Slides of lecture at Psycogeometry conference in Prague 2013.

1. Praha, March 16, 2013.
Neuroscientific perspectives in
Psychogeometry
Benedetto Scoppola, Universita’ di Roma “Tor
Vergata”

2. Summary
- Thanks
- Neuroscience and Mathematics
- A comparison with the Montessori
Method
- How did she come to it?
- Further observations

3. Neuroscience and
Mathematics
Many of the following ideas on number
perception come from the book “The sense of
number” (“La bosse des maths”) by S.
Deahaene.
There are two different ways of studying how
the brain perceives mathematics. The first
way is based on cognitive tests.

4. The representation of the
numbers on the line
If one is asked to press a button with the right
hand if a number appearing on a screen is
bigger than 5 and to press a button with the
left hand otherwise, the time needed to give
the exact answer is much longer when the
numbers are closer to 5 (say, 4 and 6). This is
considered an evidence of the geometrical
representation of the numbers in our brain.

5. Proportionality (1)
Let us try with another small experiment…
87 261

6. Proportionality (2)
Let us try again…
3 9

7. Solution
Here is the correct point for both problems

8. The ability to assign a space
proportional to the differences seems to
be crucial for the math developing. For
educated persons it is easy for small
numbers, difficult for larger numbers.
“Better” students perform better.
Hence, let us start a short list of facts,
suggested by neuroscientific research:

9. Facts
• The brain represents the numbers on a line.
Proportionality is crucial in math education.

10. Another experiment:
Tap with your right hand on your right
leg if the right set is more numerous,
with your left hand on left leg otherwise.

14. We are able to immediately detect the
larger between various sets of objects
when the sets are very small (up to
three) or when the number of objects is
very different in the sets. This process
is called subitization.
We are very slow to detect small
differences between numerous sets.

15. Facts
• The brain represents the numbers on a line.
Proportionality is crucial in math education.
• The brain perceives exactly small quantities.
Larger quantities are perceived
approximately.

16. Another techniques: PET and
FMR
The Positrons Emission Tomography and the
Functional Magnetic Resonance are
sophisticated diagnostic techniques. They are
able to identify the areas of the brain involved
in a specific activity.
Subitization and spatial perception, together
with the refined control of the hands, are
mainly located in the right hemisphere. These
abilities are present in many animal species,
they are clearly an advantage in natural
selection.

17. Is this enough to perceive
Mathematics?
Obviously not!
In mathematics we are able to treat
exact quantities, also very large ones,
and to describe exactly abstract
geometrical objects. How can we do
this?

18. Language and symbols
The ability to associate a symbol to an
object or a concept is (almost) specific
to humans. This ability has been
selected by the fact that language
improves the possibility to communicate
and hence to adapt to the environment.
This ability is located in the left
hemisphere of the brain.

19. Correct perception of
Mathematics
Mathematical concepts (geometrical and
arithmetical) are correctly perceived if the
symbolic area of the brain communicates with
the perceptive area. This is evident by PET
and FMR analysis. Maths panic is mainly
originated by the fact that the mathematical
concepts are treated only by the linguistic-
symbolic area of the brain.

20. Facts
• The brain represents the numbers on a line.
Proportionality is crucial in math education.
• The brain perceives exactly small quantities. Larger
quantities are perceived approximately.
• The ability to treat large quantities depends on the
interaction of the perceptive area with the symbolic
area. Such areas are far apart.
• The perceptive area is very close to the area that
moves the hands.
• The perceptive area treats both quantities and
shapes.

21. A comparison with Montessori
Method (1)
• The brain represents the numbers on a line.
Proportionality is crucial in math education.
Think also of the spindle boxes.

22. A comparison with Montessori
Method (2)
• The brain perceives exactly small quantities. Larger
quantities are perceived approximately.
“Children perceive clearly small numbers,
because they know they have one nose, two
hands, five fingers. […] With number rods we
want to give order to vague concepts acquired
empirically”
M. Montessori, Psicoaritmetica

23. A comparison with Montessori
Method (3)
• The ability to treat large quantities depend on the
interaction of the perceptive area with the symbolic
area. Such areas are far apart.
“To fix this set of notions of fundamental
importance well, we have to add to this lesson
also the knowledge of the numerical symbols”
M. Montessori, Psicoaritmetica
(about number rods)

24. Another example…

25. A comparison with Montessori
Method (4)
• The perceptive area is very close to the area
that moves the hands.
This seems to be related with the
Montessorian concept of peripheral
education, leaving the centre free…

26. Peripheral education (1)
“The process for achieving this result differs from
usual. It is not about fixing our mind on an idea, but
about handling an object and examining it with our
senses, moving it continuously and reproducing it
with sensitive images (drawings, papers, paper
works, etc.). The mind thus comes into contact and
lingers on the object through the periphery, taking in
everything that the object can give us. The hand
touches the evidence and the mind discovers the
secret.”
From Psicogeometria

27. Peripheral education (2)
“Therefore, ours is a peripheral education that
replaces the old-style central education. The
centre is left free to unfold in keeping with its
natural energy. We neither need to know it,
nor do we need to propose clear and precise
fulfilment of its needs.
What is necessary is to respect it.”
From the introduction of Psicogeometria

28. Peripheral education and
Neuroscience
The accent on the peripheral education
means that the teaching of maths has
to start from the perceptive area.
Remember that it is accepted by
modern research that the origin of the
maths panic is a teaching of maths
addressed only to the symbolic
(language) area.

29. A comparison with Montessori
Method (5)
• The perceptive area treats both quantities
and shapes. Hence an interaction between
arithmetic and geometry is crucial.
There are many arithmetical ideas in
Psicogeometria: for instance fractions are
treated in a geometrical sense.

30. Geometrical ideas in
Psicoaritmetica
• Number rods: numbers and sums are
presented in a geometrical sense.
• Hierarchies (positional notation) are
presented in a geometrical way.
• Product, distributive property and the square
of binomials and of trinomials are presented
as in Euclid’s Elements.
• Square roots are computed in an abstract
geometrical sense!

31. We have seen a list of facts, discovered
by recent neurological research.
Montessori knew them all…
How did she come to it?

32. Three options
• She actually came from the distant
planet Zorg, where pedagogy is far
more advanced than on Earth.
• Neuroscientists are Montessorian.
• She had different driving idea(s),
helping her to find correct answers to
teaching questions.

33. Montessori’s driving ideas
• Maths teaching based on history.
• Observation!

34. “Material” geometry and Greek science:
Theorems and formulas are proved by
geometric material: an example.

37. This “material” theorem proves that
Montessori was deeply inspired by
Euclid’s Elements:

38. Connections with Greek
science (2)
Another example: Montessori’s counters.
'The elements of numbers are the even and
the odd, and of these the latter [odd] is
limited and the former [even] unlimited. […]
and numbers, as I have said, constitute the
whole universe.'
Aristotle, Metaphysics (1.5 987a13-19)
• • • • • •
------→ ----→ •
• • • • • •

39. A general principle
The way man introduced for the first
time a mathematical concept is a
natural way to introduce the same
concept to children.
Unstated in Montessori’s books.
Stated (continuously?) in her lessons…

40. From lesson 31, Rome course
May 5, 1931
“Up to a certain epoch arithmetic and
geometry were blended together. Then
they had to be divided. But the simpler
and clearer thing is the origin of things:
as I used 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.”

41. Further observations…
Montessori gave enormous importance
to the idea of discovery as the driving
force of learning.
This is strictly tied to the other
characteristic of Montessori thought: the
existence of sensitive periods.

42. Discovery and sensitive periods
“Those working in education, who have managed to arouse
interest that leads to an action performed with all one’s effort
and effective enthusiasm, have succeeded in waking the man.”
“It is evident how a certain thing may not interest a six-year-old
child, who understands but remains indifferent and is therefore
inattentive and unmotivated, yet, when presented in the same
way to a four-year- old child, he understands and responds
actively.”
“Existing interests are the foundation for further interests -
logically connected to them. Increasingly extensive knowledge
can gradually arrange itself around a primitive nucleus, as
mental development takes place.”
From the introduction of Psicogeometria

43. Synapses formation
In the first three years of life an enormous
amount of synapses are generated in the
brain. Only half of them survive after
adolescence. They are selected by brain
activity.
Synapses connecting different regions are
selected in different ages. The degree of
interconnection increases.
The plasticity of the brain decreases.

44. Conjecture
Is the existence of sensitive periods for
some activity connected to the proper
period of selection of synapses which
connect the areas involved in that
activity? Does our brain give us the
interest and satisfaction of discovery for
the right activities in each period of the
life? We are working on it.

45. Observations
• The earlier synapses (under three years of
age) connect visual and sensorial areas with
areas of perception. Very close brain areas.
• The earlier sensitive periods described by
Montessori in Psicogeometria are related to
activities involving in particular the right
hemisphere. Close brain areas.
• The activities of subsequent sensitive periods
involve the connection between perceptive
and symbolic areas. Far brain areas.

46. Another (sad) result
It seems that the effects of correct
choices are not permanent.
Recent results on Petriccione’s test
(a test with a great piagetian flavor…)
show that when a certain activity is
interrupted also the ability produced by
that activity is interrupted.

47. Petriccione’s test

48. Conclusions
• Recent techniques give us the possibility to
understand better the value of teaching
methods.
• Montessori suggests very good activities from
a neurological point of view.
• The more we understand, the greater the
teacher’s responsibility. And, on the other
hand, the beauty of teaching emerges…

49. Thank you