Slides of lecture at Psycogeometry conference in Prague 2013.

1. Praha, March 16, 2013.
From Euclid to Montessori,
the construction of rational
geometry
Benedetto Scoppola, Universita’ di Roma “Tor
Vergata”

2. Summary
- Two quotes
- ….

3. 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.

4. What about the origin of
mathematics?
This turns out to be a quite subtle
question…

5. Second quote (A. Einstein)
Everything Should Be Made as Simple as
Possible, But Not Simpler (1948)

6. Summary
- Thanks
- Two quotes
- The “origin of things”: Platonic and
Euclidean attitude
- Montessori and Greek science
- Pedagogical outcomes

7. 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.

8. Suggestions
Two books:
The Forgotten Revolution: How Science
Was Born in 300 BC and Why it Had to
Be Reborn
by Lucio Russo
Euclid’s Elements, online version
by D.E. Joyce

9. 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?

10. 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.

11. 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.

12. 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”)

13. 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”.

14. 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.

15. The basic element
(in cardboard)
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.

16. regular solids:
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
The basic structures to construct

17. We can’t construct a solid starting from
an hexagonal shape, because it is flat.
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.

18. Starting from a triangular shape we
have a tetrahedron
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.

19. Starting from a square shape we have a
octahedron
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.

20. Starting from a pentagonal shape we
have a icosahedron
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.

21. With the same principle we can
construct the esahedron (a.k.a. cube)
and the dodecahedron, with pentagonal
faces. And that’s it.
TIFF (Uncompressed) decompressor
are needed to see this picture.
QuickTime™ and a

22. 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.

23. Skepticism
One 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.”

24. 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.

25. Postulates
Postulate 1
To draw a straight line from any point to
any point.

26. 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.

27. Postulate 3
To describe a circle with any center and
radius.
The third postulate states that we have
a compass

28. Postulate 4
That all right angles equal one another.
This is a postulate about the possibility
to translate angles

29. Postulate 5
That, 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.

30. 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.

31. Theorems
Theorem 1: To construct an equilateral
triangle on a given finite straight line.

32. This is also in Psicogeometria

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

34. 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.

35. 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.

36. Proof of theorem 47

37. Summary
Platonic attitude Euclidean attitude
• Start from an • Start from five
undefined number of postulates, true by
evident truth definition
• Deduce new results, • Deduce theorems by
using evident truth means of well
as tool to find them defined logic rules
• Positive attitude • Skeptical attitude

38. 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.

39. Euclideans won.
This edition of the
Euclid’s Elements
became the
textbook for all the
Italian high
schools.

40. Look at page
7 of
Montessori’s
textbook!

41. 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

42. “Material” geometry :
Theorems and formulas are proved by
geometric material.

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

46. 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).

47. 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

48. 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.”

49. 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”.

50. 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.

52. 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.

53. 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.

54. 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.

55. 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.