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Pointing on some neuroscience results, concerning the presence in our brain of two distinct prelinguistic systems for processing numbers, we compare different views on the cognitive roots of numbers, in epistemology, psychology and math education. The conclusion is that the two distinct processes of counting and measuring should be better integrated in early math education. To this purpose, an example is reported and commented, of a revealing cognitive behaviour of very young children.

Our study focuses on a practical mathematics teaching method applied in Austrian secondary schools at the beginning of the twentieth century. The method, proposed by Josef Jacob, the Director of a “Gymnasium” in Vienna, was inspired by the cognitive theories of Ernst Mach, who recognized this by writing the preface to the volume in which the method is described. We present its theoretical framework and give some examples of application, in particular to infinitesimal calculus.

We present a research focused on the question of changing mathematics secondary school curricula in the period of transition from the school regulations of the Habsburg Empire to the ones of the Kingdom of Italy (1918–1923). Besides teaching programmes, schools in the Habsburg Empire differed from those in the Kingdom of Italy in multiple aspects, ranging from the administrative rules to the juridical status of teachers. Regarding mathematics, there were considerable differences in content and time-tables, but the main difference was in teaching methods and was due to deep-set school principles. The question of changing mathematics programmes was at that moment of great interest to mathematics education in Italy, where a reformist current supported a less theoretical and more practical teaching. This current was a part of a larger European movement for renewal which leapt to the fore at the Fourth International Congress of Mathematicians held in Rome on April 6–11, 1908, when was created the CIEM-IMUK presided by Felix Klein. Our research deals in particular with the work of the Trieste Section of the “Mathesis” Society (founded on June 15, 1919) in preparing mathematics new curricula and selecting the most adapted Italian textbooks.

This study focuses on the work of the “Mathesis” Society, Section of Trieste (Italy), in the period of transition, in the region Venezia Giulia (now belonging to Italy, Slovenia, and Croatia), from the school regulations of the Habsburg Empire to the ones of the Kingdomof Italy (1918-1923). Before 1918, in the two countries the differences in math teaching methods and conceptions concerning education and curricula were considerable. The research, supported by documents of the Trieste Section of “Mathesis” hitherto not yetexplored, shows that, in spite of their strong Italian feelings, the mathematics teachers of the Italian language secondary schools in Trieste, trained at the Austrian universities on the teaching methods proposed by Felix Klein, did not accept passively the changes enforced on school curricula, until the Gentile Reform (carried out by the first Fascist government) obliged them to.

This paper reports on the third part of a long term experiment concerning the introduction of 12-13 years old pupils to the concept of randomness and probability. In the first parts, the pupils analysed the concept of randomness, in daily life and with LEGO-RCX robots. The third part is based on the Random Garden Game (within the programming environment ToonTalk). The pupils are guided towards the concepts of equivalence of sample spaces and of classical probability, through temporary and personal meanings, corresponding to an evolution by means of class discussions.

In Austria, in the first decade of the 20th century, a reform in school curricula gave the opportunity to introduce the first elements of differential and integral calculus in the secondary school teaching. In some Italian speaking secondary schools of Trieste, a city which at that time belonged to the Habsburg Empire, these subjects were introduced in the teaching programmes. At the end of the First World War Trieste was annexed to the Kingdom of Italy. In our research we examine an example of the didactic methodologies used in Trieste before the First World War to teach these subjects and describe briefly the situation which occurred there in the period of transition from the regulations of the Habsburg Empire to those of the Kingdom of Italy.

We recall a Houdé’s experiment in which people performing logical tasks like Wason test are observed by means of functional brain imaging techniques, and some correlated neurobiology theoretical constructs. Then we discuss possible outcomes of these results on models of cognitive dynamics and, more generally, on Mathematical Education. Finally, we show how the above theoretical issues, together with some new experimental data, support our resonance model of cognitive dynamics.

In these years many language-based theories are being extensively employed to analyze students’ and teachers’ mathematical behaviours. In this framework, starting from a long term classroom and research experience, we claim that when students are allowed to freely choice their own linguistic tools for specific goals (e.g. to understand or to communicate), the functions of different language components considerably vary and their effectiveness increase, with a particular role always played by figural component. Our claim is supported by the analysis of a protocol on an emblematic classroom activity, according to which we structure our teaching strategies at all levels. Finally, starting from our analysis of the observed cognitive behaviours of our students, we give arguments about the necessity of refining and enlarging the theoretical setting

The paper describes the new postgraduate training programme for prospective secondary school teachers, set up in Italy in these years, comparing it with what happens in other countries. In particular, the experience done in the first years in mathematics area is analysed. Several papers devoted to these questions, published in Italy in the years 2000-2003, are also reported and reviewed

The paper is a survey of the Italian researches on the teaching of "rational" geometry. Before collecting and describing 75 contributions to the theme, a general view is offered of the principal research lines, with particular emphasis on the "Italian case" for Geometry. Here, as a long lasting tradition, famous mathematicians (e.g., Peano, Enriques) devoted their attention to didactic questions as well as to theoretical ones. As a consequence, a whole research domain has been developed in the years, called "rational geometry". Since in the last, say, two decades, the international educational research has followed different paths, in the paper questions are also raised about the uncertain borders between educational and geometrical research domains.