Integrating history of mathematics with foundational contents in the education of prospective elementary teachers

The proposal presented here relates to the integration of the mathematical syllabus content, including foundational issues, with selected aspects of history of mathematics in the education of prospective elementary teachers. An analysis of the needs and cultural profile of prospective elementary teachers in Italy was the original motivation of this study. The new syllabus has been applied for several years in Rome, with good results in mathematical learning and self confidence regarding the performance as a future mathematics teacher. Students also show an ability to use history of mathematics in the classroom with good feedback from the children. The historical contents were used in Spain to offer a stronger mathematical background to a group of teachers and prospective teachers collaborating in an experimental workshop for children with Down s Syndrome. Once again the result was improved appreciation of their own skills and greater autonomy and spirit of initiative. 1 The historical background: from optimism to an age of uncertainty in children's mathematical education and its impact on primary school teachers' education Both in Italy and in Spain prospective elementary school teachers show fear and rejection in the face of mathematics, which is largely explained by a cultural atmosphere of pessimism about the difficulties of the task of teaching maths to children and also by confusion about how to carry out such a task. A loss of certainty about methods and contents of children's mathematical education spread throughout much of the industrialized world after the Second World War. In a context in which a long, solid tradition of calculations and elementary training was thoroughly applied around the world, the pestalozzian optimism of many innovators between the second half of the 19th century and the first decades of the 20th century, most of them mathematicians (Frobel,  was overwhelmed by a wave of pessimism. The crisis of European science due to political evolution and the war was one main cause, because the network of international contacts, cultural centres, publishing houses and general scientific journals had suffered a severe blow. Deep contributions based on the practical experience of educators when confronted with a didactical tradition rooted in the medieval abacus schools had marked an age of confidence in the powers of the mathematical mind to be exploited in the "preparation of the child for science" However, after many editions and translations of Laisant's Initiation to mathematics and Mace's Grand papa's arithmetic, these booklets addressed to educators became forgotten books. In his 1938 Mathematics in society and culture Federico Enriques, referring to Johann Pestalozzi and Friedrich Frobel's inspiring contributions, wrote "the formative value of mathematics can also be seen in the first grades of children's education and working class education; because mathematical intelligence is quite precocious" referring to Johann Pestalozzi and Friedrich Frobel's inspiring contributions. However, 1938 was the year of the race laws in Italy and Enriques was expelled from the university. Even Maria Montessori's optimism in her Psychoarithmetics and Psychogeometry, published in 1934 in pre civil war Spain was confined to the circle of montessorian schools in the following decades. The impact of the psychological experimental approach, from Edward Thorndike to Piaget's prominent books with Alina Szeminska and Baber Inhelder on number and concepts of space in the child can be summarized in the transition from an approach of confidence in children's intuition to a focus on the difficulties of learning mathematics dominant through the 1950s to the 1960s. The above mentioned innovators, pedagogues and mathematicians had criticized rote learning and boring and mechanical methods of 19th century primary schools and they tried to put forward a different style of teaching specially designed for children. Alternatively, the psychological world believed it had found an obstacle to learning mathematics in the young child's mind, a kind of illogical child, not able to make logical inference similar to how Levy Bruhl had spoken of a primitive illogical mentality. In the 1970s and 1980s Edinburgh psychologists led by Margaret Donaldson offered a radically different vision of the strength of children's mind (Donaldson 1978, Hughes 1986) Nevertheless, in the meantime such a context of pessimism and loss of almost every deep rooted certainty on children's mathematical education led to questioning of the on going education of young future primary schools teachers. This was the case in many European countries in Western Europe and in the United States, in contrast with the continuity in China or the USSR and Eastern European countries. The long term effects of this sharp contrast was aptly described by Liping Ma in her 1999 Knowing and Teaching Elementary Mathematics: teachers understanding of fundamental mathematics in China and the United States, which considers both the effects on children's education and on the cultural and emotive position of primary teachers. The spread of psychological views on early maths education combined with new math. Meanwhile, ideas on children's first steps in mathematics had deeply changed. For the first graders, new math meant sets, trying to find intruders in sets, exercises of equipollence, and 0 as cardinal of the empty set; for pre schoolers, down and up, inside and outside and those so called "topological notions" Geometry had never had a real relevance in primary education, but innovators had shared the feeling of the crucial role that geometry could have in transforming and improving primary mathematical education. All of this was completely 1 Montessori wasn't a mathematician but a medical doctor, however, in Rome she had attended the discussions of mathematicians deeply concerned with basic concepts of mathematics and geometry, and her optimism was a legacy of the indefatigable Seguin, the author of the 1875 Report on education. forgotten as a result of the rejection of Euclidean geometry throughout the compulsory and secondary education system. What kind of teacher should have taught new math to children? It's hardly surprising that this new trend had a deep impact on the image and profile of primary teachers. Rene Thom had perceptively anticipated this long term effect in his 1972 lecture at the Exeter conference on "Developments in mathematical education" (International Commission for Mathematical Instruction, Howson 1973) emphasizing the fact that the change in mathematical programmes or contents in primary school was intended also, if not mainly, to force primary teachers to change their pedagogical methods, in order to obtain a heuristic teaching that was direct, free and constructive, that could awaken the child's interest. Twenty years later, in 1991 he remembered as a witness the dramatic impact in the 1970s France: Old white haired teachers, who taught elementary calculations with sticks, have been seen to be forced to come up to date. They were told: Sirs, what you do is ridiculous; you know nothing about set theory, and one cannot do arithmetic without understanding it. And those old teachers were forced to sit down at school desks to listen to pretentious youngsters who explained to them that they had understood nothing about numbers!" (Thom 1991) Thom himself, wrote, that he had been a modernist ante litteram (he entered the Paris Ecole Normale Superieure in 1943) and was fond of the foundations of mathematics, logics, and set theory, but there have been excesses of zeal and a deep misunderstanding: that "by making the implicit mechanisms or techniques of thought conscious and explicit, one makes these techniques easier" (Thom 1973, p. 197) However, this is exactly what became the norm and set theory, Boolean algebra and topological structures were considered as the basic mathematical background of a teacher in order to work with illogical children. The impact on primary teacher's education, life and work, and self image as a result of the coalition of psychology with "modernism" in mathematics, as Thom put it, was severe. In many countries around the world this education lasted until very recently outside the university, in secondary education institutions, which moreover were considered of lower rank if compared with the classical language high school. In addition, his or her point of view on mathematics was that of a young student, not that of a prospective teacher. Eventually the idea of post secondary normal schools devoted to the primary teachers education spread, or alternatively there were specific degrees in primary education in colleges of education. A modern math teacher for children aged 3 to 10 should have a psycho pedagogical background, for some; or in other cases should learn logic and the foundations of mathematics; or a mixture of both new elements depending on the institutional set. 2 The challenge of self confidence among prospective teachers 2 In fact, this did not get rid of geometry, because 3D physical didactical materials designed and commercialized in those years (Dienes, Cuisenaire) were eventually introduced in primary schools to try to solve an entangled situation, which implied appealing to children's geometrical intuition (for example comparing and adding solids) In Italy and Spain, young students, mainly women, who enter the university to become teachers, have an experience of math courses in compulsory and secondary school lasting about twelve years. This experience often involves feelings such as annoyance or distress and a vision of mathematics reduced to pure calculation and mechanical procedures. An easy way to face up to this situation is to give up to teach more mathematics to them, and focus on courses on mat