Praha, March 16, 2013.

Neuroscientific perspectives in Psychogeometry

Benedetto Scoppola, Universita’ di Roma “Tor Vergata”

Summary

- Thanks

- Neuroscience and Mathematics

- A comparison with the Montessori Method

- How did she come to it?

- Further observations

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.

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.

Proportionality (1)

Let us try with another small experiment…

87 261

Proportionality (2)

Let us try again…

3 9

Solution

Here is the correct point for both problems

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:

Facts

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

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.

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.

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.

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.

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?

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.

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.

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.

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.

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

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)

Another example…

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…

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

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

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.

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.

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!

We have seen a list of facts, discovered by recent neurological research.

Montessori knew them all…

How did she come to it?

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.

Montessori’s driving ideas

• Maths teaching based on history.

• Observation!

“Material” geometry and Greek science:

Theorems and formulas are proved by geometric material: an example.

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

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)

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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…

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.”

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.

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

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.

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.

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.

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.

Petriccione’s test

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…

Thank you