Emma Castelnuovo, formázott: Betűtípus: 14 pt a mathematics teacher committed to national and international reforms formázott: Középre zárt formázott: Betűtípus: 14 pt Fulvia Furinghetti formázott: Középre zárt formázott: Középre zárt Abstract Emma Castelnuovo (1913-2014) is an important character in the history of mathematics education. As a secondary teacher she designed and carried out a programme for the middle school (11–14) based on an intuitive approach fostered by the use of concrete materials. In this way she promoted intuition, exploration, awareness of the links of mathematics with reality, and an active learning. In the 1950s and 1960s, a period of the great ferments in education, her work was very appreciated and she was invited to participate in the milestone events of those years such as the first meetings of CIEAEM, the Royaumont seminar, and the first ICME conferences. She contributed to the first issues of the new journal Educational Studies in Mathematics, addressed specifically to mathematics education themes. For all these reasons she may be considered not only a very good teacher, but also a pioneer case of teacher-researcher.

Key words: Emma Castelnuovo, reform movements, active learning, intuition, history of mathematics, teacher-researcher, profession of teacher.

Introduction

The International Commission on Mathematical Instruction (ICMI) has decided in the past to create two awards to recognize outstanding achievements in mathematics education research: the Felix Klein Award, honoring a lifetime achievement, and the Hans Freudenthal Award, recognizing a major cumulative program of research. In order to reflect a main aspect of ICMI, not yet recognized in the form of an award, in 2013 ICMI has decided to create a third award to recognize outstanding achievements in the practice of mathematics education. This award was named after Emma Castelnuovo. To understand why the name of this Italian secondary teacher is associated to this particular award we outline the main aspects of her work in school and for school. Her life and career are intertwined with the events that changed the approach to the problems of mathematics education in the second half of the twentieth century.1 Emma Castelnuovo was born in the 12th of December 1913 and died in Rome the 13th of April 2014. She grew up in an exceptional mathematical environment. It is well known that in the first half of the twentieth century an important field of mathematical research – algebraic geometry – was flourishing in . Emma’s father Guido (1865-1952) and her uncle (1871- 1946), the brother of her mother Elbina, were prominent researchers in this field, both full professors in university. They were also very committed to mathematical education. In 1908 Guido was the chairman of the fourth International Congress of Mathematicians in Rome, when the Commission’s direct parent of ICMI was founded and later on in 1912-1920 and 1928-1932 he served the Commission as a vice-president. In the years 1911 to 1914 he was president of the Italian association of mathematics teachers Mathesis and editor of Bollettino della Mathesis, the official journal of the association. Enriques was awarded honorary membership of ICMI during the tenth International Congress of Mathematicians in Oslo (1936) for his special activity in mathematics education. He was president of the Italian association of mathematics teachers Mathesis from 1919 until 1932, editor of Bollettino della Mathesis in 1919-1920 and later on of Periodico di Matematiche, the important journal which became the official organ of Mathesis in 1921. Both these mathematicians published articles in the journals addressed to mathematics teachers and participated in the discussion about Italian mathematics programmes. As we will see in the following, they were both influential in shaping Emma’s view of mathematics teaching, see (Gario, 1913). This rich mathematical milieu fostered contact with other important Italian mathematicians of the period as well as visiting researchers from abroad. Emma graduated in mathematics at the University of Rome in 1936. She won the competition for a permanent position in state schools in 1938, but because of the racial laws of 1938 which had forbidden Jewish persons to have a position in state schools, her career as a secondary teacher began in 1945 in a middle school of Rome (pupils’ age 11 to 14), where she remained until her retirement in 1979. During the Second World War she taught in the Jewish school of Rome to Jewish students, who were not accepted in state schools because of the racial laws. Just after the Second World War Emma with a university professor and a young colleague organized a successful series of talks held by mathematicians, physicists, philosophers, and educators. Many teachers attended these talks. This initiative was a good sign for the re-birth of the Italian school and culture after the War.

Looking for a new way of teaching

Already at the beginning of her career Emma was looking for a way of teaching aimed at actively involving students. As she explains in the articles (Castelnuovo, 1946; 1989) she found the answer to this wish in the treatise of geometry Élements de géometrie by Alexis-Claude Clairaut, published in Paris in 1741 and translated into Italian in 1751. The contents of this book are quite close to Euclid’s text, but they differ in the presentation since the treatise was written with didactic intentions, proposing problems to be solved and trying to guide students in the discovery, see (Barbin, 1991). In the preface of the book Clairaut complains about the usual method of teaching geometry, which starts in an abstract way with a long list of definitions, axioms, theorems. Then he proposes a method that he supposed may have been that followed by geometry's first inventors, attempting only to avoid any false steps that they might have had to take. The problems he treats are the measurement of lands. The pedagogical value of the book has been questioned, see (Glaeser, 1983), but the proposed approach inspired Emma to renewing her teaching, see (Castelnuovo, 1946; 1989). In Emma’s interpretation of Clairaut’s project the winning ideas are: intuition, real problems, and history. The problem of the balance between intuition and rigor had been already considered by her father Guido at the beginning of twentieth century. In an international conference organized by the Commission founded in Rome he discussed how rigor was treated in the most important treatises for secondary school, see (Castelnuovo, 1911). In planning new mathematics programmes for secondary school Guido stressed the danger of a too rigid adhering to the Euclidean tradition and supported linking mathematics to applications and real life. In this way he advocated the introduction of new topics such as probability in the mathematics programs, see (Castelnuovo, 1919). Some decades later Emma succeeded to have the elements of probability in the new Italian programmes for middle school launched in the 1970s. The words “intuitive/intuition” and “real/reality” became quite common in the titles of her books and articles. The textbook Geometria intuitiva, per le scuole medie inferiori (Intuitive geometry for lower secondary schools), first published in 1948 had various editions till 1964 and was translated into Spanish and English. It launched Emma at international level so that she was invited to join working groups and meetings, see (Menghini, 2013). Emma’s new way of teaching geometry is based on the use of concrete materials, on looking at objects, discovering geometric properties, and in manipulating changing figures. Rigor is not the starting point of learning, but a point of arrival reached through learners’ active involvement, which begins from the concrete and arrives at the abstract. This path fosters continuity in the learning from the early grades to university. The formative value of mathematics is not antithetic to the value of mathematics as a utilitarian discipline. All the teacher has to have is an active and emotional involvement, see (Furinghetti & Menghini, to appear). The idea of dynamic patterns in geometry and of the power of visualization used to discover and reinforce important mathematical concepts marks the activity of important educationalists in the 1950s. In this concern Rogers (to appear) discusses the work of Caleb Gattegno, Zoltan Dienes, and the British Association of Teachers of Mathematics (ATM). Since 1949 Emma was in contact with teachers of the École Decroly, where Paul Libois used concrete materials. From Libois she took also the idea of her successful mathematical exhibitions, which were an efficient means for linking mathematics with reality, and for visualizing concepts. The preparation of exhibitions was also an efficient means for creating a community of practice where young teachers and students were involved to pursue common goals, see (Menghini, to appear). Through her intuitive geometry Emma realized the ideas of good teaching that her uncle Enriques illustrated in a famous article of 1921 entitled “Dynamic teaching”, see (Enriques, 1921). Enriques was not only a paramount mathematician, but also a historian and epistemologist. On these subjects he delivered courses in university and edited journals. In many occasions Emma refers to Enriques’s view on the importance of the history of mathematics in building knowledge and claims to apply that view in her teaching. For her, history goes back to original ideas and restores intuition against the formalism appearing in the finished theories. I think that Emma’s attitude about history of mathematics in the classroom is close to the guided re-invention proposed by Hans Freudenthal

Urging that ideas are taught genetically does not mean that they should be presented in the order in which they arose, not even with all the deadlocks closed and all the detours cut out. What the blind invented and discovered, the sighted afterwards can tell how it should have been discovered if there had been teachers who had known what we know now.... It is not the historical footprints of the inventor we should follow but an improved and better guided course of history. (pp. 101, 103)

The international dimension

For many decades of the twentieth century the issues concerning mathematics education were discussed inside the community of professional mathematicians, see (Furinghetti, 2007). In the 1950s some events prepared the terrain for changing this pattern. The main changes happened in the 1960s with the creation of a specific journal dedicated to mathematics education (Educational Studies in Mathematics, in 1968) and the creation of a specific conference on this subject (ICME - International Congress on Mathematical Education, in 1969). Thus mathematics education acquired the status of an academic discipline with its own chairs. It promoted a new idea that in some important events secondary teachers were invited to contribute. This was the case of Emma who was asked by Gattegno to join CIEAEM (Commission Internationale pour l’Étude et l’Amélioration de l’Enseignement des Mathématiques - International Commission for the Study and Improvement of Mathematics Teaching). She appears in the list of the founding members of this Commisssion in (Bernet & Jaquet, 1998) together with the mathematician and pedagogue Gattegno (secretary), the psychologist Jean Piaget, the logician and philosopher Evert W. Beth, the epistemologist Ferdinand Gonseth, the mathematicians Gustave Choquet (president), Jean Dieudonné, Hans Freudenthal and André Lichnerowicz, and the secondary teachers Lucienne Félix and Willy Servais. Emma was president of CIEAEM from 1979 to 1981. The two books produced by the commission reflect the two streams of work inside the Commission. The first, (Piaget et al., 1955) treats some contents mainly from the mathematical point of view. The second, (Gattegno et al., 1958), presents some aspects of school practice with a particular focus on the use of concrete materials and mathematical films, that is contents close to Emma’s interests. She, indeed, contributed with a chapter entitled “L’Object et l’action dans l’enseignement de la géométrie intuitive” (The object and the action in the teaching of intuitive geometry, pp. 41-59). Concrete Materials became a vehicle for intuition and experiment in the classroom, see (Gattegno, 1963; Servais, 1970), and prepared the school milieu to receive subsequent innovations with mathematical technology (Ruthven, 2008). The 1950s are the years of important movements of reforms, as the well-known New Math movement in USA and the parallel Modern Mathematics in Europe, see (Furinghetti, Menghini, & Matos, 2013). All these streams of reform related to modern, or new, mathematics met in 1959 at an international seminar held in Royaumont, near Paris, see (De Bock & Vanpaemel, to appear; Furinghetti, Menghini, Arzarello, & Giacardi, 2008; Schubring, 2014). The seminar was organized by OEEC (Organisation for European Economic Co-operation), and chaired by Marshall Stone, the president of ICMI. An important role was played by members of CIEAEM, particularly by Dieudonné, who delivered a talk concerning the transition from secondary school to university. Delegates from 18 countries of Europe participated to the meeting, see Appendix. Emma was one of the two representatives of Italy and took active part in the discussion about modern mathematics. Another woman, Lucienne Félix, was invited as a guest speaker. When Freudenthal succeeded in achieving the two goals of founding a new journal and having a conference dedicated to mathematics education, Emma was an active participant in both events. She published articles in the first volume of the journal Educational Studies in Mathematics and in successive issues. She presented a talk at the first ICME in Lyon (1969) and in the third ICME in Karlsruhe (1976) presented an exhibition (comprising more than 100 posters) prepared with he middle school pupils on the theme “Mathematics in real life” , see (Castelnuovo & Barra, 1976). In Fig. 1 there are the notes on this conference that Emma gave me during my visit to her. After the Second World War international cooperation originated as a structured network of activities aimed at mutually helping people. Since the beginning, education has been one of the main themes of action for political international bodies, see (Furinghetti, 2014). Emma contributed to this movement of international cooperation: invited first by IREM (Institut de Recherche sur l’Enseignement des Mathématiques) and then by UNESCO, Emma went to four times, from 1977 to 1982, to teach in classes that corresponded to grades 6 to 8, (approximated ages 11-14) see (Lanciano, 2013). There she applied her method with the local students. In the summer of 2006, when I met Emma at her home in Rome she referred with emotion and enthusiasm to this event of her life and to the good results she had. The international role of Emma was acknowledged by her appointment as a ‘member at large’ in the Executive Committee of ICMI from 1975 to 1978.

The profession of mathematics teacher

Emma was a very special teacher for many reasons. Beside her extreme creativity in designing teaching sequences and her freedom in interpreting the programmes in order to pursue her didactic aims, it is remarkable the way she was able to establish international contacts and to emerge on the international stage of mathematics education. This was difficult for a primary or secondary teacher and even more for a woman. As discussed in (Furinghetti, 2008), with few others she may be considered as a pioneer woman in the field. Emma’s action in mathematics education was not only directed to students, but also to her colleagues, especially young colleagues. To them she transmitted her knowledge and, most importantly, her enthusiasm and motivation. I met beginner teachers who attended the talks she delivered during the workshops carried out in a little village of Italy, see (Castelnuovo, 2008): they were emotionally involved in talking about this professional experience and, while having attended many mathematical courses in university, they seemed to have discovered new aspects of mathematics during the workshop. Outside Italy her legacy is present especially in Spanish speaking countries, see the website of the Sociedad Madrileña (SMPM). Emma had cultural influences, for example, her father and Enriques for mathematics education, Ovide Decroly, Maria Montessori and Piaget for general pedagogical issues. Nevertheless I would like to point out the main character of Emma as a secondary teacher, which is her capacity to reflect in a constructive way on her profession. She was a model of what (Schön, 1983) calls “reflective practitioner”, since she lived her experience with awareness, integrated it in her knowledge for teaching and was ready to use it for shaping her beliefs and taking decisions.

Acknowledgments: Many thanks to Marta Menghini and Leo Rogers.

Figure 1. Notes on ICME-3 by Emma Castelnuovo

1 To celebrate the 100th birthday of Emma Castelnuovo UMI (Italian Mathematical Union) has published a special issue of its journal dedicated to her (La Matematica nella Società e nella Cultura, 6). It contains articles on various aspects of Emma’s career and personality. For a full account of Emma’s life see (Menghini, 2013; to appear). In the list of the websites at the end of this article there are the addresses of the website containing the list of Emma’s publication and the website from which most texts may be downloaded. Emma’s legacy is outlined in the website by Lanciano. For the history and the main characters of CIEAEM see in the website of this Commission. For the history and the main characters of ICMI see the website designed by Furinghetti and Giacardi. References Bernet, T., & Jaquet, F. (1998). La CIEAEM au travers de ses 50 premières rencontres. Neuchâtel: CIEAEM. Castelnuovo, E. (1946). Un metodo attivo nell’insegnamento della geometria intuitiva. Periodico di Matematiche, s. 4, 24, 129-140. Castelnuovo, E. (1948). Geometria intuitiva, per le scuole medie inferiori, Carrabba, Lanciano- Roma. Castelnuovo, E. (1989). The teaching of geometry in Italian high schools during the last two centuries: some aspects related to society. In C. Keitel, P. Damerow, A. Bishop, & P. Gerdes (Eds.), Mathematics, education and society. Science and technology education, Document Series N. 35 (pp. 51-52). Paris: UNESCO. Castelnuovo, E. (2008). L’officina matematica: ragionare con i materiali. F. Lorenzoni (Ed.). Molfetta: La Meridiana. Castelnuovo, E., & Barra, M. (1976). Matematica nella realtà. Torino: Boringhieri. Castelnuovo, G. (1911). La rigueur dans l’enseignement mathématique dans les écoles moyennes. L’Enseignement Mathématique, 13, 461-468. Castelnuovo, G. (1919). La riforma dell’insegnamento matematico secondario nei riguardi dell’Italia. Bollettino della “Mathesis”, 11, 1-5. De Bock, D., & Vanpaemel, G. (to appear). Modern mathematics at the 1959 OEEC Seminar at Royaumont. In K. Bjarnadóttir, F. Furinghetti, J. Prytz, & G. Schubring (Eds.), “Dig where you stand” 3. Uppsala: Department of Education, Uppsala University. Enriques, F. (1921). L’insegnamento dinamico. Periodico di Matematiche, s. 4, 1, 6-16. Freudenthal, H. (1973). Mathematics as an educational task, Dordrecht: Reidel. Furinghetti, F. (2007). Mathematics education and ICMI in the proceedings of the International Congresses of Mathematicians. Revista Brasileira de História da Matemática Especial no 1 - Festschrift Ubiratan D’Ambrosio - (December) Publicação Oficial da Sociedade Brasileira de História da Matemática, 97-115. Furinghetti, F. (2008). The emergence of women on the international stage of mathematics education. ZDM, 40, 529–543 Furinghetti, F. (2014). Part IV, Chapter XXIII. History of international cooperation in mathematics education. In A. Karp, & G. Schubring (Eds.), Handbook on history of mathematics education (pp. 543-564). New York - Heidelberg - Dordrecht - London: Springer. Furinghetti, F., Matos, J. M., & Menghini, M. (2013). From mathematics and education, to mathematics education. In M. A. Clements, A. Bishop, C. Keitel, J. Kilpatrick, & F. K. S. Leung (Eds.), Third International Handbook of Mathematics Education (pp. 273-302). Dordrecht-etc: Springer. Furinghetti, F., & Menghini, M. (to appear). The role of concrete materials in Emma Castelnuovo’s view of mathematics teaching. Educational Studies in Mathematics, 87. Furinghetti, F., Menghini, M., Arzarello, F., & Giacardi, L. (2008). ICMI Renaissance: the emergence of new issues in mathematics education. In M. Menghini, F. Furinghetti, L. Giacardi, & F. Arzarello (Eds.), The first century of the International Commission on Mathematical Instruction (1908-2008). Reflecting and shaping the world of mathematics education (pp. 131-147). Rome: Istituto della Enciclopedia Italiana. Gario, P. (2013). Le radici del pensiero didattico di Emma. La Matematica nella Società e nella Cultura, 6, 7-33. Gattegno, C. (1963). The Cuisenaire material is not a structural apparatus. In C. Gattegno, For the teaching of mathematics, Vol. 3 (pp. 103-106). Reading: Educational Explorers. Gattegno, C., Servais, W., Castelnuovo, E., Nicolet, J. L., Fletcher, T. J., Motard, L., Campedelli, L., Biguenet, A., Peskett, J. W., & Puig Adam, P. (1958). Le matériel pour l’enseignement des mathématiques. Neuchâtel: Delachaux & Niestlé. Glaeser, G. (1983). À propos de la pédagogie de Clairaut. Vers une nouvelle orientation dans l’histoire de l’éducation. Recherches en Didactique des Mathématiques, 4, 332–344. Lanciano, N. (2013). Emma in Niger. La Matematica nella Società e nella Cultura, 6, 105-108. Menghini, M. (to appear). The commitment of Emma Castelnuovo to creating a new generation of mathematics teachers. In K. Bjarnadóttir, F. Furinghetti, J. Prytz, & G. Schubring (Eds.), “Dig where you stand” 3. Uppsala: Department of Education, Uppsala University. Menghini, M., with the collaboration of M. Barra, R. Bolletta, L. Cannizzaro, N. Lanciano, M. Pellerey, D. & Valenti (2013), Emma Castelnuovo: la nascita di una scuola. La Matematica nella Società e nella Cultura, 6, 45-80. OEEC (1961). New thinking in school mathematics. Paris: OEEC. French edition: Mathématiques nouvelles. Paris: OECE. Piaget, J., Beth, E. W., Dieudonné, J., Lichnerowicz, A., Choquet, G., & Gattegno, C. (1955). L’enseignement des mathématiques. Neuchâtel: Delachaux et Niestlé. Rogers, L. (to appear). Epistemology, methodology, and the building of meaning in a new community of mathematics educators in England 1950-1980. In K. Bjarnadóttir, F. Furinghetti, J. Prytz, & G. Schubring (Eds.), “Dig where you stand” 3. Uppsala: Department of Education, Uppsala University. Ruthven, K. (2008). Mathematical technologies as a vehicle for intuition and experiment: A foundational theme of the International Commission on Mathematical Instruction, and a continuing preoccupation. International Journal for the History of Mathematics Education, 3(1), 91–102. Schön, D. A. (1983). The reflective practitioner. How professionals think in action. New York: Basic Books. Schubring, G. (2014). The road not taken – The failure of experimental pedagogy at the Royaumont Seminar 1959. Journal für Mathematik-Didaktik, 35, 159-171. Servais, W. (1970) The Significance of Concrete Materials in the Teaching of Mathematics. in ATM Mathematical Reflections (pp. 203-208). Cambridge: C.U.P.

Websites CIEAEM. History. http://www.cieaem.org/?q=node/18) Fontanari, C., Emma Castelnuovo. http://www.science.unitn.it/~fontanar/EMMA/emma.htm Furinghetti, F., & Giacardi, L., The first century of the International Commission on Mathematical Instruction (1908-2008). The history of ICMI. http://www.icmihistory.unito.it/ ICMI. http://www.mathunion.org/icmi/activities/awards/emma-castelnuovo-award/ Lanciano, N. Mathematics, imagination and reality. The legacy of Emma Castelnuovo. https://www.researchitaly.it/en/understanding/project-and-success-stories/interviews-and-life- stories/mathematics-imagination-and-reality-the-legacy-of-emma-castelnuovo/ Menghini, M. with the collaboration of M. Barra, L. Cannizzaro, N. Lanciano, & D. Valenti. http://www1.mat.uniroma1.it/ricerca/gruppi/education/scanner%20emma/Pubblicazioni_Emma_ Castelnuovo.htm Sociedad Madrileña SMPM… http://www.smpm.es

Appendix. List of participants and guest speakers at the Royaumont seminar (Source OEEC, 1961, pp. 213-219)

Scheiwein, Erwin (Austria) Studzinsky, Hermann (Austria) Ballieu Robert (Belgium) Van Hercke, Jean J. (Belgium) Burwell, James Hugh (Canada) Rindung, Ole (Denmark) Huisman, André (France) Theron, Pierre (France) Athen, Hermann (Germany) Schoene, Heinz (Germany) Georgontelis, Kanellos (Greece) Sotirakis, Nicolaos (Greece) Campbell, John (Ireland) Forde, John Martin (Ireland) Campedelli, Luigi (Italy) Castelnuovo, Emma (Italy) Kieffer, Lucien (Luxembourg) Michels, Marcel (Luxembourg) Leeman, Henri Theodoor Masie (The Netherlands) Vredenduin, Pieter G. J. (The Netherlands) Gjelsvik, Ingvald (Norway) Johansson, Ingebrigt (Norway) Piene, Kay (Norway) Frostman, Otto (Sweden) Sandgren, Carl Lennart (Sweden) Pauli, Laurent (Switzerland) Saxer, Walter (Switzerland) Kodamanoglu, Nuri (Turkey) Hope, Cyril (UK) Land, Frank William (UK) Stone, Marshall H., President of the seminar (USA) Tucker, Albert W. (USA) Djerasimovic, Bozidar (Jugoslavia)

Guest speakers at the seminar Begle, Edward Griffith (USA) Botsch, Otto (Germany) Brunold, Charles (France) Bundgaard, Svend (Denmark) Bunt, Luke N. H. (The Netherlands) Choquet, Gustave (France) Dieudonné, Jean, chairman of Section I (France) Fehr, Howard F. (USA) Félix, Lucienne (France) Maxwell, Edwin Arthur (UK) Rourke, Robert E. K. (USA) Servais, Willy (Belgium) Wall, William Douglas (UK)

Organization for the European Economic Co-operation Gass, James R. Metzger, Helmut Vincent, Louis