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In the literature we can find many examples showing the difficulty of students in solving mathematical problems. In particular, according to many authors, the critical aspect is precisely the ability to "making sense" of the problems to be solved (e.g. see de Corte, Greer, Verschaffel, 2000). From our point of view, it is not likely to gain an improvement of students' skills if we do not act adequately on teacher training. A first necessary condition is that teachers must be able to adequately solve the tasks that are usually proposed to students. For this reason, in the "Mathematics teaching" courses for the primary school prospective teacher training program at the University of Milano-Bicocca we discuss and promote a PBL (Problem-Based learning) model. These courses combine theoretical lectures with practical activities (for example the didactic pedagogical laboratories described in previous IATED conferences). One of the main topics of such courses is precisely a discussion of the idea of "problem", as opposed to the idea of "exercise" (Schoenfeld, 1992), and of the didactic potential of a teaching practice that makes good use of problems. At the end of the course we tested prospective primary school teachers with a classic problem on proportionality combined with a typical examples of pseuso-proportional problems, for which the inadequacy of the proportional model should emerge. We proposed the task to 57 prospective teachers, asking them to give an explicit and detailed resolution of the problems and to carry out an analysis of the teaching potential of such problems for primary school students. In this paper, we present a qualitative analysis of the answers collected. From such an analysis, inconsistencies between the theoretical model described at lectures and the practical application emerge for many students. In some cases students explicitly admit to have adapted their answers to the "didactic contract". The observations collected allowed us to rethink some aspects of the course, and we will experiment some changes in the next academic year.

In questo articolo descriviamo una serie di attività-gioco riguardanti la crittografia elementare, sfruttando un ambiente di gioco matematico web “pronto all’uso” da noi elaborato su WIMS. Nello specifico, proponiamo problemi di codifica e decodifica di messaggi segreti, prendendo ispirazione dal Codice di Cesare, e un gioco ispirato a tecniche di crittografia moderne, quali il codice a correzione di errore di Hamming, presentato come un gioco di magia. Inoltre, condividiamo i risultati della somministrazione delle attività sopra descritte a quattro classi di scuola primaria (due terze e due quarte).

The importance of an active engagement of students in the learning process is nowadays recognised as a central aspect for a successful teaching practice. Many researchers highlight that teaching through games is a very effective way to involve pupils, especially in primary school. Indeed, the methodology of Game Based Learning not only stimulates students’ interest, but increases their motivation and satisfaction, making it easier to understand and remember new concepts. This is particularly relevant for Mathematics, where learning a new rule through experimentation leads to a better understanding. Such a methodology becomes even more powerful when combined with the use of the so-called “new technologies”. We developed a proposal for a web-based mathematical “turnkey” game environment for primary school pupils, using the WIMS system. The WIMS system (WWW Interactive Multipurpose Server, https://wims.matapp.unimib.it/), conceived by Xiao Gang and made public in 1998, is now in widespread use throughout France and is slowly spreading to other countries. It is available in various languages, and it is recognized for its vitality and effectiveness as a server for publishing interactive learning objects. One of its advantages is that its activities can be offered to the user with no need to install any additional software on the local pc (or on any mobile device) other than a standard web browser. Also, teachers can create virtual classes to propose activities to students and actively monitor their works. Obviously, WIMS can be a very useful tool for distance learning. The aim of this work is to present an activity on elementary cryptography through games. Our proposal consists of essentially two parts. The first activity challenges the students with the problem of ciphering and deciphering secret messages, taking inspiration from Caesar code. The second part involves a game inspired by modern cryptography techniques, such as Hamming codes for error correction. It is presented as a magic trick: one volunteer is asked to think about a number from 0 to 15 and to answer 7 yes/no questions regarding it. The interesting part is that the volunteer is allowed to lie once. The “mathe-magician” will then guess not only the number, but also the eventual lie. It is a very intriguing game (for adults as much as for kids), that stimulates pupils in asking the question “How does it work?”. This is enough to capture the students’ attention, enabling the teacher to introduce them to the inner mathematical motivations of the game, such as the binary code. The applets we will present are publicly available on any WIMS server (e.g. at the URL https://wims.matapp.unimib.it/test/wims.cgi?module=E4/game/oefcesare.it) in Italian and French (and will soon be available in English as well). In our presentation we will also share the outcomes of a trial testing of such activity with four classes of Italian students, for a total of 74 pupils of 8 and 9 years old, focusing on a qualitative analysis of the reactions of the pupils and their teachers. Moreover we will outline how the WIMS contents were designed to support the explanation of how the mathematics behind the magic works.

Communication in mathematics can be particularly difficult because multimodal registers are needed in order to develop mathematical ideas (usually a simultaneous use of written texts, formulas, schemas, diagrams, geometric figures, graphs, and so on is essential). Effective communication of mathematics need to switch from one register to another, from one representation to another, as much as possible. One could go as far as to say that the true mathematical competence is knowing how to use the right representation at the right time. For this reason it turns out that old style pen and paper techniques are the best suited for carrying out a mathematical discourse. Mathematicians are known to like to use blackboards, both when they work on their researches and when lecturing. From a didactic point of view, students as well can take advantage of the possibility of writing by hand on paper both when they carry out their studying and, for example, when they are under examination. This is particularly true with written exams that contain open-ended questions and ask to provide justifications for the given answers, and in case of oral exams, which are the type of exam that best allows for verifying the capacity of building a mathematical discourse. With the outbreak of the COVID-19 emergency, in spring 2020 all exams needed to be held remotely and pen and paper turned out to be not available. Some technical solutions available were in fact not usable due to the lack of technical equipment on the part of the students. E.g. to effectively use an online whiteboard you need to have a digital pen: you simply cannot write mathematics with just the mouse. By means of the WIMS (WWW Interactive Multipurpose Server) platform, that we introduced at ICERI19, it was possible to build an online work environment that made it easier for students to work in mathematics, with particular attention to the task of conducting a mathematical discourse. We will describe in detail the setup used for the geometry course for prospective primary school teachers, showing the effectiveness of WIMS: - in complementing lectures by posing engaging problems , e.g. in order to let student experiment with the graphic properties of the geometric objects being studied; - as a tool to replace the written exam, allowing for the conceiving of an open-ended questions test, with automatic verification of the answers; - as a drawing board to be used during oral exams, so to let students easily provide drawings and representation to support their discourse. Some videos of students' use of these tools are available (e.g. see https://www.youtube.com/watch?v=okq5EOKNNRc&t=42s), which also show the achievement of a good level of ICT skills.

As in other papers presented at previous IATED conferences, we start from the assumption that the use of active learner-centered methotologies at school, especially with the use of pedagogical laboratories, is essential for an effective teaching of mathematics. In order to stimulate the use of active methodologies in future teachers, the pre-service teacher training program at the university of Milano-Bicocca includes compulsory laboratories, that, at least for the math discipline, are implemented according to Problem-based learning (PBL) principles. The design of such laboratories is conceived so to allow for group work and to promote intensive peer interaction between students. The lab class consists of at most 30 students, who are split into smaller groups. Each group is assigned the problem and has to tackle it, trying to provide a solution, being given the necessary amount of time. The tutor walks around the classroom among the desks, observing the students, listening to their argueing and answering their questions when necessary. With the outbreak of the COVID-19 emergency, the laboratories scheduled for April and May 2020 had to be redesigned in order to be remotely held. An analysis of the laboratories realized in the last 10 years has been carried out in order to highlight the essential features that contributed to the effectiveness of such laboratories. Then a deep study was made in order to identify which of the available ICT tools could allow to reproduce these characteristics online. The following features were considered essential: - Group work and cooperation: sharing ideas is the foundation of the learning process in a PBL environment. An online working environment that allowed for a easy interaction had to be selected. Such an environment had to be used both for synchronous and for asynchronous interactions, and for multi-hands cooperative writing of documents. - The role of the problem: in a PBL environment the problem is the source of the learning process. An analysis has been carried out in order to understand which of the problems used in the previous experiences would better exploit the use of ICT. Some of these had to be adapted, e.g. introducing some meta questions, in order to make better use of the new online environment. We will describe in detail the design and implementation of a laboratory on Symmetry held remotely on April and May 2020, highlighting the ICT tools chosen in order to better enhance the laboratory's PBL setting.

As in other papers presented at previous IATED conferences, we start from the assumption that the use of active learner-centered methotologies at school, especially with the use of pedagogical laboratories, is essential for an effective teaching of mathematics. In order to stimulate the use of active methodologies in future teachers, the pre-service teacher training program at the university of Milano-Bicocca includes compulsory laboratories, that, at least for the math discipline, are implemented according to Problem-based learning (PBL) principles. The high number of students requires the implementation in parallel of many laboratories each year. These laboratories can be very different one from the other, thus making the offer for students extremely heterogeneous. Therefore we felt the need to set up a monitoring of the effectiveness of the various laboratories. The evaluation model that we will describe is based on an analysis of the careers of students who have attended these laboratories during the 2017-18 academic year. Data on about 380 students were collected and for each student the grades in university mathematical exams were taken into account. By comparing the students' performance before and after attending the laboratories, we created 4 indices that allow for a ranking of some characteristic each laboratory. Differences in "performance" between the different laboratories emerged. Clearly a quantitative analysis of this type needs further investigation (e.g. we plan to take a similar analysis for the laboratories held in the 2018-19 academic year, as soon as students' grades will be available). However, this kind of analysis lays the groundwork for a subsequent detailed examination of the causes of these performance differences, by looking closely at each lab implementation (e.g. problems chosen for the lab, tutor personal performance, time schedule, number of participants...).

The reflection on the long persistence of unsatisfactory results, has led us to upset the most common idea of fraction: fraction-of-something. To this, we have: (a) corrected the “primitive intuition” of fraction by constructing an “intuitive representation”, (b) set the familiarization of children with fractions as a goal, and (c) practiced the idea of fraction as megaconcept. In our enquiring we have focused on the Pythagorean statement: the comparison is a logos. This latter is like a modern act of mathematisation of the comparison and it is the starting point of our didactic practice. It has the following features: (a) it is imposed, (b) it is a leap, (c) it is elementary, (d) it is an axios, (e) it has the characteristic of resence/absence along the path towards the megaconcept.

Many researches highlight the positive aspects of the use of games in teaching, that, especially in mathematics, can lead to a more effective learning process. In particular, playful environments can better shape those stages of mathematical learning that require familiarization with procedures and too often lead learning into tedious mnemonic efforts by pupils. Since 2008, we have been involved in primary school prospective teacher mathematical training. Most of our classes are devoted to promotion of active learning (exp. PBL) and we stress on the fact that in order to design and direct such kind of activities a flexible mathematical knowlege is essential for the teacher. In our lectures, we analyse games and problems, with particular attention to the potential of the "new technologies". We developped a proposal for a web-based matematical "turnkey" game environment for primary school pupils, using the WIMS system. In widespread use throughout France, the WIMS system (WWW Interactive Multipurpose Server, https://wims.matapp.unimib.it/) is gradually revealing its vitality and effectiveness as a server for publishing interactive learning objects. WIMS was conceived by Xiao Gang and made public in 1998. Installed on a WWW server, it provides standard LMS-like facilities (virtual classes and students' management, discussion forum, repository of interactive documents, exercises, quizzes, ...). Its main strenght is its flexibility and the capability of interacting with sofwares of various kinds. This capability, together with the possibility to use random parameters, allows for the creation of a large variety of exercises with automatic scoring. WIMS's activities can be offered to the user with no need to install any additional software on the local pc (or on any mobile device) other than a standard web browser. From July 2019, in the WIMS distribution an example class "Play with math" is available, with a selection of mathematical games suitable for primary school children. This example class provides a "ready for use" environment for any teacher who wants to try it (and who can therefore adapt the virtual space for his own students so to monitor their progresses). We will illustrate some example taken from this class, so to show the potentiality of WIMS in providing interactive exercises that are significative from the mathematical (and therefore from the teaching of mathematics) point of view. The full example class is available at this url: https://wims.matapp.unimib.it/wims/wims.cgi?lang=en&+module=home&+user=anonymous,1097,anonymous We will also report the first reactions of pre- and in-service teachers that experimented this platform and that have highlighted some its strengths and weaknesesses. Finally we will discuss a few issues that a research on the effectiveness of such games should focus on.

A long tradition of educational research has promoted the use of active learner-centred methodologies at school, especially with the use of pedagogical laboratories. When mathematics is concerned, it seems that PBL (Problem-Based learning) is the most suitable approach for a more effective long-life learning. Nevertheless these recommendations do not seem to find their proper location in school practice, where self-perpetuating "traditional" methods impose themselves. In order to discourage the misuse of such so called "traditional" methods, many authors observe that teacher training should adopt, at least in part, the exact methods that are pleaded in the lectures. This very notion inspires the teacher training program at the University of Milano-Bicocca, during which prospective primary school teachers are required to attend PBL laboratories, that accompany the "theoretical" discussion on the potential of this instructional method. This contribution will present the structure of one of these laboratories and the achievements obtained by the students. The laboratory we will describe is specifically aimed at stimulating the students in the understanding, through their experience, of how PBL really works and to train them to design PBL activities they could use in a maths laboratory once they become teachers. A qualitative analysis of the final essays of 80 students attending the laboratory will be provided with particular attention to: - the ability to observe the activation of their own mathematical knowledge by mean of the PBL approach; - the ability to transpose the lab activities, without distorting their essential mathematical content, to a level suitable for primary school children.

Zero è un numero? Perché 1 + 1 fa 2? Come si fa a decidere se un numero è divisibile per 2? E per 3? Perché non si può dividere per zero? Perché “meno per meno fa più”? Cosa sono le radici quadrate? Che cos’è il numero “pi greco”? I quadrati sono rettangoli? Perché l’area del rettangolo si ottiene facendo “base per altezza”? I ricordi degli argomenti matematici affrontati a scuola ci mettono spesso in difficoltà e ci fanno sentire il bisogno di sciogliere i dubbi rimasti irrisolti. Il volume, rivolto agli studenti di Scienze della formazione primaria e a chiunque desideri rivedere criticamente le proprie conoscenze matematiche, ripercorre “da un punto di vista superiore” i temi di matematica propri della scuola primaria (numeri, forme, misura e previsioni), approfondendo i modi in cui il pensiero matematico si sviluppa, dall’analisi di un problema alla costruzione di una soluzione.

Il nostro contributo ha la finalità di presentare una doppia sperimentazione di tipo laboratoriale iniziata nel precedente anno scolastico che vede coinvolti alcune classi della scuola primaria (II, III e IV) e gli studenti di alcuni laboratori di matematica di Scienze della formazione Primaria dell’Università degli Studi di Milano-Bicocca. Il contenuto di questo laboratorio è un progetto di ripensamento delle attività di familiarizzazione dei bambini della scuola primaria con il concetto di frazione e la proposta di un nuovo approccio alle frazioni per i futuri maestri al fine di favorire una “doppia concettualizzazione” attraverso la sperimentazione di attività su questo tema.

Nella sua "militanza" matematica, Mimmo ha regalato a studenti e colleghi mille cose belle: animazioni virtuali e conferenze, corsi e immagini. Non di tutto è rimasta traccia pubblica. Qui, allora, ci piace dare conto di una delle sue realizzazioni che ancora oggi è a disposizione di tutti.

We will briefly describe the contents of the course in geometry for prospective teachers at University of Milano-Bicocca (from the point of view of the "geometry of transformations") and review the "resources" that can be used (in such teacher training courses, but also in the teachers' classroom activities). In particular, we will focus on electronic resources and we will describe what is available in WIMS.

We wish to introduce WIMS, showing its potentials as a tool for the teaching of mathematics (and beyond)

Il poster presenta l'integrazione di WIMS (WWW Interactive Multipurpose Server) con Shibboleth, realizzata presso l'Università di Milano-Bicocca al fine di gestire l'autenticazione e l'autorizzazione degli utenti istituzionali sulla piattaforma.

Largamente diffuso in Francia, WIMS sta lentamente mostrando la sua vitalità e la sua efficacia come server di pubblicazione di esercizi interattivi. Inizialmente progettato per costruire esercizi di matematica, le sue potenzialità sono poi state sfruttare per molte altre discipline.In questo articolo ci si propone di illustrare le potenzialità di WIMS e darne un esempio di utilizzo come supporto didattico ai corsi erogati nell’ambito del progetto di e-learning dell’Università di Milano-Bicocca.

Although since a long time the University of Milano-Bicocca hosts a mirror site of WIMS, only recently we have experimented a consistent use of WIMS as a support for our mathematics classes. In particularly we have been proposing different WIMS activities within the primary school teacher training program. We will describe such activities, focusing on different aspects: use of WIMS to provide material for self-study activities for the students enrolled in the "traditional" university courses; use of WIMS to provide home assignments for the students enrolled in the "e-learning" courses; use of WIMS' exams facilities for testing students' proficiency.

One of the critical factors for the success of the PBL approach is the selection of suitable problems. Typically problems must be engaging, not of immediate solution and usually should rely on disciplinary competences superior to the one that the teachers are supposed to teach. The purpose of this poster is to show examples of problems we use for our PBL teacher training session.

Starting from the assumption that problem-based learning is an instructional approach suitable for transmitting a real understanding of mathematics, we have to regret that such a methodology is not common in real teaching practice at school, as teachers usually rely on self perpetuating "traditional" methods. We would like to describe experiences conducted by the Center "matematita" (Interuniversity Research Center for the Communication and Informal Learning of Mathematics) in order to support teachers in adopting a problem-based learning approach. These actions include: - training courses for pre-service and in-service teachers; - organizing problem-based mathematical laboratories in school (both in primary school and at a higher level); - designing interactive exhibitions; - organizing web-based mathematical game contests; - publishing books and magazines. It is our believe that the role of University Research is crucial in carrying on synergical actions both on the teachers and on the pupils in order to improve the effectiveness of the teaching of mathematics.

Introduzione al Problem-based learning in matematica e descrizione della progettazione dell'esperienza di didattica per problemi condotta in alcune scuole primarie dal Gruppo di ricerca sull'uso del gioco nella Didattica della matematica del Centro matematita.