Emma Castelnuovo and the Use of Concrete Materials (O Experiments, O …)

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 Rome 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 Italy. Emma’s father Guido (1865-1952) and her uncle Federigo Enriques (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...

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