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Doing mathematics implies three levels of manipulation: manipulating the abstract, manipulating symbols and manipulating logic. Teaching mathematics therefore involves the teacher proposing situations in which pupils can explore a small part of mathematics through these manipulations. In doing so, the pupils work on both the heuristics enabling them to confront themselves with a real mathematical research and knowledge in construction. Through examples of didactic situations of problem solving, this article aims to show how problems can be drivers of mathematics learning., Comment: in Italian language

In October 2017 the Italian National Institute of Statistics (ISTAT), Italy's body for official statistics, has published the book of fairy tales Le streghe di Bayes (The witches of Bayes) written by ISTAT staff members with the commendable aim of introducing statistical and probabilistic reasoning to children. In this paper the fairy tale which gives the name to the book is analyzed in a dialog between three teachers with different background and expertise. The outcomes are definitively discouraging, especially when the story is compared to the appendix of the book, in which the teaching power of every story is indeed explained (as a matter of fact, without the appendix the fairy tale of the witches seemed to be written with the purpose of make the 'Bayesians', meant as the villagers from 'Bayes', ridiculous). In fact the fairy tale of the witches does not contain any Bayesian reasoning, the suggested decision strategy is simply wrong and the story does not even seem to be easily modifiable (besides the trivial correction of the decision strategy) in order to make it usable as a teaching tool. As it happens in real dialogues, besides the fairy tale in question, the dialogue touches several issues somehow related to the story and concerning probability, inference, prediction and decision making. The present paper is an indirect response to the invitation by the ISBA bulletin to comment on the fairy tale., Comment: 48 pages, 14 figures, in Italian, be published in Progetto Alice, Rivista di matematica e didattica

In this paper we make some considerations about using a dynamic geometry software for teaching history of mathematica at university level. After a short introduction to the software GeoGebra, we discuss four activities. The first is an exploration of the menu of the tools menu following the first part of the first book of the elements of Euclid. The second is an activity about pentagons and pentagrams which aims at finding a more natural way of discovering their properties than that suggested by Euclid as an illustration of the method of synthesis and analysis. The third is an activity on circle inversion and its connection with polarity and the construction of the fourth harmonic point after three given point on a line. The activity continues with the reading of Poncelet's ideas about ideal secants of conics and how to use GeoGebra to make sense of them. The fourth activity shows how to use the possibility to move objects in GeoGebra to make sense of the geometric intuition behind Cremona and Hirst letters about the Bellavitis-Hirst inversion of a circle. We discuss the reasons why we believe that these kind of activities may help students to read and understand original writings in projective geometry and why we believe that they may be very effective in the training of prospective teachers of mathematics., Comment: 38 pages, in Italian, 44 figures

The purpose of this paper is to prove directly, by an elementary method, the Poisson probability law. This proof is offered as an alternative to the more usual derivation from binomial distribution in the limit of small probabilities. The same proof is then applied to the solution of a problem in statistical mechanics. --- Lo scopo di questo articolo \`e dimostrare direttamente, con un metodo elementare, la legge di probabilit\`a di Poisson. Questa dimostrazione \`e proposta in alternativa alla pi\`u consueta derivazione dalla distribuzione binomiale nel limite delle piccole probabilit\`a. La stessa dimostrazione viene quindi applicata alla soluzione di un problema di meccanica statistica., Comment: 14 pages, 1 figure, in Italian, submitted to VSMP-Bulletin

The aim of this article has been to put together various aspects of the personality of the Italian mathematician Francesco Severi: the mathematical, the political, the institutional, academic and philosophical one. We tried in particular to elucidate if the events related the First World War represented a turning point in the life and scientific production of Severi., Comment: This paper is in Italian

The content of this paper is the noted transcription of the documents (232 letters and 25 documents) which make up the Fondo Gian Antonio Maggi stored at the Library of the Department of Mathematics of the University of Milan. The documents cover the period 1885-1937. La corrispondenza qui riprodotta (232 lettere e 25 documenti) compone il Fondo Gian Antonio Maggi conservato presso la Biblioteca del Dipartimento di Matematica (Universita' degli Studi di Milano). I documenti coprono il periodo 1885-1937., Comment: 324 pages, 79 figures, in Italian

The collaboration of mathematics in the two musical systems of Pythagorean sounds and of equal sounds is evident and opportune for generating the elements and managing their relationships. The only essential notion for a rational intervention in the two sound worlds is that of distance between sounds. The scale of natural sounds does not seem to dialogue directly with a purely mathematical module that articulates its definition and is generally presented as a manual adjustment of the ancient Pythagorean sounds. The model of proportions that formed and inspired the architecture of the sixteenth century, an era in which the natural scale emerges, provides a clear idea for systematically adjusting natural sounds; on the other hand, the generation of sounds based and started exclusively on the respect of the proportions proposes interesting natural sound systems. The object is not so much to illustrate the harmonic device of the already formed natural scale, as to implement the formation of sound elements that respond to the unitary principle of proportion and combine harmoniously., Comment: 20 pages, 4 figures, in italian

These are the lecture notes (in Italian) of a course held in Perugia, Italy, during the summer 2002. They concern the basic facts on the iterative solution of linear systems. The course is self-contained and requires only basic knowledge of numerical linear algebra. This is a corrected version of the original lecture notes, and are dedicated to the memory of Cecilia Magherini., Comment: 150 pagine. This is the corrected version of the lecture notes (in Italian). In memory of Cecilia Magherini

The content of this paper is the noted transcription of the 83 letters written by Enrico Betti between 1860 and 1886 stored at the Istituto Mazziniano, Museo del Risorgimento in Genova and now available at www.luigi-cremona.it The letters are addressed to: Luigi Cremona (79), Eugenio Beltrami (2), Pietro Blaserna (1) and one son of Gaetano Giorgini (1). ----- La corrispondenza qui riprodotta, composta dalle 83 lettere scritte da Enrico Betti tra il 1860 e il 1886, si trova a Genova (Istituto Mazziniano, Museo del Risorgimento) e ora anche sul sito www.luigi-cremona.it Destinatari sono: Luigi Cremona (79), Eugenio Beltrami (2), Pietro Blaserna (1) e uno dei figli di Gaetano Giorgini (1)., Comment: 56 pages, 1 figure, in Italian

The content of this paper is the noted transcription of the 48 letters that Celeste Clericetti wrote to Luigi Cremona and the 10 letters that Cremona wrote to Clericetti between 1871 and 1887. The correspondence is stored at the Istituto Mazziniano, Museo del Risorgimento in Genova and is now available at www.luigi-cremona.it ----- La corrispondenza qui riprodotta, composta da 48 lettere scritte da Celeste Clericetti a Luigi Cremona e 10 lettere scritte da Cremona a Clericetti tra il 1871 e il 1887, si trova a Genova (Istituto Mazziniano, Museo del Risorgimento) e ora anche sul sito www.luigi-cremona.it La trascrizione ha un ampio apparato di note e comprende i cenni biografici di tutti i nomi citati nelle lettere., Comment: 90 pages, 6 figures, in Italian

The theme of the influence of Bourbaki's ideas and methodologies on disciplines that transcend the mathematical field has been frequently addressed and, in particular, the reflection on the relationship between Bourbaki and Oulipo has animated a passionate debate within literary criticism. The purpose of this contribution is to investigate the relations between Le Lionnais, Queneau and Roubaud - and more generally Oulipo - and Bourbaki. Through a comparative examination between the two groups, the existence of an undeniable charisma of the "polycephalic mathematician" on the artistic-literary movement was highlighted., Comment: in Italian

This paper deals with the celebrated Euclidean theorem about isosceles triangles, comparing different proofs., Comment: in Italian

To contrast the phenomenon of false positives in the diagnoses of dyscalculia in Italy, among 3rd grade children, a 3-year project (2011-2014), built upon a collaboration between psychologists and mathematics educators, was carried out. Within the project specific teaching strategies for preventing and addressing learning difficulties in arithmetic arising at the beginning of primary school were designed and tested in schools nation wide. This paper presents the project's background and the theoretical foundations of the didactical material designed, providing also some examples. In particular, prototypical examples will show how the activities, grounded within a kinesthetic-tactile and visual-spatial approach, are designed to lead to students' interiorization of part-whole relations, and to their thinking about multiplication through structured diagrams. Qualitative data confirming effectiveness of the proposed didactical strategies will also be discussed., Comment: in Italian

This short paper focuses on Schr\"oder's contribute towards a structural view of group theory., Comment: in Italian

In this paper, our aim is to discuss the outcome of a ten-year experimental workshop activity carried out by the Center `matematita' (http://www.matematita.it/) with classes of pre-university students of any age. We analyse the potential benefits of this way of teaching and learning mathematics, both from the point of view of students and teachers., Comment: In Italian

The theory of numbers was supposed to be the less useful branch of mathematics. At the same time, cryptography was thought to be a military or a diplomatic issue. In this note we show how the two concepts are today strictly related and how this link affects our daily activities., Comment: 34 pages, in Italian. Extended version of a paper published by the Italian magazine Progetto Alice, 2010-III, Vol. XI, n.33, 427-462

This paper collects some reflections about an apparent incongruity between the usual (third-person) understanding of the probability of an event calculated for an extended period of time in the future (e.g., the expected probability of a driver to meet with a car accident in the next $M$ years) and the subjective perception of the same probability/risk that the person involved in that event has, instant by instant during that period of time. Similarities with the classical Zeno's paradoxes come to mind., Comment: 7 pages, no figures. English translation of v2 (in Italian)

In this paper we present three simple applications of probability and highlight and discuss their paradoxical flavour., Comment: Plain LaTeX, 9 pages, one figure. Text in Italian

The equable, Pythagorean and natural scales are built on the basis of a mathematical logic., Comment: in Italian

Recently the media broadcast the news, together with illustrative videos, of a so-called Japanese method to perform multiplication by hand without using the multiplication tables. "Goodbye multiplication tables" was the headline of several websites, including important ones, where news are however too often `re-posted' uncritically. The easy numerical examples could induce naive internauts to believe that, in a short future, multiplications could be really done without the knowledge of multiplication tables. This is what a girl expresses, with great enthusiasm, to her father. The dialogues described here, although not real, are likely and have been inspired by this episode, being Maddalena the daughter of the author. Obviously the revolutionary value of the new method is easily disassembled, while its educational utility is highlighted to show (or remember) the reasoning on which the method learned in elementary school is based, although mostly applied mechanically., Comment: 14 pages, 15 figures, in Italian, published in Progetto Alice, Rivista di Matematica e Didattica

We discuss several aspects of infinity in the history of mathematics., Comment: in Italian

At the beginning of the 900s Guido Castelnuovo turned his attention toward methodological, didactical, historical and applicative issues. He was an active member of the ICMI (International Commission on Mathematical Instruction) and of the Mathesis, an Italian association of mathematics teachers. He also was the author of the mathematics curricula for the new Modern Lycee and he taught the first University courses devoted to preservice training of mathematics teachers. His course on Precision and Application Mathematics, inspired by Felix Klein, shows Castelnuovo's point of view on mathematics and presents a higher point of view on the programmes of the Modern Lycee., Comment: in Italian

We discuss a non-intuitive situation concerning percentages., Comment: in Italian

Here we present the transcription of the handwritten notes, carefully taken by an anonymous student, of the last lecture course taught by Guido Castelnuovo in 1922-23. The title of the course is "Curve algebriche piane e sghembe" ("Plane and space algebraic curves"): indeed, the language of both the original text and our brief introduction is Italian. The manuscript belongs to the legacy of Alfredo Franchetta (1916-2011) and the typesetting is due to Natascia Zangani. This edition is our modest contribution to celebrate the 150th birth anniversary of Guido Castelnuovo (1865-1952)., Comment: in Italian

The aim of this report is to describe a field trial conducted in Italy during the academic year 2009-10, with primary and secondary school teachers (grades K-8), in order to test prototypes of "ready to use" kits to be used to implement PBL laboratory sessions in the classroom. This work is meant to provide a sort of snapshot of the classroom implementation of the teaching recommendations (so that any readers can conduct their own analysis). The kits were designed by matematita (Interuniversity Research Center for the Communication and Informal Learning of Mathematics, http://www.matematita.it). I wish to thank the "Ufficio Scolastico Regionale della Lombardia" for sponsoring the trial., Comment: The reports by the participating teachers are in Italian. Translations may be provided on request

Lectures notes (in italian) of some arguments of classical analysis, with exercises. A particular emphasis to functional analysis and elementary operator algebra theory is given, by means of exercises and examples., Comment: 207 pages, in Italian; some minor corrections made; the exposition has been improved and the bibliography has been updated