Emma Castelnuovo was born in Rome on 12 December 1913 and died in Rome on 13 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 Italy. Emma's father Guido Castelnuovo and her uncle Federigo Enriques, 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 . 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.
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  and  that 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 Clairaut's 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 . 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, and 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 , but the proposed approach inspired Emma to renewing her teaching, see  and .
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 rigour 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 rigour was treated in the most important treatises for secondary school, see . 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 programmes, see . 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 , 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 .
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. Rigour 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 .
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. With this concern Rogers  discusses the work of Caleb Gattegno (1911-1988), Zoltán Pál Dienes (1916-2014), and the British Association of Teachers of Mathematics (ATM). Since 1949 Emma was in contact with teachers of the École Decroly, where Paul Libois (1906-1991) 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 .
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 was not only a paramount mathematician, but also a historian and epistemologist. On these subjects he delivered university courses and edited journals. On 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.For many decades of the twentieth century the issues concerning mathematics education were discussed inside the community of professional mathematicians, see . 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 Commission in  together with the mathematician and pedagogue Gattegno (secretary), the psychologist Jean Piaget (1896-1980), the logician and philosopher Evert Willem Beth (1908-1964), the epistemologist Ferdinand Gonseth (1890-1975), the mathematicians Gustave Choquet (president), Jean Dieudonné, Hans Freudenthal and André Lichnerowicz, and the secondary teachers Lucienne Félix (1901-1994) and Willy Servais (1913-1979). 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, , treats some contents mainly from the mathematical point of view. The second, , 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  and , and prepared the school milieu to receive subsequent innovations with mathematical technology .
The 1950s are the years of important movements of reforms, as the well-known New Math movement in the United States and the parallel Modern Mathematics in Europe, see . All these streams of reform related to modern, or new, mathematics met in 1959 at an international seminar held in Royaumont, near Paris, see ,  and . 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 participated in the meeting.
Emma was one of the two representatives of Italy and took an active part in the discussion about modern mathematics. Another woman, Lucienne Félix (1901-1994), 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 the middle school pupils on the theme "Mathematics in real life", see .
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 . Emma contributed to this movement of international cooperation: invited first by the French IREM (Instituts de Recherche sur l'Enseignement des Mathématiques) and then by UNESCO, Emma went to Niger four times, from 1977 to 1982, to teach in classes that corresponded to grades 6 to 8, (approximated ages 11-14) see . 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 in her life and to the good results she had obtained.
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.
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 that 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 so for a woman. As discussed in , along with few others, she may be considered as a pioneer woman in the field.
Emma's contributions to 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 : 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 Madrilena (SMPM).
Emma had cultural influences, for example, her father and Enriques for mathematics education, Jean-Ovide Decroly (1871-1932), Maria Montessori (1870-1952) and Jean 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 Donald Alan Schön (1930-1997) 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.
Article by: Fulvia Furinghetti, University of Genoa.