Benedetto Scoppola: Neuroscientific perspectives in Psychogeometry
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 MathematicsMany of the following ideas on numberperception come from the book “The sense ofnumber” (“La bosse des maths”) by S.Deahaene.There are two different ways of studying howthe brain perceives mathematics. The firstway is based on cognitive tests.
The representation of the numbers on the lineIf one is asked to press a button with the righthand if a number appearing on a screen isbigger than 5 and to press a button with theleft hand otherwise, the time needed to givethe exact answer is much longer when thenumbers are closer to 5 (say, 4 and 6). This isconsidered an evidence of the geometricalrepresentation of the numbers in our brain.
Proportionality (1)Let us try with another small experiment… 87 261
SolutionHere is the correct point for both problems
The ability to assign a spaceproportional to the differences seems tobe crucial for the math developing. Foreducated persons it is easy for smallnumbers, 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 rightleg if the right set is more numerous,with your left hand on left leg otherwise.
We are able to immediately detect thelarger between various sets of objectswhen the sets are very small (up tothree) or when the number of objects isvery different in the sets. This processis called subitization.We are very slow to detect smalldifferences 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 treatexact quantities, also very large ones,and to describe exactly abstractgeometrical objects. How can we dothis?
Language and symbolsThe ability to associate a symbol to anobject or a concept is (almost) specificto humans. This ability has beenselected by the fact that languageimproves the possibility to communicateand hence to adapt to the environment.This ability is located in the lefthemisphere of the brain.
Correct perception of MathematicsMathematical concepts (geometrical andarithmetical) are correctly perceived if thesymbolic area of the brain communicates withthe perceptive area. This is evident by PETand FMR analysis. Maths panic is mainlyoriginated by the fact that the mathematicalconcepts 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)
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 fromusual. It is not about fixing our mind on an idea, butabout handling an object and examining it with oursenses, moving it continuously and reproducing itwith sensitive images (drawings, papers, paperworks, etc.). The mind thus comes into contact andlingers on the object through the periphery, taking ineverything that the object can give us. The handtouches the evidence and the mind discovers thesecret.”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 NeuroscienceThe accent on the peripheral educationmeans that the teaching of maths hasto start from the perceptive area.Remember that it is accepted bymodern research that the origin of themaths panic is a teaching of mathsaddressed 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, discoveredby 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 thatMontessori was deeply inspired byEuclid’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)• • • • • •------→ ----→ •• • • • • •
A general principleThe way man introduced for the firsttime a mathematical concept is anatural way to introduce the sameconcept 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 importanceto the idea of discovery as the drivingforce of learning.This is strictly tied to the othercharacteristic of Montessori thought: theexistence 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-oldchild, who understands but remains indifferent and is thereforeinattentive and unmotivated, yet, when presented in the sameway to a four-year- old child, he understands and respondsactively.”“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 formationIn the first three years of life an enormousamount of synapses are generated in thebrain. Only half of them survive afteradolescence. They are selected by brainactivity.Synapses connecting different regions areselected in different ages. The degree ofinterconnection increases.The plasticity of the brain decreases.
ConjectureIs the existence of sensitive periods forsome activity connected to the properperiod of selection of synapses whichconnect the areas involved in thatactivity? Does our brain give us theinterest and satisfaction of discovery forthe right activities in each period of thelife? 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) resultIt seems that the effects of correctchoices are not permanent.Recent results on Petriccione’s test(a test with a great piagetian flavor…)show that when a certain activity isinterrupted also the ability produced bythat activity is interrupted.
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…