## Emma Castelnuovo, The Teaching of Geometry

Emma Castelnuovo, |

My subject concerns Italian high schools during the last two centuries. As this period is very long, I will emphasize only those points which, in my opinion, are rather original.

First of all I will begin talking about a French book of the 18th century which had a great influence on our teaching of geometry. The book is 'Eléments de géométrie' by A C Clairaut, the great mathematician and astronomer; it was written in 1741. The idea of Clairaut was to present geometrical properties in such a way, that beginners would be stimulated to investigate. He says, in his preface, that beginners are not able to follow a book like Euclid^{-}s 'Elements', starting with axioms and abstract properties. He bases his work on examples taken from agriculture, the economy of his country being essentially agricultural: the problem of how to measure fields of different shapes, ...

But Clairaut^{-}s 'Elements' never influenced the teaching of mathematics in France. On the contrary, this book, translated into Italian, was used for many years as a textbook in some technical schools of Lombardia, a region in northern Italy. Without any doubt the book was used because of a strange coincidence: The governors of Lombardia thought that Clairaut^{-}s book would be very suitable for the technical-agricultural schools because many examples were connected with the measurement of fields. It is interesting to observe that Clairaut had foreseen this misunderstanding; in his preface he says, with humor: "I do hope that people do not think that this book has been written for students of agriculture; the problems of measuring fields only represent a stimulus to investigate further." However, Clairaut^{-}s 'Elements', interpreted rightly or wrongly, were widely used in our country in the first half of the last century.

The unification of Italy in 1861 led to the establishment of national curricula in our schools. The curricula of mathematics in secondary schools were published in 1867 by three distinguished mathematicians: Luigi Cremona, Enrico Betti, and Francesco Brioschi. The ideas inspiring these curricula are clearly described by their preliminary directives; they say: "The fundamental aim of mathematics is to accustom pupils to reasoning and deduction." It is clear that the application of mathematics were excluded. I would like to underline that, even without an explicit declaration, these curricula were conceived for students of a high social level. But a very unusual fact happened in Italy: In 1881, that is 15 years after the publication of new curricula, a very open-minded Minister of Education, a doctor named Guido Baccelli (1830-1916), declared that the curriculum of geometry was too abstract for pupils 11 to 14 years old. He succeeded in convincing the mathematicians to introduce into the junior secondary school the so-called "Intuitive Geometry." He thought, particularly, of lower social level children. Thanks to this Minister, Italy was the first country to introduce Intuitive Geometry. But this new course was not very original: It reproduced the senior course of geometry leaving out the traditional proofs.

At the same time, curricula of senior schools have always been inspired by Euclid, even if, at the beginning of this century, some mathematicians declared the absurdity of such a purism in a modem society. I would like to quote the declaration of Guido Castelnuovo, pronounced in 1912 during a national congress on the teaching of mathematics. He says:

In our schools we drive students to idolize a perfection which is illusory, instead of encouraging them to work with approximations. We must accustom them to compare theory with practice in order to prepare people able to participate in the life of our country.

But voices like this always remained unheard at the political level.

After the last war many initiatives flourished on education. For what concerning the teaching of geometry in junior secondary schools, a critical study of Clairaut^{-}s old book gave brilliant ideas. The examples on fields of different shapes and their measurement were replaced by up-to-date examples and problems, and geometry was strictly connected to the other branches of mathematics in order to investigate practical and theoretical problems. In 1979 a new reform of the first secondary cycle officially recognized the value of these experiments. A geometry motivated by various real problems became a part of a brilliant and widespread course of mathematics. A "dynamic" course whose principal aim is to sensitize pupils to the function concept.

I have to point out that teachers have always had full freedom in their teaching. It is this freedom, perhaps, that saved us from the so-called "modern mathematics."

When, in the 60s most countries were introducing at all levels mathematics inspired by set theory, Italy was continuing with its curricula and methodology, completely oblivious to a fashion which was invading the world. Why? Some foreign mathematicians say that our behavior was due to the influence that Galileo^{-}s thought has had on Italian people: Galileo never proceeded without experimenting or without foreseeing the result which one experiment could produce. I really should like to think that this interpretation reflects the truth!

JOC/EFR July 2014

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