidence in the truth had a deep crisis. From a philosophical point of view this is the period of skepticism. The crisis was not confined to the Greek world. For instance, this is the period in which the Qoheleth was composed:

“Vanity of vanities! All is vanity.”

Skepticism and math

This cultural attitude had a great and positive impact on the science. Euclid wrote a compendium of all the known geometry starting from an idea of truth defined inside the theory.This is the concept of postulate.We should define few postulates and consider them as the definition of the context in which our theory is valid, and then we have to deduce everything.

Postulates

Postulate 1

To draw a straight line from any point to any point.

Postulate 2

To produce a finite straight line continuously in a straight line.

Hence the first two postulates state that we have a (ungraded) ruler.

Postulate 3

To describe a circle with any center and radius.

The third postulate states that we have a compass

Postulate 4

That all right angles equal one another.

This is a postulate about the possibility to translate angles

Postulate 5That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

The V postulate was widely discussed by mathematicians: for centuries they tried to prove it starting from the other postulates. It sounds like a theorem, actually.In the XIX century it was realized that it is essential to define the space (euclidean) on which the geometry is defined. On the sphere, for instance, it is not true.

TheoremsTheorem 1: To construct an equilateral triangle on a given finite straight line.

This is also in Psicogeometria

Theorem 2To place a straight line equal to a given straight line with one end at a given point.

Comment on theorem 2

You see how everything has to be proved in Euclid’s Elements: since we have only the possibility to draw straight lines, we want to be sure to have the possibility to transport straight lines in the plane, in order to be able to measure them. And we can use only the postulates and Theorem 1! This is the so called rational geometry.

Theorem 47

This is the famous Pythagora’s theorem: In right-angled triangles the square on the side opposite the right angle equals the sum of the squares on the sides containing the right angle. Note that theorem 46 is: To describe a square on a given straight line.

Proof of theorem 47

Summary

Platonic attitude• Start from an

undefined number of evident truth

• Deduce new results, using evident truth as tool to find them

• Positive attitude

Euclidean attitude• Start from five

postulates, true by definition

• Deduce theorems by means of well defined logic rules

• Skeptical attitude

Montessori and Greek science

Did Montessori know all this?

She had a technical education (she studied in the so called Technical Institute) in the newborn Italian nation.

In the ‘60 of the XIX century a wide discussion aroused on the geometry programs of the high school. Two options: Euclidean or Projective geometry.

Euclideans won. This edition of the Euclid’s Elements

became the textbook for all the

Italian high schools.

Look at page 7 of

Montessori’s textbook!

Montessori knew well Euclid

Since we have to suppose that she was a brilliant student, we can safely assume that she knew the difference between Platonic and Euclidean attitude quite well.

She surely knew the Elements: for instance the Proposition 47 as been translated in terms of material

“Material” geometry :Theorems and formulas are proved by

geometric material.

This “material” theorem is a translation of the proof in Euclid’s Elements

A comparison between Montessori and Euclid

As we have seen, the content of Psicogeometria is clearly inspired by the Elements. If you remember the first quote: “Up to a certain epoch arithmetic and geometry were blended together…” you may understand that many arguments in Psicoaritmetica are also inspired by Euclid (for instance the number rods).

However we have to say that Montessori decided to present the geometry to the children with platonic attitude. She was conscious of the difference:

“That which we are about to describe is not an elementary, systematic study of geometry. We only offer the means to prepare the mind for systematic study.”

From Psicogeometria, chapter 2

Then she says:

“The discovery of relationships is certainly most likely to arouse real interest. The theorem itself is not interesting to a child […] However, discovering a relationship oneself, formulating a theorem and possessing the words to describe it correctly, is something truly able to fire the imagination.”

Pedagogical outcomes

The first important point that this complex story tells us is the fact that the discovery comes before the construction of the rational geometry.

Hence the children have to discover things with a material, platonic attitude, and then we have to propose them the rigorous constructions. By the way the last chapter of psicogeometria is “reasoning”.

To educate a skeptic mind

Another very important point to be remembered regards the role of a skeptic attitude in the education. Montessori is the first who realized explicitly that children have to do long works because they want to convince themselves of the things.

To be skeptical implies a lot of work!

An educator prepares the child to develop a rational mind when he/she exploits the tendency to do hard work to be convinced of non intuitive things.

When a child is not convinced of something this represents a great educational opportunity.

Euclid should be studied…

The proposal of Psicogeometria is so interesting and modern because the author knew well the end point of the educational process.

In primary school we can not teach theorem 2 about the translation of segments, but we have to know it if we want to “prepare the mind” to the rational thought.

Why we should “prepare the mind to the rational thought”?This turns out to be the last and more important question: geometry is interesting in itself, but from an educational point of view it is a terrific way to educate children to be rational.

Irrationality brings to fundamentalism and to violence. Maybe this is why Montessori wrote this last quote.

Peace“The education has today, in this particular period, a really enormous importance. And this increasing importance can be said in a single sentence: the education is the weapon of the peace”Maria Montessori, Education and Peace, 1937

My best wishes to be peace constructors.