Philosophy of Mathematical Education Journal 11 (1999)

Some answers to the Discussion Document for the ICMI Study on The role of the history of mathematics

LUCIA GRUGNETTI, FRANCESCO SPERANZA

Italians are particularly interested

In Italy, there is a tradition of historical studies. But there is another special reason for our interest. In 1998, Italy "will come back into Europe" as far as teacher training is concerned. Since 1922, the neo-idealistic slogan "He who knows can teach" has been dominant, and this implies that everyone with a university degree can teach.

However, a law passed in 1990 established a two-year postgraduate course for "abilitazione", a compulsory qualification for those aspiring to teach. There is almost universal agreement about the presence of specific didactics, history and epistemology in the curriculum. We are therefore particularly interested in the results of this ICME Study. The central issue is: What history (and what epistemology) are suitable for didactics?

In these notes, epistemology will be considered as being almost equivalent to philosophy .

Didactics, history and epistemology of mathematics

Our central thesis is this: In teacher training, didactics, history and epistemology (of mathematics) must form a "virtuous circle" in which each justifies and strengthens the others (cf 1 , 2 and 6; cf also our reply to question 10). In high-school mathematics teaching, there must be a historical-epistemological component and, symmetrically, a mathematical component in the teaching of philosophy.

In the Eighties, many "mathematics didacticians" in many countries felt the need to give a more reliable foundation to educational research through a philosophical reflection. What philosophy is suitable for this purpose?

Philosophy must explain mathematical thought (not only at the level of research, but also as far as teaching is concerned), and its development in the past: it needs history . This means that we must avoid the identification "philosophy of mathematics = mathematical logic". Our philosophy must guide and explain educational choices; it must help in a correct planning of teaching (here, we can see a case of Ferdinand Gonseth's principe d'idoneité) It must be open to new reflections (Gonseth's principe de revisibilité).. Such a philosophy can be a non-absolutist one (Confrey, Ernest). They can be traced back to a tradition connected to the non-Euclidean and Einstein's "revolutions" (Gauss, Riemann, Clifford, Poincaré, Enriques, Einstein, Weyl, Bachelard, Gonseth, . . . ), and taken up again by Popper and his school (Kuhn, Feyerabend, Lakatos).

What history is suitable? There is a history of documents and a history of ideas . The latter needs the former, but didactics and epistemology need the latter. We may find examples of such approaches in the writings of Koyré, Mach, Enriques, Crombie, De Santillana, Kuhn, . .; they show great syntheses, comparisons between different moments of scientific and philosophical thought, interdisciplinary connections, and sometimes also heterodox historiographical principles: "The history of scientific thought must be rewritten, taking into account the development of science" (Bachelard). "History is constructed a priori , except to change the constructions if they do not correspondence to the texts and documents . . ." (Enriques).

In these notes, we shall develop the historical and philosophical perspectives together. History needs some suitable philosophical premises: "History without philosophy is blind; philosophy without history is void" (Kant).

Answers to some questions in the document

1. How does the educational level of the learner bear upon the role of history of mathematics? Students who will explicitly meet these problems must be, or become, able to understand the "cultural values" of mathematics. The historical-philosophical dimension of mathematics allows the development of these values. It would be interesting to research how students in their first year of schooling construct an "indirect awareness" of such values.

We fear that teachers and students at all grade levels are dazzled by technological novelties. Another danger is the reduction of history to a few anecdotes; if these are not put into a cultural context, they are (nearly) useless.

It is important to place mathematics in the general context of knowledge ; Unfortunately, the contrary choice is widespread today (see, for example, the Introduction to Bourbaki's Elements ).

2. At what level does history of mathematics as a taught subject become relevant? A separated teaching of history and philosophy of mathematics is suitable only at university. We must take into account, also, to opinions of some (e.g. Lakatos) who maintain the insertion of historical and philosophical subjects into "normal" courses; but this choice, in our academic organisation, would no longer leave room for historical and epistemological research.

Epistemological reflections (in mathematics or other specific areas) can begin with the "formal thought" stage (at approximately fourteen years of age). Historical subjects can begin before this stage as well; but history cannot be reduced to tales.

3. What are the particular functions of a history of mathematics course or component for teachers? Teachers, and also those who will not teach these subjects explicitly, must have in their curricula a historical-epistemological education; maybe the best solution would be a partial integration of both. In our opinion, this must be valid for all mathematicians.

The role of history and philosophy in education must be treated differently for different school levels; a postgraduate course can make the planning of these subjects easier. "The history of foundations and the ideas of proof" would be a good subject, but it is not the only one.

L. Grugnetti & F. Speranza (1994). Teachers' training in Italy: the state of the art ; Proc. First Italo-Spanish bil. symp., 205-210

F. Speranza (1996). Perché l'epistemologia e la storia nella formazione degli insegnanti? Università e scuola, 1, 70-72

4. What is the relation between historians of mathematics and those whose main concern is in using history of mathematics in mathematics education? In Italy, historians of mathematics are partly uninterested in educational problems, and are partly also "didacticians". Epistemologists of mathematics (with a pre-eminently mathematical training) are mostly interested also in didactics, because the recent revival of epistemology had also an educational motivation. Generally speaking, didacticians of mathematics are at least aware of the importance of history and epistemology. Those who are also historians or epistemologists are a relevant minority.

There could be a difficulty: the "paradigms" of historical and epistemological research accepted by the respective communities might not be suitable for educational research. It is thus very important that there be educational researchers engaged also in historical and/or epistemological research.

In university organisation, there is a unique group for didactics, history and epistemology of mathematics. This is relevant, for example, for the selection of teachers (for both permanent and temporary positions), and encourages interdisciplinary studies.

5. Should different parts of the curriculum involve history of mathematics in a different way? Generally speaking, yes . For instance, geometry can provide many suggestions because of its long history and the changes over the centuries. There is need for a reflection about its role because its weight in most curricula has, in the last few years, been heavily reduced (also in Italy, where it was once a "national glory". The role of the historical and epistemological dimension is fundamental to a "return to geometry".

F. Speranza (1994). The role of non-classical geometries for a radical renewal of mathematics teaching . Proc. First Italo-Spansich bilateral symposium, 249-256

F. Speranza (1995); Geometry and the development of our culture ; In ICME Study: C. Mammana (Ed.), Perspectives on the teaching of geometry for the 21st century . Pre-proc. Catania Conf., Dept. of Math., Univ. of Catania, 242-245

6. Does the experience of learning and teaching mathematics in different parts of the world, or cultural groups in local contexts, make different demands on the history of mathematics? We remember A. Toynbee's idea of "civilisation" (A Study of History): science has been (nearly) exclusively the domain of the Hellenic and Western ones of other . Nevertheless, it can be interesting to study some basic conceptions of other civilisations (for instance, about the concepts of space and time), in order to reflect on how scientific thought could develop.

We remember, too, the thesis of Sapir-Whorf: every language (or, better still, every group of languages) contains in itself a conception of the world. For example, the use of spatial metaphors to express non-spatial concepts seems characteristic of certain languages; on the other hand, some people have no adequate idea of the future because, in their languages, there is no future tense. Nevertheless, when an individual belonging to a population with a radically different language from our own learns one of our languages, he/she is able to learn our basic concepts.

8 and 9. What are the relations between the role or roles we attribute to history and the ways of introducing or using it in education? What are the consequences for classroom organisation and practice? History and epistemology can give us meaningful suggestions for a "more humane" teaching of mathematics. Pupils must understand the principal problems connected with the historical-developmental construction of mathematics, and its role in the culture. Let us think of the concept of space, the axiomatizations of geometry, the development of analysis, non-Euclidean geometry, and algebraic and logical symbolism. A (non-compulsory) list of meaningful subjects, adequately justified, could be prepared.

F. Speranza (1988), History, Epistemology, Didactics: some noteworthy cases. Proc. First Italo-German bil. symp. on didactics of math., Pavia, 95-107.

L. Grugnetti (1994), Relations between history and didactics of mathematics. Proc. 18th PME Conference, vol 1, Lisboa, 121-124.

10. How can history of mathematics be useful for the mathematics education researcher? We can complete Kant's aphorism: "Didactics without history and philosophy is blind". These disciplines give didactics its basic references (Let us remember Piaget's épistemoloie génétique). Obviously, other references can be drawn from "educational sciences" such as pedagogy, psychology, general didactics, sociology, and so on.

"From the greatest men of the past we can receive help and stimulation more fruitful than from the best men of our age ... Devoid of the methods, which they created (and which we do not understand without a compete knowledge of their works), they succeeded in ordering and mastering the subjects of their research, giving it a conceptual form" (Mach).

"Only via their ruptured history do present theoretical concepts receive their empirical content" (Bachelard). 11. What are the national experiences of incorporating history of mathematics in national curriculum documents and central political guidance? In Italy, at the beginning of this century, many mathematicians were sensitive to historical and epistemological problems. So-called "neo-idealism", supported by Fascism, "sanctioned" a sharp division between the two "cultures". In our high schools, the "Gentile reform" (1923) was marked by non-idealistic principles. "Humanistic disciplines" are taught from a historical point of view; in mathematics, this viewpoint is ignored(and the past is tacitly considered a series of misconceptions). Moreover, "reflections" on sciences (and on scientists) were explicitly banned by Gentile.

In the last few years, there has been a revival of interest in historical and epistemological themes, especially in the "Nuclei di Ricerca Didattica", university-school groups, supported by Ministry of Universities and the National Research Council).

Future researches

We are planning researches on the understanding of historical development of some meaningful subjects such as: - the concept of limit, - the passage from Euclidean geometry (considered as a unshakable truth) to a multiplicity of geometries (non-Euclidean geometry, Klein's Erlangen program). Some relevant ideas, as "epistemological/didactical obstacle", or "revolution" could be useful.

© The Authors 1999