1. Introduction
What are the relations among mathematics, its history and its epistemology (i. e. philosophy of scientific knowledge)? It is a debate that crops up periodically and which sometimes extends to science in general.
Of course, the scientists who deal with these problems have historical or philosophical sensitiveness, or both, and tend to exalt the interaction between these disciplines (to the extent of proclaiming their fusion; cf. for example Enriques, Gonseth, Weil).
On the contrary, the traditional training of scientists ignores this type of problems: it is said that science must not "look back", if it wants to progress, which means that it must ignore its own history (however, those who proclaim that take sides in a philosophical problem: as Aristotle said, there is no way to avoid philosophizing). Maybe the solution is finding a balance point between the need for specialization, typical of modern science, and a general vision. And what about the training of teachers?
Here we must speak about the relation of history and epistemology with didactics: the witnesses in favour of this interaction are richer, and in Italy they are even "institutionalized". In fact, the historical, epistemological and didactic disciplines taught at the university (as for mathematics and physics as well) are grouped into a single <<disciplinary sector>>, which means acknowledging the existence of a common base of contents and the interchangeability of teachers. They <<characterize>> the didactic "branch", that is the one aimed (even if not exclusively) to the training of teachers. There are also plans to include them in the post-graduate courses for the training of teachers.
Italy has its own tradition on the subject, which in itself gives some explanations of these facts, which we wish to analyse briefly.
2. The Italian tradition
First of all, we see that, strictly speaking, "history" is the totality of the facts happened, of written works and of opinions; what is written on these subjects, then, should more properly be called "historiography". However, we will not always follow this distinction.
Towards the end of the 19th century, people already wondered about the meaning of history and epistemology in didactics. A Society of teachers of mathematics existed, and reviews mainly addressed to teachers as well.
The first important society of secondary school teachers of mathematics was Mathesis (yet exiting). It was founded in 1895 and its motto was <<To turn the progress of Science to the advantage of the school>>. Its organ was the Bollettino della Mathesis (1895-1898, 1909-1920); another journal related to it was the Periodico di Matematica per l'insegnamento secondario (1886, in 1921 it became Periodico di Matematiche and is still published). In that period many other didactic journals were founded. We list here those that were published the longest: Rivista di Matematica elementare (1874-1885), Rivista di Matematica, founded by Peano (1891-1906), Il Pitagora (1894-1919), Bollettino di Matematica e Scienze Fisiche e Naturali, for the primary teaching (1899-1916), Il Bollettino di Matematica (founded in 1902, changed in Archimede in 1949 and still published).
The problem regarding the foundations of the didactics of mathematics, and the role of history and philosophy in particular, was the object of a heated debate during the first decades of the twentieth century, until neo-idealistic philosophy, which had become "philosophy of the State", emarginated mathematics and science from the ambit of culture (we must wait until the seventies/eighties to find a similar debate).
Today, a lot of the writings on this subject, which date back to that period, are still interesting to read and can offer starting points for reflection, at least in some precincts (we must overcome the mentality, which is typical of the scientific ambit, according to which an article loses interest automatically, after a certain number of years, and is not even mentioned in bibliographies).
As a rule, these starting points are useful for the cultural training of teachers: more seldom, they are real didactic indications: on the contrary, this dimension appeared when the debate was resumed, since 1980 onward.
For example, in 1896 Vailati wrote <<... a wrong remark, an inconclusive argument of a scientist in times gone by can be worthy of consideration just as an ingenious insight or discovery, if they equally help us throw light on the causes that have quickened or delayed progress>>[1].
In 1914, Enriques writes << A dynamic vision of science brings naturally to the ground of history. The rigid distinction between science and history of science is founded upon the idea of the latter as pure literary erudition; in this sense history brings to the theory an extrinsic addition of information about chronology and bibliography. But the historical comprehension of the knowledge reaches a very different meaning. It aims to discover the purchase in the ownership and then it uses the ownership to clarify the path of the idea and it conceives the purchase as something going farther than the limits temporarily reached. This kind of history becomes an integrating part of science and has its place in the exposition of doctrines.>>[2]
3. Some "philosophies of historiography"
At this point, we think it is useful to remind you of the "philosophies of history" of a few great thinkers: they are a little heterodox in comparison with the most widespread historiography of science, but they can suggest ways of using history in scientific teaching, and clarify the value of epistemology and history in scientific training.
According to Enriques, historiography is based on an a-priori process, that is on hypotheses, which must be followed by a control on texts and documents, just as physical science.
Bachelard believes that the history of science is basically different from political history: in science there is progress (which is part of the statute itself of science), and history must explain this progress. "The present clarifies the past and the latter enlightens the present".
Lakatos maintains the rational reconstructions of history, that is saying how science should have developed, to compare real history with rational reconstructions.
According to Popper, when we are in front of an authoritative tradition, we must take it into account, if there are no good reasons to reject it.
4. Epistemologies for didactic research
In advanced Countries the didactics of mathematics experienced a strong development in the sixties and seventies: starting from the eighties, the need of a thorough analysis of the basic reasons of this development was felt, which renewed the interest in the history and philosophy of mathematics (or better still, a real re-launching of the latter).
It was natural to turn to epistemologies "suitable" for explaining and helping didactics: in this connection, those of Enriques (E) (1871-1946), Bachelard (B) (1884-1962), Gonseth (G) (1895-1975) (who highlighted the idea of "suitability" - a philosophy that wants to explain certain facts must be 'suitable' for explaining them) and (more well-known, although less developed because of his untimely death) Lakatos were particularly interesting.
National traditions had their influence (for example, the Italians turned to Enriques , the French to Bachelard and so on). Lakatos (L) (1922-1974) interested scholars in many countries.
Let us sum up a few essential points:
1) Fallibilism also in mathematics (E, G, L)
2) Knowledge is a continuous approximation (E, B, G)
3) Epistemological obstacles (B: he refers to experimental science and excludes mathematics)
4) The importance of history
5) The revaluation of the elementary aspects of mathematics (E, G).
6) A basic analogy between the philosophy of science and the philosophy of mathematics (the theoretical aspects of science and the empirical aspects of mathematics are highlighted).
From a programmatic point of view, these epistemologies are open to revision ('principle of revisability', G) to take new developments into account.
Once, the history of mathematics was mainly regarded as a continuous development, unlike what happened, of course, with experimental science. For some years, people have been wondering whether "revolutions" have taken place also in mathematics and whether epistemological obstacles have been met (in particular, a few French teachers of mathematics use this idea for mathematics education, following G. Brousseau).
Broadly speaking, we see that a type of historiography that proposes some "facts" or characters, only, is not particularly suitable for didactics. However, an organization from the point of view of "the history of ideas" (Bourbaki) will be necessary.
Another kind of problems results from the interactions with psychology. There have been epistemological currents that have explicitly rejected any reference to the mind (Frege, Russell "the question of the mind is irrelevant", Popper "knowledge without a knowing subject"). On the contrary, to a different extent, Enriques, Bachelard and Gonseth have pointed out the interest that epistemology must have for cognitive processes. According to Enriques <<Logic is the totality of the laws that regulate a mental process, which can be represented in the static form of symbolism in fiction, only. Therefore, explaining logical relations means acknowledging the operations of the mind which are of use in meaning>>[3].
Thus, the bases for the enhancement of psychology have been laid, both as foundations for epistemology, and to give an epistemological basis to cognitive psychology. In this sense, the work of Piaget, who lays "genetic epistemology " at the centre of his system, must be considered pioneering.
At present, different currents (both post- and anti-Piaget) are engaged in the use of instruments that are both epistemological and psychological (just think of the different forms of constructivism, from von Glasersfeld to Ernest.).
5. The history and epistemology of mathematics for didactics in Italy
Now we wish to present a short review of opinions coming from the community of the Italian scholars of didactics in the period 1988-1995.
According to Furinghetti and her group (in Genova) is important to realize an integration of the history of mathematics in school mathematics.
An implicit use of history in designing teaching is introduced by the research group of Modena (coordinated by M. Bartolini Bussi). Early historico-epistemological studies of this research group resulted in the adoption of a particular context which could have emphasised the dialectical feature of mathematical experience as well as offered opportunities to develop meaningful historical studies in the classroom: it is the context of mathematical machines . For example, some elements of the study of conic sections are considered in the project Mathematical Machines in High School. The leading motives of this teaching experiment can be described under the following keywords: geometry, history, machines.
The research group of Genova (coordinated by P. Boero) considers, particularly, the following uses of the history of mathematics in the teaching of mathematics: 1) as a source of ideas, for the teacher, on "fields of experience" in which to construct mathematical skills and concepts, through suitable teaching itineraries and situations; when used this way, the pupil does not necessarely have to receive explicit information on the historic background used for the teaching project; 2) as a field of study for pupil, in order to work on particular mathematical objects (concepts and formalisms), to analyse their historical evolution and to translate from one formalism to another; 3) as an opportunity to set off developing mathematical discussions and demostrations, based on questions which have arisen in the course of the history of mathematics. Moreover, Boero is particularly interested in epistemological problems related to cognitive psychology, the same as F. Arzarello.
Arzarello considers the root of some modern algebraic concepts, especially that of ideal. He emphasizes the contraposition between the "computational" approach by Kronecker and the "abstract" one by Dedekind. Some consideration from the given epistemological analysis are drawn for the teaching of algebra.
From the research group of Cagliari (L. Grugnetti, working also in the research group of Parma) comes an example of the history of mathematics also in view of an interdisciplinary teaching: the Liber Abaci by Leonardo Pisano (Fibonacci) as a source of problems (from XIIIth century) which concern different teachers and subjects: Italian and Latin (what kind of language is that of the Liber Abaci?), general history (The development of the Middle Ages in Europe and Islam), geography (the West, the Middle East "Islam"), mathematics (the pupils' strategies for solving some questions from Liber Abaci; Leonardo Pisano's strategies: why he solved his problems by those strategies?). L. Grugnetti is also trying to link the history of mathematics - in a path fraught with difficulties and obstacles - to the difficulties that students can meet with throughout their personal course.
Interdisciplinarity is also the leading motivation for the interest of the research group of Rome (M. Menghini). The purpose of interdisciplinarity is not to show how mathematics can be applied to other disciplines, nor to examine a certain period in different fields, but to show how different disciplines help all together to form a branch of knowledge. And this can be observed better "from a certain distance", when the concept is clearer; also because the sistematization process can be very long. M. Menghini, L. Cannizzaro (Roma) e P. L. Ferrari (Genova) back up explicity a constructivist outlook on mathematics and on its teaching. More precisely in (Menghini, 1991 and Menghini, 1992) <<the change of meaning of a theorem in the course of history is linked to didactical questions tied to the "contraposition" between constructive procedure and proof>>.
For their part, the researchers of the group in Palermo analyze some aspects of Euclidean geometry and elementary aspects of analysis from the point of view of epistemological and didactic obstacles.
According to Speranza, the meaning of history and its link to disciplines are in the (explicit or implicit) philosophies which the historical presentation is based on. He suggests to mainly highlight the "revolutionary" aspects of the development of mathematics and the epistemological obstacles met. In this sense, he has analyzed the contribution of a few great thinkers (Enriques, Bachelard, Lakatos).
Speranza is also trying to use these epistemological themes to propose "new" approaches to mathematics, in particular to geometry, to which he assigns a fundamental role both in the development of the philosophical and mathematical thought, and in teaching/learning. The analysis of the concept of space is used to highlight cultural interactions also with figurative arts (in the past and nowadays, too). As far as epistemology is concerned, its "programme" basically aims at carrying out a thorough analysis of the philosophical themes "suitable" for didactics. In particular, these philosophies are useful to interpret the various aspects of geometry .
On the contrary, Bernardi and Marchini are clearly interested in epistemological problems linked to logic (symbolism, set theory, definitions and demonstrations...).
Also other Italian research groups are begining to work on the role of the history and of the epistemology of mathematics in mathematics education. The different points of view allow a dynamic debate on this subject and point out a growing up of an important field of mathematics education. Since 1989 a national group about "Epistemology of Mathematics" exists; it is largely composed by persons who are also researchers in didactics of mathematics.
References
Arzarello, F.: 1994, 'Symbols, computations and concepts in algebra: historical roots of a cognitive obstacle', in Malara, N. A. and Rico, L. (eds.) Proc. First Italian-Spanish Symposium in Mathematics Education, Modena, Italy, 213-220.
Bagni, G. T.: 1993, 'Spunti storici per la didattica della matematica: la prospettiva e le geometrie non euclidee', in D'Amore, B. (ed.) Atti degli Incontri con la Matematica n. 7, Castel San Pietro Terme (Bologna), Pitagora Editrice, 107-110.
Bagni, G. T.: 1994, 'L'approssimazione di [[pi]], i poligoni regolari e la circonferenza', L'insegnamento della matematica e delle scienze integrate, 17B (4), 347-355.
Bartolini Bussi, M. and Pergola, M.: 'History in the mathematics Classroom: Linkages and kinematik Geometry', to appear in Jahnke, H. N., Knoche, N. and Otte, M. (hrsg.), Geschichte der Mathematik in der Lehre: Goettingen: Vandenhoek & Ruprecht.
Bartolini Bussi, M. and Pergola, M.: 1994, 'Mathematical Machines in the classroom: the History of Conic Sections', in Malara, N. A. and Rico, L. (eds.), Proc. First Italian-Spanish Symposium in Mathematics Education, Modena, Italy, 233-240.
Bernardi, C.: 1992, 'Nessuno ci scaccerà...', in Speranza, F. (ed.) Epistemologia della Matematica - Seminari 1989-91 , CNR-TID project, FMI series, 10, 89-100.
Bernardi, C.: 1993, 'Il formalismo matematico fra astrazione e realtà', Epsilon, VI, n. 14, 9-13.
Boero, P. and Garuti, R.: 1994, 'Approaching rational geometry: from physical relationships to conditional statements', in Proc. PME XVIII , Lisbon, 1, 121-124.
Campedelli, M. G.: 1992, 'Il metodo delle coordinate: storia e didattica', N uova Secondaria X, (1), (2), (3), 75-77, 71-72, 73-74.
Cannizzaro, L.: 1988, 'Verso il concetto di funzione: pluralità di impostazione e di sviluppi. Spunti di riflessione teorica, storica e didattica a margine del progetto RICME ', La Matematica e la sua Didattica, 1, 27-31.
Cannizzaro, L.: 1994, 'Mathematical models of real world phenomena: some historical developments as a guideline for identifying cultural aims', in Bazzini, L. and Steiner , H. G. (eds.) Proc. Second Italian -German Bilateral Symposium on Didactics of Math.ematics , Bielefeld, IDM Materialen und Studien Band 39, 115-126.
D'Amore, B.: 1988,'Tra lingua e matematica: esistono basi epistemologiche del rigore?', La Matematica e la sua didattica, 2, 3, 24-32.
D'Amore, B., Plazzi, P.: 1990, 'Intuizione e rigore nella pratica e nei fondamenti della matematica', La Matematica e la sua didattica, 4, 3, 18-24.
D'Amore, B., Sandri, P.: to appear, 'Bernhard Bolzano: pagine di didattica della matematica', in Speranza, F. (ed.) Epistemologia della Matematica -Seminari 1994-95, CNR
Dell'Aquila, G., Ferrari, M.: 1994, 'La lunga storia dei numeri interi relativi. (in tre parti)', L'Insegnamento della Matematica e delle Scienze Integrate, 174 (2) (3)(4), 145-157, 247-266, 335-362.
Di Leonardo, M. V., Marino, T.: 1991, 'Una dimostrazione di alcune proprietà del segmento parabolico note ai tempi di Archimede', L'Educazione Matematica, XII (1), 45-59.
Di Leonardo, M. V., Marino, T., Spagnolo, F.: 1994, 'Alcune osservazioni didattiche ed epistemologiche sul postulato di Eudosso-Archimede ed il metodo di esaustione', La Matematica e la sua Didattica, 9, 25-37.
Ferrari, P. L.: to appear, 'The constructivist paradigm and the philosophy of mathematics, in Actes de l'Université d'Eté sur l'Histoire des Mathématiques, Montpellier, France.
Furinghetti, F.: 1992, 'The ancients and the approximated calculation: some examples and suggestions for the classroom', The mathematical Gazette, 76 (475), 139-142.
Furinghetti, F.: 1993, 'Insegnare matematica in una prospettiva storica', L'Educazione Matematica, 14 (3), 123-134.
Furinghetti, F., Somaglia, A.: 1995, 'La storia come strumento di lavoro in classe', in E. Gallo, E., Giacardi, L. and Pastrone, F. (eds.), Conferenze e Seminari 1994-1995 della Mathesis Subalpina, 134-145.
Grugnetti, L.: 1989, 'The Role of the History of Mathematics in an Interdisciplinary Approach to Mathematics Teaching', ZDM , 4, 133-138.
Grugnetti, L.: 1992, 'Une expérience interdisciplinaire fondée sur l'histoire des mathématiques', PLOT , 3ème trimestre , 17-21.
Grugnetti, L.: 1994, 'Relations between history and didactics of mathematics', Proc.PME XVIII, Lisbon, 1, 121-124.
Grugnetti, L.: 1994, 'La storia della matematica nella didattica: riserva di spunti, metodologia esemplare, o scelta filosofica?', in D'Amore, B. (ed.) Atti Incontri con la Matematica n. 8, 57-65
Grugnetti, L.: 1994, 'Il concetto di funzione: difficoltà e misconcetti', L'Educazione Matematica, XV (3), 173-184.
Lanciano, N.: 1989, 'Ver y hablar como Tolomeo y pensar como Copernico', Enseñanza de las Ciencias, 7, n. 2, 173-182.
Lanciano, N.: 1989 'Due numeri fra cielo e terra:cinque e sette', L'Educazione Matematica, IX (1), 43-59.
Lucchini, G.; 1993, 'Matematizzazione e de-matematizzazione', in Marchini, C. (ed.)Seminari di didattica. Università di Lecce, 145-163.
Maraschini, W. and Menghini, M.: 1992, 'Il metodo euclideo nell'insegnamento della geometria', L'Educazione Matematica XIII (3), 161-180.
Marchini, C.: 1992, 'La teoria alternativa degli insiemi, una nuova proposta per i Fondamenti della Matematica', in Atti Convegno per i sessantanni di Francesco Speranza, 157-170.
Marchini, C.: 1992, 'Le definizioni e le convenzioni, un problema didattico', in C. Marchini (ed.)Seminari di didattica. Università di Lecce, 125-143.
Menghini, M.: 1989, 'Dopo una conversazione con H. G. Steiner', Epsilon 4 (II), 20-23.
Menghini, M.: 1989, 'Form in Algebra: reflecting with Peacock, on Upper Secondary School Teaching', For the Learning of Mathematics, 14 (3), 9-14.
Menghini, M.: 1991 'Problematiche didattiche attuali e sviluppo storico dell'analisi matematica', L'Insegnamento della Matematica e delle Scienze Integrate, 14 (10), 909-929.
Menghini, M.: 1992, 'Piano affine e "costruttivismo"', La Matematica e la sua Didattica, 4, 5-13.
Navarra, G.: 1990 'E se una triandria di poliferi ammirasse una scultura perilitica? Cronaca di un'esperienza didattica condotta in una scuola media sulle etimologiche dei termini scientifici', Scuola e Didattica, n. 10, 79-82.
Navarra, G : 1994 'Dalla moltiplicazione "a Gelosia" ai bastoncini di Genaille, in Basso, M. et al. "Numeri e proprietà", Univ. Parma, 23-28.
Pergola, M.: 1995 'Macchine Matematiche ed altri oggetti: un progetto di ricerca didattica', in Marchini, C., Speranza, F. and Vighi, P. (eds), "La Geometria da un glorioso passato ad un brillante futuro", Atti 3deg. Incontro Internuclei Scuola Secondaria superiore Parma, Italy, 21-30.
Pergola, M. and Zanoli, C.: 1994, 'Macchine Matematiche': note su un progetto di ricerca didattica', L'Insegnamento della Matematica e delle Scienze Integrate, 17, 733-751.
Sibilla, A.: 1991, 'Riflessioni ed esperienze sull'uso della storia della matematica per l'insegnamento della matematica nella scuola media', L'Insegnamento della Matematica e delle Scienze Integrate, 14, 1103-1123.
Spagnolo, F. and Margolinas, C .: 'Un ostacolo epistemologico rilevate per il concetto di limite: il postulato di Archimede', La Matematica e la sua Didattica, 7 (4), 410-427.
Speranza, F.: 1988, 'Epistemologia e insegnamento della Matematica', L'insegnamento della Mat.ematica e delle Scienze Integrate, 11 (7/8), 705-714.
Speranza, F.: 1989, 'La matematica: problemi e teorie', EIT, Teramo.
Speranza, F.: 1989, 'History, Epistemology, Didactics: some noteworthy cases',, in Bazzini, L. and Steiner, H. G. (eds.) Proc. First Italian-German Bil.ateral Symposium on Didactics of Mathematics , CNR-TID project, FMI series, 5, 95-107.
Speranza, F.: 1990, 'Controindicazioni al riduzionismo', La Matematica e la sua Didattica, 4, f.3, 12-17.
Speranza, F.: 1990, 'Filosofia e matematica: prospettive di interazione', Cultura e Scuola, 29 (116), 174-181.
Speranza, F.: 1990, 'Il significato filosofico della Matematica e il suo insegnamento', in Atti convegno "Il pensiero matematico nella cultura e nella società italiana negli anni '90", Quad. PRISTEM n 1-Documenti, 59-66.
Speranza, F.: 1992, 'Il ruolo della storia nella comprensione dello sviluppo della scienza', Cultura e scuola, 31 (123), 201-208.
Speranza, F.: 1994, 'A la recherche d'un philosophie de la mathématique idoine pour la didactique', Tetradia didaktikéz ton maqhmatikon, Cahiers de didactique des Mathématiques, anche in (Histoire et enseignement des Mathématiques), Erasmus ICP-93-G-2011/11, Thessaloniki, 53-62 e 177-184.
Speranza, F.: 1994, 'Epistemologia', L'insegnamento della Matematica e delle scienze int.grate, 17 (5), 496-505.
Speranza, F.: 1994, 'The Idea of Revolution as an Instrument for the Study of the Development of Mathematics and for its Application to Education', in Ernest, P. (ed.) Constructing Mathematical Knowledge: Epistemology and Mathematics Education, The Falmer Press, London, 241-247.
Speranza, F.: 1994, 'The influence of some mathematical revolutions over philosophical and didactical paradigms', in Bazzini, L. and Steiner , H. G. (eds.) Proc. Second Italian -German Bilateral Symposium on Didactics of Math.ematics , Bielefeld, IDM Materialen und Studien Band 39, 163-174.
Speranza, F.: 1994, 'The Significance of History and of Non-Absolutist Philosophies of Mathematics in Mathematics Education', Perspectives, n. 53, 42-51.
Speranza, F.: 1994, 'Dalla storia dell'epistemologia: indicazioni per leggere la storia della scienza', in Lo sviluppo storico della Matematica, vol. III, D'Amore, B. and Speranza, F. (eds), Angeli, 148-158.
Testa, G.: 1995, 'Equazioni di terzo grado e numeri complessi', Didattica delle Scienze, n. 176, 43-49.
| Francesco Speranza | Lucia Grugnetti |
| Dipartimento Di Matematica | Dipartimento di Matematica |
| Università | Università |
| Via M. D'Azeglio 85/A | Via M. D'Azeglio 85/A |
| I - 43100 Parma, Italy | I - 43100 Parma, Italy |