From Euclid as Textbook to the Giovanni Gentile Reform (1867–1923)

Paedagogica Historica
Vol. 42, Nos. 4&5, August 2006, pp. 587–613
From Euclid as Textbook to the
Giovanni Gentile Reform (1867–1923):
Problems, Methods and Debates in
Mathematics Teaching in Italy
Livia Giacardi
Taylor and
Paedagogica
10.1080/00309230600806880
CPDH_A_180631.sgm
0030-9230
Original
Stichting
2006
000000August
4/5
42
[email protected]
LiviaGiacardi
Article
Paedagogica
(print)/1477-674X
Francis
Historica
2006LtdHistorica
(online)
The earliest legislation aimed to give comprehensive organization to the Italian education system was the
Casati law, from the name of the then Minister for Education Gabrio Casati who drafted it. Promulgated
by King Vittorio Emanuele II on 13 November 1859, the new law was designed to reorganize the school
system in Piedmont and Lombardy, and was gradually and with difficulty extended to the other Italian regions.
All legislation regarding education in Italy had been based on this law until 1923, when Giovanni Gentile,
a prominent figure among Italian Idealist philosophers, introduced the reform that brought important changes
to the school system, while maintaining various key features. To give a comprehensive view of the history of
the teaching of mathematics in secondary schools in Italy over this 60-year period, we must look beyond the
legislative, political and social factors, however important these undoubtedly are, to examine other factors
not comprehensively studied as yet. On the one hand, we need to consider the role of mathematicians involved
in advanced research; on the other hand, the role played by many different factors: secondary school teachers
and their associations; the textbooks; journals concerning the teaching of mathematics; publishers’ initiatives;
conferences on teaching methods and practices; debates on methodology; international influences; teacher
training. The purpose of this article is to give a general overview of the period, while describing in detail certain
decisive moments, in order to show clearly the effects some decisions had, the debates they gave rise to, as well
as the work carried out and the methodological approaches adopted by the mathematicians involved.
L. Cremona and Euclid’s Elements as a Textbook: Mathematics as ‘Mental
Gymnastics’1
The Casati law divided secondary education into two branches: classical (consisting
of ginnasio (five years) and liceo (three years) leading on to university studies and
1
I am indebted to Gert Schubring for having invited me to contribute to the present special issue
and for his suggestions. I am very grateful to Lucia Ciarrapico and to Giacomo Michelacci for their
help in finding some archival sources. I also want to express my gratitude to Emma Castelnuovo for
her help and to Paolo Freguglia for his support in organizing the workshops on this subject at the
Domus Galilaeana in Pisa.
ISSN 0030-9230 (print)/ISSN 1477-674X (online)/06/040587–27
© 2006 Stichting Paedagogica Historica
DOI: 10.1080/00309230600806880
588 L. Giacardi
intended to form the elites of the future, and technical (lower three years and upper
three years), intended as training for trades, and not as leading to university admission (see Figure 1).
In any case, it was the ginnasio-liceo that formed the core of the secondary school
system in Italy. In the technical institutes, only the physics–mathematics stream,
created in 1860, gave access to the science faculties at universities. Despite ups and
downs it remained for about sixty years the branch of secondary education in which
Figure 1.
Organization of the Italian School System: From the Casati Act (1859) to 1877.
Figure 1 Organization of the Italian School System: From the Casati Act (1859) to 1877.
Paedagogica Historica
589
mathematics was of prime importance, and is to be credited for having educated
mathematicians of scientific standing such as Vito Volterra, Corrado Segre and
Francesco Severi.2
Important changes for the teaching of mathematics, however, resulted from the Act
of Parliament issued by the Minister for Education, Michele Coppino, on 10 October
1867.3 The mathematics syllabi and instructions on teaching methods were actually
the brainchild of the geometry scholar Luigi Cremona who reintroduced Euclid’s
Elements as the textbook to be used in the classical secondary schools. Indeed, he felt
that:
… in the classical secondary schools, mathematics should not be seen merely as a set of
propositions or theories having their own intrinsic value, which young people are required
to learn in order to apply them in real life; Rather, it is principally a means to develop
general knowledge, a kind of mental gymnastics aimed at exercising the faculty of
reason…. As for the teaching of geometry: if we wish to make it as educationally effective
as possible, we have only to follow in our own schools the example of the English, and
reintroduce Euclid’s Elements, which are universally acknowledged to be the most perfect
model of rigorous reasoning.4
Coppino’s law laid down general guidelines for the curriculum that extended over the
fifth year of the ginnasio, and over the first and second years of the liceo, and particularly stressed the importance of the teaching method:
When taught with the same method used by the ancient Greeks [writes Cremona],
geometry is easier and more interesting than the abstract science of numbers…. The
teacher is advised to follow Euclid’s method, because it is the best possible way to create
in young minds the habit of constant rigor in their reasoning. Above all, he must not
pollute the purity of the geometry of ancient times by transforming geometrical theorems
into algebraic formulae, thus replacing the concrete magnitudes with their measurements.5
Likewise, arithmetic was to be taught by deductive and demonstrative methods.
There was also the recommendation, little more than a suggestion in the algebra
curriculum, that students should be guided towards the fruitful method of limits and
the concept of function; this suggestion remained, however, vague and isolated.6
Cremona’s principal concern was the education of the future elite. He believed the
role of the ginnasio-liceo was not to furnish students with a mass of knowledge, but
rather to provide a method to deal effectively with problems.
2
Cf. Ulivi, Elisabetta. “Sull’insegnamento scientifico nella scuola secondaria dalla legge Casati
alla riforma Gentile: la sezione fisico-matematica.” Archimede 4 (1978): 166–82. For an overview of
the legislative measures concerning the teaching of mathematics in secondary schools, cf. Vita,
Vincenzo. I programmi di matematica per le scuole secondarie dall’unità d’Italia al 1986. Rilettura storicocritica. Bologna: Pitagora, 1986.
3
The text of this Act can be viewed on the website (http://www.dm.unito.it/mathesis/
documents.html) together with the most significant legislative measures regarding the teaching of
mathematics in the Italian secondary school from 1867 to 1923.
4
Cf. “Istruzioni e programmi per l’insegnamento della matematica nei ginnasi e nei licei.”
Supplemento alla Gazzetta Ufficiale del Regno d’Italia, Firenze, 24 October 1867.
5
Cf. “Geometria.” Ibid.
6
Cf. “Aritmetica ragionata e algebra.” Ibid.
590 L. Giacardi
Just one year after the Coppino reform came into force, a new textbook was
published: Gli Elementi di Euclide con note aggiunte ed esercizi ad uso de’ ginnasi e de’ licei
a cura di Enrico Betti e Francesco Brioschi. Although Cremona was not mentioned by
name as one of the authors, his was the guiding hand behind the book, and in this he
was helped by a teacher from Pavia, Giacomo Platner, as can be seen from his ample
correspondence with Betti, which remains as an invaluable record, and enables us to
reconstruct the origins of the book and to establish the key role played by Cremona.7
The writers’ aim was to provide teachers with a completely Italian textbook in keeping
with the new curricula and to: ‘… do away with the myriad of shoddy little manuals,
produced purely for profit, which infested the very schools where textbooks needed
to be rigorous in their science and solid in their teaching method’.8 Moreover,
Cremona and his co-writers intended to oppose the methodological approach adopted
in the Eléments de géometrie (1794) by Adrien Marie Legendre and, in particular, the
deliberate use he made of arithmetic and algebra in his treatment of geometry. This
aspect of Legendre’s work had been further exacerbated by Marie A. Blanchet in his
expanded edition of the Eléments, and by various Italian translators and imitators.9
The text presents Euclid’s Elements translated into Italian without, however,
providing any valid didactic method or style, either from the point of view of the
language adopted, or from the content. The language is purely Euclidean, with no
concession made to algebraic symbolism, or sometimes even to clarity. It was therefore only to be expected that when the book was published it elicited much negative
response: teachers did not like its complicated language, while mathematicians saw in
it an unwelcome return to the past and thus a refusal to acknowledge the new discoveries in the field of geometry.
The debate that arose after the publication of the Betti-Brioschi text broadened
when Giuseppe Battaglini, one of the main promoters of the propagation of nonEuclidean geometry in Italy, published the translation of an article by an English
mathematician, J. M. Wilson, in his journal, the Giornale di matematiche.
In direct opposition to the prevailing opinion in England, Wilson severely criticized
Euclid’s Elements from both a scientific and the didactic point of view. His criticisms,
which were neither particularly original nor new, were the following: Euclid did not
accept the ‘hypothetical constructions’; he ‘avoids the method of superposition as
much as possible’; he separated geometry from arithmetic; lastly, his theory of
parallels was ‘faulty’, and his theory of proportions obsolete (‘the fifth book is dead’).
From the didactic point of view, Wilson observed that the language was obscure and
7
Cf., for example, Gatto, Romano. “Lettere di Luigi Cremona a Enrico Betti (1860–1890).” In
La corrispondenza di Luigi Cremona (1830–1903), edited by M. Menghini. Milano: Quaderni
PRISTEM 9 (1996): 7–90.
8
Brioschi, Francesco, and Luigi Cremona. “Al signor Direttore del Giornale di matematiche ad
uso degli studenti delle Università italiane.” Giornale di Matematiche 7 (1869): 51–54. An extract of
this letter was translated into French by J. Hoüel and published in Nouvelles Annales de Mathématiques 2, no. 8 (1869): 278–83.
9
Cf. Schubring, Gert. “Neues über Legendre in Italien.” In Mathematik im Fluß der Zeit, edited
by W. Hein, and P. Ullrich. Algorismus 44 (2004): 256–74.
Paedagogica Historica
591
complicated, that the treatment did not encourage discovery (‘it is unsuggestive’),
that it was seriously limited by its exclusion of all arithmetical applications and, most
importantly, that a textbook written so long ago ‘could not be a fit introduction to the
science at the present time’. After these comments, he concluded categorically:
‘Euclid is antiquated, artificial, unscientific and ill-adapted for a textbook’.10
Response from Cremona and Brioschi was immediate and took the form of a letter
to the editor that appeared in the 1869 issue of the Giornale di matematiche. The two
authors examined each of Wilson’s criticisms of the Elements, but were actually unable
to produce entirely convincing arguments. They were, in the end, forced to concede
‘that in some points [the Elements] should be revised and simplified, but not distorted
by mixing geometry with arithmetic’.11
The Giornale di matematiche published other comments for and against Euclid’s
Elements as a textbook, and the debate continued to create interest for several years
among academics and, in equal measure, among teachers. This becomes clear as we
read through the correspondence, published and unpublished, of the Italian mathematicians of that time (Battaglini, Beltrami, Cremona, Forti, Genocchi, Bellavitis
etc.),12 and also from the prefaces to the elementary geometry textbooks, of which
many were published in the second half of the nineteenth century.
For Cremona the return to Euclid was a means, through a compromise, of
relaunching mathematics teaching. His real hope was for a much deeper reform, as is
clearly perceptible from his correspondence and from an article he wrote in 186013 in
which he set forth some of his ideas on the teaching of elementary geometry, outlining
among other things a ‘dynamic teaching’ method based on the concept of transformation. Again it was Cremona who wrote Elementi di geometria projettiva ad uso degli
10
Wilson, J. M. “Euclid as a Textbook of Elementary Geometry.” Educational Times (1868):
125–28. Translated by R. Rubini under the title “Euclide come testo di geometria elementare.”
Giornale di matematiche 6 (1868): 361–8.
11
Brioschi, Francesco, and Luigi Cremona. “Al signor Direttore del Giornale di matematiche ad
uso degli studenti delle Università italiane.” Giornale di Matematiche 7 (1869): 53.
12
Cf., for example, the letters of Hoüel to Genocchi in Fenoglio, Lorenza, and Livia Giacardi.
“La polemica Genocchi-Beltrami sulle superficie pseudosferiche: una tappa nella storia del concetto
di superficie.” In Angelo Genocchi e i suoi interlocutori scientifici. Contributi dall’epistolario, edited by A.
Conte, and L. Giacardi. Torino: Deputazione subalpina di storia patria, 1991: 155–209; Giacardi,
Livia. “La corrispondenza fra Jules Hoüel e Luigi Cremona (1867–1878).” In La corrispondenza di
Luigi Cremona (1830–1903), edited by A. Millan Gasca. Vol. I. Quaderno della Rivista di Storia
della Scienza, no. 24, Università di Roma “La Sapienza” (1992): 77–88; Calleri, Paola, and Livia
Giacardi. “Le lettere di Giuseppe Battaglini a Jules Hoüel (1867–1878). La diffusione delle
geometrie non euclidee in Italia.” Rivista di Storia della Scienza 2 (3.1) (1995): 127–209; Giacardi,
Livia. “Scientific research and teaching problems in Beltrami’s letters to Hoüel.” In Using History to
Teach Mathematics: an International Perspective, edited by V. Katz. Washington, DC: Mathematical
Association of America, 2000: 213–23; cf. also the letters of G. Bellavitis and A. Forti to J. Hoüel,
Paris, Archives, Académie des Sciences, Correspondance adressée à J. Hoüel.
13
Cremona, Luigi. “Considerazioni di storia della geometria, in occasione di un libro di geometria elementare pubblicato a Firenze.” (1860). In Opere matematiche. Milano: Hoepli. Vol. I (1914):
176–207.
592 L. Giacardi
istituti tecnici (1873) after projective geometry was included in the syllabus of the
physics–mathematics stream in 1871.14
Mathematical Textbooks for Secondary Schools
In any case the reintroduction of Euclid and the heated debate that ensued15 played
a catalyzing role in arousing the Italian secondary school from the state of inertia into
which it had fallen. As Enrico d’Ovidio and A. Sannia wrote, ‘it was like a surgical
operation: extremely painful, but curative’.16 Not only did it focus attention on
important issues regarding the teaching of geometry (the need for a thorough analysis
of foundations; the role to be played by the ‘rigid motion’ in the study of geometrical
problems; the usefulness, in geometrical exposition, of a previous theory of real
numbers; and, lastly, the relationship between rigorous reasoning and intuition), but
it also encouraged specialized publications covering this field. On one hand we have
the founding of the first journals dealing with problems concerning the teaching of
mathematics,17 some of which survive to this day (Periodico di Matematica, first
published in 1886, for example); on the other hand the publication of a great number
of high-quality teaching textbooks written by the most eminent Italian mathematicians of the time (Betti and Brioschi, D’Ovidio, De Paolis, Veronese, De Franchis,
Enriques, and Amaldi, etc.). Through their books, these scholars compared and
contrasted different methodological approaches, thus stimulating debate on the
teaching of mathematics.18
14
Cf. Di Sieno, Simonetta. “Cremona e la formazione tecnica preuniversitaria nella seconda metà
dell’Ottocento.” In Conferenze e seminari, 2004–2005. Torino: Associazione Subalpina Mathesis,
2005: 57–68.
15
For further details on the Coppino Act and the heated debate it provoked, cf. Giacardi, Livia.
“Gli ‘Elementi’ di Euclide come libro di testo. Il dibattito di metà Ottocento in Italia.” In Conferenze
e seminari, 1994–1995. Torino: Associazione Subalpina Mathesis, 1995: 175–88 and Schubring,
Gert. Analysis of Historical Textbooks in Mathematics. Rio de Janeiro: Pontifícia Universidade
Católica do Rio de Janeiro, Departamento de Matemática, 1997: 81–90.
16
Sannia, Achille, and Enrico D’Ovidio. Elementi di Geometria. Napoli: Pellerano, 1895: V.
17
Cf. Furinghetti, Fulvia, and Annamaria Somaglia. “Giornalismo matematico ‘a carattere
elementare’ nella seconda metà dell’Ottocento.” L’insegnamento della matematica e delle scienze
integrate 15 (1992): 816–52.
18
Cf. Maraschini, Walter, and Marta Menghini. “Il metodo euclideo nell’insegnamento della
geometria.” L’educazione matematica 3 (1992): 161–81; Mammana, Carmelo. “I Grundlagen der Geometrie e i libri di testo di geometria in Italia.” Le Matematiche 55 (2000, Supplemento 1): 225–51;
Mammana, Carmelo, and Rossana Tazzioli. “The Mathematical School in Catania at the beginning
of the 20th century and its influence on didactics.” In Proceedings. Histoire et épistémologie dans l’éducation mathématique. Louvain: Univérsité Catholique, 2001: 223–32; Giacardi, Livia. “I manuali
per l’insegnamento della geometria elementare in Italia fra 800 e 900.” In TESEO, Tipografi e editori
scolastico-educativi dell’Ottocento, edited by G. Chiosso. Milano: Editrice Bibliografica, 2003:
XCVII–CXXIII.
Paedagogica Historica
593
The textbook by Achille Sannia and Enrico d’Ovidio (1868–1869)19 follows the
Euclidean method while improving it where it shows weaknesses, and while adding
supplementary topics and exercises to prepare the student for the more advanced
levels of geometry. Aureliano Faifofer’s Elementi di geometria ad uso dei licei (1880) is
notable for the method adopted in treating the theory of the equivalent figures following the guidelines laid down by Duhamel.20 Riccardo De Paolis’s textbook Elementi
di geometria (1884)21 marks the beginning, in Italy, of ‘fusionism’, the name given to
a teaching method where related subjects of plane and solid geometry are studied
together, properties of the latter being applied to the former in order to gain the
maximum benefit. The textbook by Michele De Franchis (1909)22 is notable for its
rigorous approach to the theory of congruence, an approach that introduced the
‘group of motions’. This book, and the textbook by Giuseppe Veronese (1909), were
clearly influenced by studies on the foundations of geometry, which, through the
works of Pasch, Peano, Pieri, Enriques and Veronese himself, culminated in David
Hilbert’s work Grundlagen der Geometrie (1899).
Of the many textbooks published in this period, that which gives most importance
to the demands of teaching method is undoubtedly Elementi di geometria ad uso delle
scuole secondarie superiori (1903).23 Federigo Enriques, an eminent figure in the Italian
school of algebraic geometry, wrote this acclaimed textbook together with Ugo
Amaldi. The scientific and methodological bases for it, as Enriques himself states,
derive from his Questioni riguardanti la geometria elementare (1900), a collection of
papers on problems of elementary mathematics seen from a higher point of view. The
book was written with the contribution of Enriques’s friends and of the members of
his school, and was also influenced by Felix Klein and the many initiatives he carried
forward in Göttingen in support of teaching reform and teacher training.24 In the
Elementi, the subject is approached via the rational-inductive method, in an attempt to
overcome the defect typical of Euclidean exposition, which ‘by presenting
19
Sannia, Achille, and D’Ovidio Enrico. Elementi di Geometria, Napoli: Stab. Tip. delle Belle Arti,
1868–69 (II ed. 1871, III 1876, IX 1895), cf. the review of J. Hoüel. “Revue bibliographique.”
Bulletin des Sciences Mathématiques et Astronomiques 1 (1870): 329–30.
20
Cf. Duhamel, Jean-M.-C. Des méthodes dans les sciences de raisonnement. Paris: Gauthier-Villars,
1878. Vol. II. Note sur l’équivalence: 445–50.
21
De Paolis, Riccardo. Elementi di geometria. Torino: Loescher 1884, cf. Candido, Giacomo. “Sur
la fusion de la planimétrie et de la stéréométrie dans l’enseignement de la géométrie élémentaire en
Italie.” L’Enseignement mathématique a. I (1899): 204–15, in particular 206.
22
De Franchis, Michele. Geometria elementare ad uso dei Licei e dei Ginnasi superiori. Milano–
Palermo–Napoli: Remo Sandron, 1909.
23
Enriques, Federigo, and Ugo Amaldi. Elementi di geometria, ad uso delle scuole secondarie
superiori. Bologna: Zanichelli, 1903; Elementi di geometria elementare ad uso dei ginnasi superiori.
Bologna: Zanichelli, 1904, textbooks republished several times.
24
Cf. Schubring, Gert. “Pure and Applied Mathematics in Divergent Institutional Settings in
Germany: the Role and Impact of Felix Klein.” In The History of Modern Mathematics, edited by D.
Rowe and J. McCleary. London: Academic Press, 1989. Vol. II: 171–220; Gario, Paola. “Quali
corsi per la formazione del docente di matematica? L’opera di Klein e la sua influenza in Italia.”
Bollettino della Unione Matematica Italiana, Sez. A, s. 8, IX-A (2006): 131–41.
594 L. Giacardi
coordinated in a deductive system propositions analysed thoroughly in their logical
connections, conceals the progress towards discovery under a dogmatic form’.25 The
exposition proceeds according to the following structure: on the basis of a series of
observations, the authors enunciate certain postulates, from which the theorems
depending on them are developed by logical reasoning, and continuously they go
back to observations or explanations of intuitive nature.
Textbooks for geometry, above all else, influenced the debate on methodology,
since geometry, more than any other area of mathematics, brings into focus the
problems of methodology inherent in the teaching of mathematics, clarifying the delicate relationship between training and giving information. It is, in any case, worth
mentioning two algebra textbooks with different methodological approaches – one by
Cesare Azelà, the other by Giuseppe Peano – which influenced the subsequent
mathematical literature. The Trattato di algebra elementare (1880) by Arzelà, Professor
of Analysis at the University of Bologna, was one of the most widely adopted
textbooks in secondary schools. Written for the physics–mathematics stream of the
technical institutes, it featured a methodological approach different from that of
Joseph Bertrand, whose textbook, translated by Enrico Betti, was at that time very
widely used in Italy; in fact, the core concept behind the presentation of the material
was not equation but rather function.26 His attention to issues of teaching methodology can be seen in the way he deals with irrational numbers, which are introduced as
the common limit of two convergent series, this method being linked to the measurement of magnitudes by means of approximation by defect and by excess. Peano’s
Aritmetica generale e algebra elementare (1902) reproduced whole sections of Formulaire
Mathématique, and featured the systematic use of logical symbols which ‘contribute
not only brevity, but also precision and clarity’ (p. III).
The Royal Commission for Secondary School Reform: the Proposals of G.
Vailati and Humanitas Scientifica
The years from the Unification of Italy up to the early twentieth century were a period
of great political and social ferment. Italians were also making advances of considerable importance in scientific research, achieving international recognition at the
highest levels with the successes of the Italian school of algebraic geometry, and with
Peano’s studies on logic.
Towards the end of the nineteenth century, studies on the foundations of mathematics created a common area of interest between elementary mathematics and
advanced research. As a result, certain mathematicians committed to pure research
were also personally involved not only in preparing school textbooks, but also, on the
25
Enriques, Federigo. “Sull’insegnamento della geometria razionale.” In Questioni riguardanti le
matematiche elementari. Bologna: Zanichelli, 1912. Vol. I: 24.
26
Cf. Gavagna, Veronica. “Cesare Arzelà e l’insegnamento della matematica.” Bollettino di storia
delle scienze matematiche XII, no. 2 (1992): 251–77.
Paedagogica Historica
595
politico-cultural side, in developing an improved framework of laws on education,
and in teacher training.
The mutual interchange between universities and secondary schools was a further
source of enrichment: university teachers often began their careers as secondary
school teachers (Cremona, Betti, D’Ovidio, De Paolis, for example), while the most
distinguished secondary school teachers taught courses at university (Lazzeri,
Faifofer, Bettazzi, Vailati). This enabled them to incorporate the experience of
teaching on two different levels into their daily work.
The appearance of a considerable number of textbooks and articles on teaching,
now written by Italians, was just one of the signs of this atmosphere of renewal. The
Scuole di Magistero (teacher training schools) were established (1875),27 and the first
teachers’ associations were founded. In 1895–1896 in Turin Rodolfo Bettazzi
founded the Associazione Mathesis, an association with the specific goal of ‘improving
the school system and the training of teachers in both scientific and methodological
aspects of mathematics’. Under the leadership of its presidents, including Bettazzi
himself, Severi, Castelnuovo and Enriques,28 this association was often to make its
voice heard on issues regarding legislation for secondary schools.
Strangely enough, this commitment on the part of mathematicians did not
correspond to a significant improvement in the quality of mathematics teaching
during the last 20 years of the nineteenth century.29 We need only consider the series
of legislative measures enacted between 1881 and 1904 to see how the role of mathematics was progressively weakened, both in the curriculum content and in the
number of lesson hours allocated.
The official reports on the examinations for the licenza liceale (secondary school
diploma) reveal that the teaching of mathematics, and of other subjects as well, was
considered inadequate in many cases.30 Furthermore, the comparative survey of the
curricula and timetabling in classical secondary schools (liceo-ginnasio) in Italy and
other European countries, commissioned by the Ministry of Education in 1887,31
clearly shows the defects of the Italian liceo-ginnasio system when compared with
27
Cf. Nastasi, Pietro. “La Mathesis e il problema della formazione degli insegnanti.” In La
Mathesis, PRISTEM/STORIA 5, Milano: Springer-Verlag Italia, 2002: 59–119; Giacardi, Livia.
“Educare alla scoperta. Le lezioni di Corrado Segre alla Scuola di Magistero.” Bollettino della Unione
Matematica Italiana s. VIII, VI-A (2003): 141–64.
28
Rodolfo Bettazzi was president of the Mathesis Society from 1896 to 1900 and from 1902 to
1904, Francesco Severi was president from 1909 to 1910, Guido Castelnuovo from 1911 to 1914,
Federigo Enriques from 1919 to 1946.
29
Cf., for example, Scarpis, Umberto. “L’insegnamento della matematica nelle Scuole classiche,
I. I successivi programmi dal 1867 al 1910.” In Atti della Sottocommissione italiana per l’insegnamento
matematico. Bollettino della Mathesis III (1911): 25–33; Scorza, Gaetano. “L’insegnamento della
matematica nelle Scuole e negli Istituti tecnici.” Ibid., 49–80.
30
Cf., for example, Ministero della Pubblica Istruzione, Bollettino Ufficiale VIII (May 1882):
375–98 and IX (April 1883): 264–84.
31
Cf. “Esame comparativo dei programmi nelle scuole secondarie classiche.” Bollettino Ufficiale
dell’Istruzione XIII (October 1887): 193–241. Valentino Cerruti, professor of Mechanics at the
University of Rome, was the president of the Commission.
596 L. Giacardi
German schools: first and foremost, the excessive number of lesson hours devoted to
the native language and the lack of foreign languages teaching; furthermore, with
regard to the teaching of mathematics, the absence of subjects like analytical geometry and differential calculus (which had been among the complementary subjects in
Prussia since 1882), the poor coordination between mathematics and physics teaching, and, finally, the adoption of a teaching method that was purely rational, allowing
very little room for practical application (see Table 1).
To revitalize a school system that no longer responded to the demands of the Italian
society, in 1904 Minister Orlando found no better solution than to issue a decree that
allowed students of second-year liceo the option of choosing between Greek and
mathematics, ‘releasing congenitally incapable students from a useless burden’.32
This decision, which was severely criticized by the various teachers’ organizations,
was abolished only in 1911.33
At the beginning of the twentieth century, 40 years after the Unification of Italy,
several factors showed that reform was a pressing need: the evident deficiencies in
secondary school teaching, the changed social and historical context, the influence
of reform movements in other European countries – especially Felix Klein’s movement in Germany and Gaston Darboux’s in France34 – the increasingly active
participation of teachers in the political issues of education, and, finally, the
remarkable increase in the number of pupils enrolled in secondary schools (from
18,231 to 94,572 between 1861 and 1901). Pressed by all these factors, the Minister Leonardo Bianchi in 1905 appointed a Royal Commission for the reform of the
secondary school system.35 This body was made up of professors, teachers and
Ministry inspectors, and its task was to conduct a comprehensive inquiry into
upper secondary schools, and to present proposals regarding the most urgent and
necessary changes to be made. Despite obstacles and opposition, including internal
conflict within the board, in February 1908 the Commission presented a draft law
proposing, on the one hand, a professional technical school with three-year courses
enabling entry to the technical institutes, and, on the other hand, a three-year
course for the lower secondary school common to all schools (scuola media unica)
excluding Latin as a subject, which was to grant students access to the three different branches of the liceo (upper secondary school): classico (with Latin and Greek),
scientifico (with two modern languages and a wider science syllabus), and moderno
(with Latin and two modern languages) (see Table 2).
32
“Programmi di matematica per i ginnasi ed i licei.” Bollettino Ufficiale del Ministero dell’Istruzione
Pubblica XXXI, II, no. 52, Roma 29 December 1904: 2851.
33
Cf. in this regard the wide debate on this matter during the Congress of the Associazione
Mathesis held in Milan in 1905 on 21–22 April, in Bollettino della Mathesis (1905): 37–91.
34
Cf. Belhoste, B., H. Gispert, and N. Hulin. Les sciences au lycée. Un siècle de réformes des mathématiques et de la physique en France et à l’étranger. Paris: Vuibert, 1996, in particular the articles of B.
Belhoste, M. Artigue and G. Schubring.
35
Cf. Bollettino della Mathesis 1905–1906, 1907–1908.
–
–
–
27
24
9
10
4
2
–
196
–
14
(14)
20
13
4 1/2
5
8
–
–
141
19
–
–
31
34
10
8
(1)*
3
5
274
21
23
78
42
9 years
Leipzig
Humanistisches
Gymnasium
Saxony
17
–
–
27
35
8
8
(1)*
6
8
264
18
25
74
38
9 years
Darmstadt
Humanistisches
Gymnasium
Assia
21
–
–
28
34
10
8
(1)*
4
6
263
19
21
77
40
–
–
–
35
23
10
8
8
–
–
198
–
47
47
20
8 years
Ginnasio
Liceo
Frankfurt am
Main
Humanistisches
Gymnasium
9 years
Italy
Prussia
*In Germany philosophy, where taught, is generally incorporated into lesson time dedicated to the Native Language.
Source: “Esame comparativo dei programmi nelle scuole secondarie classiche”. Bollettino Ufficiale dell’Istruzione XIII (October 1887): 204
16
26
50
28
–
18
37 1/2
21
Religion
Native Language
Latin
Greek
Foreign Languages:
a) French
b) English
c) German
Geography and History
Mathematics
Natural History
Physics
Philosophy
Calligraphy
Drawing
8 years
7 years
Vienna
Gymnasium
Lycée
(humanités
classiques)
Subjects
Austria
France
Table 1 Comparison of Timetables and Subjects in the ‘Classical’ Secondary Schools in Italy and Europe, 1887 – Lesson Hours per Week
Paedagogica Historica
597
598 L. Giacardi
Table 2 Reform Project of the Royal Commission, 1909: Subjects and Lesson Timetables
Ginnasio
Subjects
I
II
III
Total
Italian
An outline of History and Political Geography
Elements of Psychology
French
Mathematics
Natural Sciences and Geography
Drawing
Lesson hours per week
9
9
9
27
5
4
3
3
24
5
4
3
3
24
5
4
3
3
24
15
12
9
9
72
Liceo Classico
Subjects
Italian
Latin
Greek
French
History
Geography
Philosophy
Mathematics
Physics
Chemistry
Natural History
Lesson hours per week
German (optional)
Total
I
II
III
IV
V
Total
5
8
–
3
3
2
–
2
–
–
2
25
–
25
4
6
5
2
3
2
–
2
–
–
2
26
–
26
4
6
5
–
3
2
3
2
–
2
–
27
3
30
4
6
5
–
3
–
2
2
3
–
–
25
3
28
4
6
5
–
3
–
3
2
2
–
2
27
3
30
21
32
20
5
15
6
8
10
5
2
6
130
9
139
Liceo Moderno
Subjects
Italian
Latin
French
German or English
History
Geography
Philosophy
Mathematics
Physics
Chemistry
Natural History
Elements of Law and Economics
Drawing
Lesson hours per week
I
II
III
IV
V
Total
5
6
4
–
3
2
–
2
–
–
2
–
2
26
4
5
2
5
3
2
–
2
–
–
2
–
2
27
4
4
2
4
3
2
3
2
–
2
–
–
–
26
4
4
2
3
3
2
2
2
3
–
–
3
–
28
4
3
2
3
3
–
3
2
3
–
2
3
–
28
21
22
12
15
15
8
8
10
6
2
6
6
4
135
Paedagogica Historica
599
Table 2 (Continued)
Liceo Scientifico
Subjects
Italian
French
German or English
History
Geography
Philosophy
Mathematics
Physics
Chemistry
Natural History
Drawing
Lesson hours per week
I
II
III
IV
V
Total
5
3
3
3
2
–
5
–
3
2
2
28
4
2
3
3
2
–
5
–
3
2
2
26
4
–
3
3
2
3
5
3
–
2
2
27
4
–
3
3
2
2
5
3
–
2
2
26
4
–
3
3
2
3
5
3
–
2
2
27
21
5
15
15
10
8
25
9
6
10
10
134
Source: Commissione Reale per l’Ordinamento degli Studi Secondari in Italia. “Orari.” In Relazione. Roma:
Ministero della Pubblica Istruzione, Tipografia L. Cecchini, 1909: 669–672.
The syllabi for mathematics, and the instructions on teaching method, original and
fresh in their approach, were written by Giovanni Vailati and expressed his own vision
of mathematics, where positivist principles, epistemological propositions from
Peano’s school and the need to render culture democratic all blended harmoniously
with pragmatism, and with his deep-rooted belief in the unity of knowledge and in the
educational importance of mathematics.
Vailati’s proposals derived from his lucid analysis of the defects in the Italian
secondary school system. First of all, he found that the teaching was based on passive
learning, which turned the schools into factories for rote learning, where the student was
too much engaged in learning (accipere), and too little motivated to try to understand
(concipere).36 There was also little interaction between humanistic and scientific
culture, with a disproportionate number of lesson hours devoted to the teaching of
Italian and Italian literature. Classes were overcrowded, and there was a lack of
facilities supporting teaching activities, such as libraries or laboratories. Finally,
schools were not supplied with good books (dictionaries, encyclopaedias, editions of
the classics, etc.).
What Vailati proposed was a ‘school as laboratory’, not in the limited sense of a
laboratory for scientific experiments but, rather:
… a place where the pupil is given the opportunity, with the guidance and prompting of
the teacher, to train himself to experiment and solve problems, to take measurements and,
36
Vailati, Giovanni. Review of C. Laisant, La Mathématique: philosophie, enseignement. In Giovanni
Vailati, Scritti, edited by M. Quaranta. Bologna: Forni, 1987. Vol. III: 261.
600 L. Giacardi
above all, to take his own measure, to test himself in his approach to obstacles and difficulties which will stimulate intelligence and encourage initiative.37
In particular, mathematics teaching should adopt an experimental and active
approach, and, since learning moves from the concrete to the abstract, pupils should
never be forced to ‘learn theories before knowing the realities to which they refer’. On
the contrary, they should be required to show that they knew how to do things, not
merely how to repeat things.
The kind of lesson most likely to achieve this objective is the Socratic lesson,
which allows teachers to guide their students towards discovery of mathematical
truths, meanwhile stimulating enquiry and reflection. In a ‘school as laboratory’
teachers must allow an important place in the learning process to moments of play,
as well as to manual activity, which provides ‘an excellent antidote to the common
misconception that one knows something simply because one has learnt certain
words’.38
The effectiveness of a teaching system that moves from the concrete to the abstract
is particularly evident in the teaching of geometry. Intuitive method was the name
usually given to the method to be followed in the first stage of teaching, but Vailati
preferred to use the term experimental or operative geometry, since this is more
indicative of the way in which it differs from the rational geometry to be studied in
the upper high school courses; drawing, the use of simple mathematical instruments
and small experiments allow students to discover some of the properties of geometrical figures, stimulating their desire to understand why this is so, making learning more
interesting.
Furthermore, deduction, maintained Vailati, should be used ‘definitely not to
demonstrate propositions that students already find quite obvious … but rather to use
these propositions to arrive at others which they do not yet know’.39 In this way,
deductive reasoning was to be seen by them as a means to discovery.
Other methodological aspects were stressed by Vailati. One of these was the
importance of showing the applications of algebra to geometry, and vice versa, in
order to make pupils appreciate immediately the underlying unity of the mathematical
disciplines, and to train them to approach any given problem with a variety of methods, choosing, as the situation requires, the best possible approach. He also considered it important to find a balance between intuition and rigour in mathematics
teaching, and to use the history of mathematics in order to achieve three goals: to
encourage dialogue between the scientific and humanistic cultures, to ‘render the
37
Vailati, Giovanni. “Idee pedagogiche di H. G. Wells.” (1906). In ibid., 292.
Vailati, Giovanni. Review of Maria Begey, Del lavoro manuale educativo. In ibid., 265.
39
Vailati, Giovanni. “Sugli attuali programmi per l’insegnamento della matematica nelle scuole
secondarie italiane.” In Atti del IV Congresso Internazionale dei matematici, 6–11 aprile 1908. Roma:
Tip. Accademia dei Lincei, 1909: 485.
38
Paedagogica Historica
601
teaching more fruitful … more effective and also more interesting’,40 and to avoid any
form of dogmatism.
Moreover, in view of the aims of the different courses of study, the concepts
of function and of derivative were introduced in all three branches of the liceo,
the concept of integral in the scientifico, while probability theory and its applications were taught in the moderno to students intending to enter the world of
work, or to continue, turning to technical studies. In the liceo classico the emphasis was on Euclidean geometry, accompanied by readings from the original writings of the great geometers of the ancient world, thus offering the students a
more complete picture of classical civilization, not limited to the fields of art and
literature.41
This reform, and especially the unification of the lower secondary school, was
considered too radical, not only by conservative thinkers but also by the majority of
the members of the national federation of secondary school teachers and even by
forward-looking men like the historian Gaetano Salvemini. He was opposed to a
common middle school (which he calls ‘minestrone-school’42) for two main reasons.
First, a school organized in this way would be ‘schizophrenic’ because it would accept
both students planning to leave school to take up a trade, and students intending to
continue on to high schools, and then to university.
Second, a common middle school would postpone the teaching of Latin,
further weakening the liceo classico whose important function was to educate those
who would take up influential roles in government and in society. Furthermore,
he was convinced that such a radical reform could not possibly be implemented
without a prior period of experimentation, or without the requisite training of the
teachers. He believed that attendance at the ginnasio-liceo schools should be
limited to students whose intellectual capacities and social class would enable
them to continue their studies successfully and to eventually assume leading positions in the state and in society. In his opinion it would have been better to
reform the system of education for trades and to upgrade elementary-school
courses.
The mathematics syllabi prepared by Vailati also attracted criticism. The 1907
article43 in particular, where he suggested experimental and hands-on teaching of
40
Vailati, Giovanni. “Sull’importanza delle ricerche relative alla Storia delle Scienze.” In Giovanni Vailati, Scritti. Vol. II: 10.
41
Vailati, Giovanni. “L’insegnamento della Matematica nel nuovo ginnasio riformato e nei tre
tipi di licei.” Il Bollettino di Matematica IX (1910): 57.
42
Galletti, Alfredo, and Gaetano Salvemini. La riforma della scuola media. Notizie, osservazioni e
proposte. Milano: Remo Sandron, 1908: 66.
43
Vailati, Giovanni. (1907) “L’insegnamento della Matematica nel primo triennio della Scuola
secondaria.” In Giovanni Vailati, Scritti. Vol. III: 302–06.
602 L. Giacardi
geometry, gave rise to a lively debate44 on methodology with Giuseppe Veronese and
Beppo Levi, who maintained that at the lower level of secondary school the teaching
of geometry should be essentially intuitive.45 Vailati’s curricula were then discussed
during the congress of the Associazione Mathesis held in Firenze in 1908. The
committee appointed to present a report evaluating the proposed programme agreed
with Vailati’s general approach but were critical of some aspects, including the
absence of any treatment of the theory of proportions, or of a rational treatment of
arithmetic, the excessive fragmentation of some parts of the programme and the
abolition of descriptive geometry.46
These debates are linked to a broader international discussion on the role of
intuition and rigour in the teaching of mathematics,47 which found its expression in
Italy in two opposing schools of thought: Corrado Segre’s (algebraic geometry) and
Peano’s (mathematical logic).
In any case, the reform was never carried through. The reorganization of the
secondary school was shelved till the 1920s when it was to be introduced in
completely different terms: the positivist, liberal-democratic culture was ousted by
the new political trends and by the prevailing idealism.
G. Castelnuovo and the Liceo Moderno: ‘Bringing Teaching into Contact
with Nature and with Life’
Part of the reform was implemented in 1911 when Minister Luigi Credaro
established a liceo moderno, which diverged from classico only after the second year
of liceo, where Greek was replaced by a modern language (German or English),
more attention was paid to scientific subjects, and elements of economics and law
were added. In mathematics greater importance was given to numerical approximations, while the concepts of function, of derivative and integral were presented
and illustrated by applications to the experimental sciences. The president of the
Associazione Mathesis, a distinguished scholar of algebraic geometry, Guido
44
Giacardi, Livia. “Matematica e humanitas scientifica. Il progetto di rinnovamento della scuola
di Giovanni Vailati.” Bollettino della Unione Matematica Italiana 3-A (1999): 339–41.
45
Levi, Beppo. “Esperienza e intuizione in rapporto alla propedeutica matematica. Lettera aperta
al prof. Giovanni Vailati.” Il Bollettino di Matematica VI (1907): 177–86.
46
Berzolari, L., E. Bortolotti, R. Bonola and E. Veneroni. “Relazione sul tema: I programmi
di matematica per la Scuola Media riformata.” In Atti del I Congresso della Mathesis Società
Italiana di Matematica, Firenze 16–23 Ottobre 1908. Padua: Premiata Società Cooperativa Tipografica: 26–33.
47
Smith, David Eugene. “Intuition and experiment in mathematical teaching in the secondary
schools.” L’Enseignement mathématique 14 (1912): 507–34, partially translated by G. Castelnuovo
in Bollettino della Mathesis (1912): 134–9.
Paedagogica Historica
603
Castelnuovo48 – who was also a member of the Comité Central of the Commision
Internationale de l’Enseignement Mathématique – was given the task of preparing
the syllabi and the instructions on teaching method for the new courses. His
interest in the school system was motivated by social factors:
Nous demandons parfois si le temps que nous consacrons aux questions d’enseignement
n’aurait pas été mieux employé dans la recherche scientifique. Eh bien, nous répondons
que s’est un devoir social qui nous force à traiter ces problèmes…. Ne devons-nous pas
faciliter à nos semblables l’acquisition du savoir, qui est à la fois une puissance et un
bonheur?49
Unlike Cremona, Castelnuovo believed that the main aim of the secondary school
was ‘to educate the future citizen’,50 not the elite:
The education system [he wrote in 1909] will not be really effective if it is not directed at
people of average intelligence, if it is not capable of creating the cultivated democracy
which is indeed the basis of any modern nation. How to find the means to disseminate this
culture, even at the cost of sometimes sacrificing depth of approach: this is the problem we
all must face and attempt to solve!51
Observing the defects of the Italian system of education (in particular, the teaching,
which was too abstract and theoretical; the division of the work, which was taken to
extremes; and the distorted view of culture on which it was based), Castelnuovo
began to reflect on teaching methods and to develop his own approach clearly influenced by Klein. In his article, Il valore didattico della matematica e della fisica,52 virtually
a manifesto of his thinking on teaching methods, the links he makes between mathematics and physics are by no means casual. In fact, he maintains that observation and
experimentation are important, and that continuous comparison and contrast
between abstraction and reality is an aid to learning. He is also convinced of the
importance of the applications ‘to illustrate the value of science’. Furthermore, he
believes that greater emphasis must be put on heuristic procedures for two reasons:
48
Cf. Castelnuovo, Guido. “I programmi di matematica proposti per il liceo moderno.” Bollettino
della Mathesis V (1913): 86–94 and “Istruzioni per lo svolgimento del programma di matematica del
Liceo moderno.” Bollettino ufficiale del Ministero della Pubblica Istruzione XL, 45 (30 October 1913):
2759–2804. Cf. also Castelnuovo’s report on liceo moderno during the CIEM Congress held in Paris
in 1914 (L’Enseignement mathématique 16 (1914): 295) and Loria, Gino. “Les Gymnases-lycées
Modernes en Italie.” Zeitschrift für mathematischen und naturwissenschaftlichen Unterricht aller
Schulgattungen begründet 1869 von J.C.V. Hoffmann 45 (1914): 188–93.
49
Castelnuovo, Guido. “Discours de M.G. Castelnuovo.” In Compte Rendu de la Conférence
internationale de l’enseignement mathématique, Paris, 1–4 avril 1914. L’Enseignement mathématique 16
(1914): 191.
50
Cf. “Istruzioni per lo svolgimento del programma di matematica del Liceo moderno.” Bollettino
ufficiale del Ministero della Pubblica Istruzione XL, 45 (30 October 1913): 2761.
51
Castelnuovo, Guido. “Sui lavori della Commissione Internazionale pel Congresso di
Cambridge. Relazione del prof. G. Castelnuovo della R. Università di Roma.” In Atti del II Congresso
della Mathesis Società italiana di matematica. Padova: Premiata Società Cooperativa Tipografica,
1909, Allegato F: 4.
52
Castelnuovo, Guido. “Il valore didattico della matematica e della fisica.” Rivista di Scienza 1
(1907): 329–37.
604 L. Giacardi
… the first, and the higher reason, is that this kind of reasoning is the most effective for
arriving at the truth, not only in the experimental sciences, but in mathematics itself….
The other reason is that this is the only form of logical procedure that is applicable in daily
life and in all the knowledge involved with the everyday world.53
His conception of the teaching of mathematics is apparent from a number of slogans
that appear often in his speeches and in his articles: ‘Reinstate the senses’, ‘Knock
down the wall separating our schools from the world outside’, ‘Bring teaching into
contact with nature and with life’.
So it is no surprise that Castelnuovo was one of the supporters of the project to
reform the secondary school proposed by Vailati, that he immediately told the teachers to begin to introduce some of these proposals without waiting for the reform to be
approved by the Ministry,54 and that he then adopted some elements of this project
in developing the syllabi for the liceo moderno.
In the instructions accompanying the syllabi, he therefore highlighted the importance of coordinating mathematics teaching with physics teaching; of eschewing
the over-refinement of modern criticism, avoiding, at the same time, the trap of
simplistic empiricism. He thought it helpful to consider the process of history leading up to the problems and their solutions, and necessary, above all, to rouse the
students’ interest by explaining the important role played by mathematics in
modern society:
But if we wish our secondary school pupil to be filled with inspiration by the new
mathematics and to glean something of the grandeur of its structure, we must introduce
him to the concept of function and explain to him, even if in a simplified form, the two
operations that form the basis of infinitesimal calculus. So, if he has the scientific spirit, he
will acquire a more correct and balanced idea of what constitutes science today…. And
even if the pupil has a mental disposition towards other subjects, in mathematics he will
meet, rather than a tedious exercise of logic, at least a series of methods and results that
can be easily applied to concrete problems.55
The syllabus for the liceo moderno began to be introduced into the schools from
1914–1915, despite the difficulties caused by the lack of trained teachers, by the
hostility of the teachers in the liceo classico schools, who sent the less able pupils to
the liceo moderno, and by the absence of funds, which made it difficult to provide
science laboratories.56 Among the various textbooks written for the new kind of
school, those worthy of note are Enriques and Amaldi’s Nozioni di matematica ad
uso dei licei moderni (1914), and Sebastiano Catania’s Corso di algebra elementare per
i licei classici e moderni secondo i nuovi programmi (1914). Although both these texts
seek to follow the guidelines laid down by Castelnuovo, they offer a diversity of
methodological approach that reveals the differences between two schools of
53
Ibid., 336.
Castelnuovo, Guido. “Sui lavori della Commissione Internazionale pel Congresso di
Cambridge”, 1909: 3.
55
Castelnuovo, Guido. “La riforma dell’insegnamento matematico secondario nei riguardi
dell’Italia.” Bollettino della Mathesis XI (1919): 5.
56
Cf. for example the Mathesis inquiry, Bollettino della Mathesis VIII (1916): 94–96.
54
Paedagogica Historica
605
thought: on the one hand, the school of algebraic geometry, on the other, Peano’s
school of logic.57
The Teaching of Mathematics in the Provinces Annexed from Austria
After the First World War, the debate over the reform of the secondary school
system was resumed with great energy, and meanwhile another problem had arisen
that required a solution: how to harmonize the mathematics programmes in Italian
schools with those in the two provinces, Trento and Trieste, recently annexed
from Austria, where the syllabi had been based on Klein’s ideas since 1908–
1909.58 At the Mathesis conference held in Trieste in 1919,59 this problem was
discussed at great length, and, voting on a motion proposed by Gino Fano, the
association passed a resolution to ensure that the Education Minister heard
representatives of teachers from the former Austrian provinces before he made any
decisions on curricula. The conference also submitted the following proposals,
which took into account the fact that the content of the Austrian syllabus was, in
some ways, more extensive and detailed, with more teaching hours allocated to
mathematics:
1. That in teaching experimental geometry in the lower secondary school the construction
and measuring of the figures should go hand in hand with the description of their properties, in accordance with the proposals of the Central Inspectorate as issued by the
Ministry of Education in 1918, and that the curriculum of the liceo classico should be
closer to that of the liceo moderno.
2. That the teaching of algebra should begin in the fourth year of the ginnasio, and should
include the theory of operations, while placing more importance on the exercises and
the practice of calculus, numerical and literal. The same approach should be adopted
for the first year of the technical institute.
3. That each class of the ginnasio should be allotted three, rather than two, lesson hours
of mathematics teaching per week.
4. That the introduction to concepts of function and to graphic representation should
begin in the ginnasio.
5. That in assigning subjects to each year consideration should be given to the requirements of teaching related subjects.
57
Castelnuovo came into conflict with Catania about the secondary school textbooks for algebra
based on mathematical logic: cf. Castelnuovo, Guido. “Risposta ad un’osservazione del Prof.
Catania.” Bollettino della Mathesis V (1913): 119–20; Catania, Sebastiano. “Sui metodi di insegnamento della matematica nelle Scuole medie.” Bollettino della Mathesis V (1913): 142–43; Castelnuovo, Guido. “Osservazioni all’articolo precedente.” Ibid., 143–45.
58
Cf. E. Beke’s report on the teaching of differential and integral calculus in secondary schools,
L’Enseignement mathématique 16 (1914): 296–306.
59
Cf. the transactions of the Congress of Trieste in Bollettino della Mathesis XII (1920): 1–62, in
particular Cantoni, Arrigo. “Programmi e metodi dell’insegnamento nelle scuole delle terre redente
e negli antichi confini d’Italia,” 8–17; Castelnuovo, Guido. “Sull’insegnamento medio delle
matematiche in Italia dal 1867 ad oggi,” 17–21 and Furlani, Giacomo. “Rapporti fra la matematica
e la fisica nell’insegnamento,” 22–31.
606 L. Giacardi
6. That the analytical plane geometry introduced in the last year of the secondary school
in the provinces annexed from Austria should be retained as an experiment.60
During the many meetings held in the various branches of the Mathesis, the
differences in method and contents were clearly shown and were substantially the
following. While in Italian schools there was a prevailing tendency to theory and
abstraction, in the schools in the former Austrian provinces more importance was
given to the mental development of students, to the demands of teaching method,
and to the application of mathematics to the other sciences. There was, moreover, a
greater differentiation between the lower school level, where teaching was essentially
experimental, and the upper level where the rational method was accompanied by a
variety of problems and applications from the other sciences. In addition, a central
role was accorded to the concept of function, which was now introduced from the first
years of secondary school. A teaching method of this kind, it was said, permitted the
students to see the purpose and the importance of what they were studying with
regard to modern developments in science, and trained them to work independently.61 In addition, the various papers on the subject stressed the similarity of
method and content to that of the liceo moderno.
The president of the Mathesis at that time, Federigo Enriques, an eminent scholar
of algebraic geometry, had made great efforts to expand the association from the
moment he took office – in fact the membership tripled in number in a very short time
– in order to increase its lobbying power with the government. His aim was to see
scientific education for the young given the importance it merited, not only in order
to advance scientific research but also to improve society as a whole.62 In view of the
wide range of his interests, historical, philosophical, cultural, etc., Enriques is such a
complex – and sometimes contradictory – figure that it is impossible to outline in brief
the epistemological views underlying his scientific work. I will therefore mention only
a few principles that inspired his commitment to education as they emerge from his
article Insegnamento dinamico (dynamic teaching) published in 1921 in the first issue
of the fourth series of the journal Periodico di matematiche, of which Enriques was then
editor.63 The main points of his thinking on education are the following: teaching
must be an active process and develop the capacity to discover; bridging the gap
between mathematics and the other areas of knowledge such as physics, biology,
psychology, physiology, philosophy and history fosters a unified vision of culture.
Logical reasoning and intuition are two inseparable aspects of the same process, and
60
Bollettino della Mathesis XII (1920): 55.
Cf. for example Furlani, Giacomo. “Relazione sull’insegnamento della matematica del prof
Giacomo Furlani alla Sezione romana della Mathesis.” Bollettino della Mathesis XII (1920): 176–82;
Verson, Adolfo. “Rapporto. Viaggio di studio compiuto a Bologna, per assistere a lezioni di
matematica e fisica presso alcune Scuole Secondarie.” Periodico di matematiche s. 4, 1 (1921): 222–
29 and Voghera, Guido. “Intorno ad un metodo di insegnamento della matematica in uso nelle
scuole delle terre redente.” Periodico di matematiche s.4, 2 (1922): 475–78.
62
Enriques, Federigo. “Ai Lettori.” Periodico di matematiche s. 4, 1 (1921): 1–5.
63
Enriques, Federigo. “Insegnamento dinamico.” Periodico di matematiche s. 4, 1 (1921): 6–16.
61
Paedagogica Historica
607
therefore it is necessary in teaching to find the correct balance between the two,
moving by degrees from the concrete to the abstract. Higher mathematics, considered
in the context of its historical development, allows greater understanding of certain
aspects of elementary mathematics, and therefore must have a key role in teacher
training.
Teaching should not be a gift from a teacher to a person who comes to hear his perfectly
prepared lessons … but rather it should be an aid given to the person who wants to
learn by himself or is, at any rate, disposed not merely to absorb passively, but to win
through to knowledge, as if it were a discovery or a product of his own spirit; there is no
gap or schism between elementary and higher mathematics, because the latter is a development of the former, as a tree develops from a seedling. And as by studying the tree we
discover new aspects of the seedling, and understand characteristics whose meaning had
escaped our understanding, so the development of mathematical problems will throw
light on the elementary theories in which they have their roots. On one condition,
however: that for every theory we study the origins, the relations, the development, not
some static formulation. Which teaching method could be more complete than to
present the complex problems and the conflicting difficulties that have exercised the
minds of all the students and scholars who ever laboured before us in the school that is
the world!64
The proposals put forward by Mathesis concerning the unification of the Italian
schools’ mathematics curricula with those of the former Austrian provinces were,
up to a point, accepted by the Ministry.65 A committee was set up with teachers
from the former Austrian provinces, and an inspector was appointed by the Ministry of Education. On 13 June 1921, they presented the Central Office for the New
Provinces with a proposal for timetables and curricula for the various types of
secondary school, which they had developed on the basis of what had emerged at
the Congress of Trieste in 1919. The Central Office for the New Provinces
informed the Minister (on 26 July 1921) of certain problems arising from the
curricula proposed: in particular, the lack of religious education,66 the difference in
curricula between the Italian-language schools and the German-speaking schools,
and the substantial increase in the total number of lesson hours. By late spring of
1922 the Ministry had prepared a draft law for the New Provinces, but in the
following autumn, after the March on Rome, Mussolini became head of government and the Fascist dictatorship began. Giovanni Gentile, the Minister of Education, managed to take advantage of the law passed on 3 December 1922 granting
full powers to the first fascist government, and in a single year introduced a total
and radical reform of the Italian education system, based on the principles of
64
Ibid., 16.
Cf. Archivio di Stato, Roma, PCM 161–2.
66
It is interesting to notice that religious instruction was part of the curriculum in Austria and
Germany, whilst in Italy it was not (see Tables 1 and 2).
65
608 L. Giacardi
pedagogy and philosophy he himself had been developing since the early years of
the twentieth century.67
Gentile’s Reform and the Dominance of Humanistic Culture
At the beginning of February 1923, Enriques, who had been in contact with the
eminent philosopher Gentile68 for some time, realized that the reform of the secondary school system was imminent, and that in general it tended to put greater emphasis
on humanistic studies. His immediate response was to call an emergency meeting of
the Mathesis’ Board of Directors with the object of presenting the minister with a
reform proposal, which took into consideration past educational experiences. At this
meeting, Castelnuovo delivered a passionate speech in which he first reminded the
assembled members of the important reforms introduced in Germany and France
and, in particular, of the important (but never completely implemented) proposals of
the Italian Royal Commission for the reform of the secondary school system, as well
as the innovations of the liceo moderno. He then expressed a hope that the coming
reform would not depreciate the educational value of science and would establish
three kinds of schools preparatory to university:
… a liceo classico where Greek would be of greater importance; a liceo moderno, with Latin
but no Greek, where particular attention would be given to economics, law and social
studies; finally, a ginnasio-liceo scientifico, without Latin … which would focus on scientific
training … in both these licei moderni modern languages would be studied.69
But since many of those present, including mathematicians of renown like Francesco
Severi and Enriques himself, were convinced of the ‘undoubted superiority as an
educational system’70 of the ginnasio-liceo classico, the assembly did not immediately
accept the proposal made by Castelnuovo, and decided first to consult with the local
branches. The legislation relevant to the secondary school system was promulgated
on 6 May 1923: Gentile rejected the democratic concept of a lower secondary school
common to all students, and separated secondary schooling into two streams; of these
the classical-humanistic stream was intended to train the elites and was to be
67
The bases of the reform were established by the Royal Decrees no. 1679 (31.12.1922) and no.
1753 (16.7.1923) (school administrative system reform), no. 1054 (6.5.1923) (secondary school reform), no. 2102 (30.9.1923) (university reform), n. 2185 del 1.10.1923 (primary school reform),
followed by other supplementary decrees. In this paper we intend merely to illustrate the influence
exerted by Gentile’s reform on the teaching of mathematics rather than to make a comprehensive
assessment of this reform. For a general commentary, cf., among others, Charnitzki, J. Fascismo e
scuola. La politica scolastica del regime (1922–1943). Firenze: La Nuova Italia, 1996.
68
Cf. Guerraggio, Angelo, and Pietro Nastasi. Gentile e i matematici italiani. Lettere 1907–1943.
Torino: Bollati Boringhieri, 1993; and Moretti, Mauro. “Insegnamento dinamico. Appunti sull’opera
scolastica di Federigo Enriques (1900–1923).” In Federigo Enriques, Insegnamento dinamico. La
Spezia: Agorà Edizioni, 2003: 15–91.
69
Cf. “Riunione straordinaria promossa dal Consiglio direttivo. Roma, 11 febbraio 1923.”
Periodico di matematiche s. 4, 3 (1923): 156.
70
Ibid., 156, 158.
Paedagogica Historica
609
considered overwhelmingly superior to the scientific-technical stream. The teaching
of Latin was introduced in all the lower levels of the secondary school system. Gentile,
in fact, eliminated the liceo moderno, as well as the physics-mathematics stream in the
upper technical schools, replacing it with a weakened liceo scientifico, with no specific
lower secondary school course to prepare its students and limited access to university
faculties. In addition, mathematics was to be taught together with physics, and the
lesson hours allocated to this combined course were generally fewer than those previously assigned to mathematics alone.
Besides the licei mentioned above, a liceo for girls was also established, clearly
intended to be a second-rate school: it offered a diploma of no value as a professional
qualification, no possibility of entering university and absolutely no teaching of the
sciences (see Table 3 and Figure 2).
Moreover, the reform established an official public examination (esame di stato) at
the end of every school cycle and put public and private schools on the same footing.
The liceo classico school-leaving examination took on the features of a real entrance
examination to university and had to provide proof that the candidate possessed a
broad humanistic culture.
The Mathesis Board of Directors went into action at once: before the end of May,
Enriques submitted a detailed statement of their position, the Promemoria del Consiglio
direttivo, to the Minister. In his statement he stressed that ‘the general impression that
the teaching of sciences was accorded diminished importance throughout the entire
bill’, and in particular called on the Minister to consider the following points. First,
the combining of mathematics and physics should not lead to a reduction in the
number of teaching hours allocated to these subjects. Second, leaving aside the
question of the intrinsic educational value of mathematics, some consideration must
be given to the needs of students intending to enrol in science faculties after completing the liceo classico. Third, the mathematics and physics programme in the liceo
scientifico should not be more limited than that of the liceo moderno or of the physics–
mathematics section of the upper technical school. Lastly, he said, it was deplorable
that in the liceo for girls there was not the slightest trace of the teaching of mathematics, physics or natural sciences.71 The Promemoria concludes with an impassioned
plea to the Minister:
Figure 2.
Organization of the Italian School System: From the Gentile Reform (1923) to 1945.
Men who have devoted their lives to science, which for them constitutes the embodiment
of their ideals and their reason for living, are deeply troubled and distressed by the threatened introduction of measures which – by relegating Italy to an inferior position with
respect to all other civilised nations – harbour the seeds of a national decline, the extent of
which, at this point in time, we cannot evaluate.72
Gentile’s response was, however, far from reassuring. Consequently, Castelnuovo
refused to take part in preparing the curricula and timetables, and in September
71
Enriques, Federigo. “Pro-memoria del Consiglio direttivo.” Periodico di matematiche s. 4, 3
(1923): 340.
72
Ibid., 341.
610 L. Giacardi
Table 3 Gentile’s Reform. Subjects and Lesson Timetables for the ginnasio-liceo
Ginnasio
Subjects
Italian
Latin
Greek
Foreign Language
History and Geography
Mathematics
I
II
III
IV
V
7
8
–
–
5
1
21
7
7
–
3
5
2
24
7
7
–
4
4
2
24
5
6
4
4
3
2
24
5
6
4
4
3
2
24
Liceo
Subjects
Italian
Latin
Greek
History
Philosophy and Political Economy
Mathematics and Physics
Natural Sciences, Chemistry and Geography
History of Art
I
II
III
4
4
4
3
3
4
3
–
25
4
4
4
3
3
4
2
2
26
3
3
3
3
3
5
3
2
25
Liceo Scientifico
Subjects
Italian
Latin
Foreign Language
History
Philosophy and Political Economy
Mathematics and Physics
Natural Sciences, Chemistry and Geography
Drawing
I
II
III
IV
4
4
4
3
–
5
3
3
26
4
4
4
3
–
5
3
2
25
3
4
3
2
4
6
2
2
26
3
4
3
2
4
6
2
2
26
Source: “Orari e programmi per le regie scuole medie.” Bollettino Ufficiale del Ministero dell’istruzione pubblica 50,
II, 17 (November 1923): 4418.
Paedagogica Historica
611
Figure 2 Organization of the Italian School System: From the Gentile Reform (1923) to 1945.
Mathesis organized a conference in Leghorn (Livorno) to discuss the reform.73 At the
conclusion of the conference a document was drawn up addressed to the Minister
with these requests: that the liceo scientifico course allow students access to other
73
Cf. “Relazione del Congresso di Livorno, 25–27 Settembre 1923.” Periodico di matematiche s.
4, 3 (1923): 454–78. In particular see papers on the reform by C. Rosati and G. Sansone.
612 L. Giacardi
university faculties in addition to science and medicine; that in the liceo classico physics
be taught together with chemistry, not with mathematics; that the total number of
lesson hours for mathematics be increased; that the number of lower level technical
schools (460 against 2140 ginnasio schools in 1923) be increased.
On 14 October 1923, the curricula and timetables for the secondary school system
were passed as law, and none of these requests was granted. The deeply humanistic
basis of this reform is evident from the incipit of the Avvertenze, which preface the
syllabi of the ginnasio-liceo:
The liceo-ginnasio must be an institution transmitting a humanistic-historical culture; it
must educate students for the highest offices in society, for the professions, for political
careers. It must give an education that begins at the roots of life, forming the man himself,
as a moral being, who has a precise place in history, and therefore knows how arduous has
been the progress of humanity from the primitive life of the cave dweller to the present
state of civilization. Civilization does not mean the technical perfections so vaunted in our
modern life, to the point of becoming the end and not a means; rather, it consists of the
profound communion between souls, of a profound sense of human liberty and responsibility, the profound consciousness of one’s own personality.74
The Assistant to the Minister was Gaetano Scorza, who was also a member of
Mathesis and one of the Italian representatives on the International Commission on
Mathematical Instruction. If one compares these curricula with those prior to the
reform, the main difference is in quality. First of all, the new curricula derive from the
epistemological view that the sciences can and must find their meaning and educational value only within the great Italian philosophical tradition. Second, the old
curricula that provided the teacher with the articulation of the subject and with helpful instructions on methodology had been replaced by examination syllabi indicating
the objectives to be reached but not the path to achieve them. The reform thus
presumed that teachers were quite capable of developing a teaching plan on their own
but paid no attention at all to the question of their professional training. The Scuole
di Magistero (teacher training schools) – which, with all their limitations, had functioned since 1875 – had been abolished by Benedetto Croce in 1920.75 With a decree
issued in March 1923, Gentile had declared the existing Istituti Superiori di Magistero
(higher Magistero institutes) for women to be equal in status to university faculties,
and had decreed that they should now admit male students as well. However, these
institutes were to limit themselves to training students to teach philosophy and
pedagogy, or for the positions of school principal or school inspector; no training for
future teachers of scientific subjects was to be provided.
74
“Orari e programmi per le regie scuole medie.” Bollettino Ufficiale del Ministero dell’istruzione
publica 50, II (17 November 1923): 4435.
75
Nastasi, Pietro. “La Mathesis e il problema della formazione degli insegnanti”. In La Mathesis,
La prima metà del Novecento nella Società Italiana di Scienze matematiche e fisiche. Note di Matematica,
Storia, Cultura 5 (2002): 59–119; Gario, Paola. “Quali corsi per la formazione del docente di
matematica? I congressi dei professori di matematica.” Bollettino della Unione Matematica Italiana,
Sez. A, s. 8 (in press).
Paedagogica Historica
613
Mathesis was not alone in criticizing. A number of scientific associations and the
Faculties of Science of certain universities immediately joined the protest movement.
The Accademia dei Lincei, the most prestigious society of Italian scientists, officially
opposed the reform in a paper entitled Sopra i problemi dell’insegnamento superiore e
medio a proposito delle attuali riforme,76 resulting from the work of a commission that
was headed by Vito Volterra and included Castelnuovo among its members. The
report drawn up by Castelnuovo is divided into two parts, the first of which concerns
university studies, the second the secondary school. Castelnuovo reiterates the points
already made at the emergency meeting of the Mathesis society and expresses the
committee’s disapproval of the predominance of the teaching of philosophy in the two
licei.
Our Committee [writes Castelnuovo] fears that the inordinate amount of space given to
philosophy in the liceo programmes could encourage a return to excessively a priori thinking and to purely verbal argumentation, against which the greatest thinkers of the Renaissance struggled so hard, and which seemed to have been brought to an end, thanks to the
triumph of our great Galileo.77
He also spoke out against integrating the teaching of different disciplines – such as
history and philosophy, mathematics and physics, natural sciences, chemistry and
geography – which might, on the one hand, lower the level of the teaching, and, on
the other, run the risk of weakening the dialogue between the various schools of
thought. In conclusion, he suggested that although official public examinations were
valid in principle, they might reduce the function of the school system to mere preparation for these examinations, and the textbooks to ‘manuals of practical tips to
obtain a passing grade’.
Other eminent scientists were also quick to add their criticisms of the reform,
criticisms that were clearly informed by concerns of a political nature regarding
Fascism. But the imbalance between classical and scientific education created by
Casati in 1859, and consolidated by Gentile’s reform, was destined to last to the end
of the century.
76
Castelnuovo, Guido. Sopra i problemi dell’insegnamento superiore e medio a proposito delle attuali
riforme. Roma: Tipografia della R. Accademia dei Lincei, 1923.
77
Ibid., 10.