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This paper discusses the results coming from a phenomenological study implemented with a sample of 110 Italian preschool teachers about the impact of the Covid-19 on preschool education. We investigate in which way the related pandemic phenomenon changed and is still changing the preschool pedagogical/didactic scenario about Mathematics playing. Through a questionnaire we qualitatively analyze and classify teachers' argumentations highlighting their teaching approaches, their attitudes, their expectations in playing Mathematics in Preschool Education.

Several researches in STEM education research highlight the advantages of an inte- grated approach to these disciplines that relates knowledge and know-how, design and implementation, theoretical and practical problems [5, 4, 6]. In some researches, the effectiveness of these approaches on students conceptual understanding and motivation and has been studied through the use of quantitative analysis tools such as cluster analysis (CLA) [1, 7]. Through CLA it is possible to characterize students analyzing the strategies they deploy to tackle, for example, questionnaires built so as to investigate the lines of reasoning implemented by them when they are proposed with problematic situations. In particular, it is possible to characterize the students in terms of a limited number, m, of typical ways of answering the questionnaire questions [2]. Each student is therefore identified by a binary vector (each component can be 1 or 0) with m dimensions. Cluster Analysis techniques allow students to be grouped into homogeneous groups based on com- mon characteristics and the representation of these groups is ideally referred to an m-sized space. However, for reasons of simplicity and clarity, it is often preferred to perform the representation of groups in three or two dimensional spaces. One of the techniques used for this purpose is Multidimensional Scaling [3] (MDS). It allows the researcher to move from the m-sized space to a space with a smaller number of dimensions that is a function of the initial m-dimensional representation, preserving the global distances between the group elements. However, the application of the MDS methodologies depends strongly on the typology of the initial data and a non-thorough knowledge of the mathematical details at their base can lead to obtaining results that are not reliable and / or of little significance. In this paper we will study a MDS methodology based on Principal Component Analysis, with particular reference to a set of binary data, highlighting how the results obtained through this methodology can be reliable and significant for the researcher in education.

Il paper mette a fuoco gli esiti dell’indagine statistica rivolta agli insegnanti di Ma- tematica di scuola Primaria del contesto italiano, circa l’adozione della Didattica a Distanza (DaD), impiegata durante il periodo di emergenza sanitaria da Covid-19. In una prima parte si inquadra la DaD generale da un punto di vista storico e teorico-pratico e, dopo, il focus diviene l’insegnamento e l’apprendimento della Matematica, attraverso l’e-learning. In una seconda parte viene presentata la ricerca e i risultati ad essa correlati. L’indagine statistica, messa in atto attraverso un questionario, qui interamente riportato, ha avuto l’obiettivo di esaminare le rela- zioni intercorrenti tra molteplici variabili che sono risultate essere rilevanti nel processo educa- tivo-didattico della Matematica. Dallo studio sono emerse molteplici problematiche didattiche, pedagogiche ed epistemologiche legate all’implementazione della DaD, all’“impreparazione” degli insegnanti in riferimento alla DaD e, in generale, all’uso delle tecnologie dal punto di vi- sta didattico. In ultimo, emerge una forte esigenza, percepita dagli insegnanti stessi, di forma- zione sui temi qui indagati e relativa ad una seria implementazione ad hoc delle metodologie e- learning nel contesto scolastico della scuola Primaria, metodologie queste ancora poco note nel panorama italiano.

La consapevolezza dell’importanza di guardare, attraverso una lente di tipo matematico, i disegni prodotti dai bambini, considerandoli come strumento diagnostico per quello che D’Amore et al. (2004) chiamano come il “Campo di Esperienza” Spazio, Ordine e Misura, ha dato spunto a questo contributo che discute una sperimentazione didattica condotta in una sezione di scuola dell’infanzia (5 anni). All’interno della stessa è stato chiesto ai bambini di rappresen- tare dei mattoncini Lego con un disegno, e di descrivere successivamente i propri elaborati rintracciando e argomen- tando in essi eventuali segni di scarto numerico-spaziali rispetto al mattoncino osservato/manipolato, attraverso visio- ni prospettiche di erenti. L’analisi qui discussa ci sembra interessante sia per i ricerca- tori in Didattica della Matematica sia per gli insegnanti di scuola dall’infanzia e primaria, che vogliono indagare il ruolo del disegno come forma espressiva e diagnostica per ap- profondire conoscenze, abilità e competenze matematiche dei loro studenti.

Since several years, politicians, researchers and educators from different countries are debating on possible reasons for the success of Confucian area students in interactional assessment projects (such as PISA). Starting from the study of teaching practices developed in this East area, in Italy we developed an educational research based on the process of cultural transposition (Mellone & Ramploud, 2015). With this term we mean the process of change that develops considering two or more cultural-educational backgrounds with the aim to maintain their differences without “translating” them from one culture to another, but rather highlighting these in order to review their meaning processes and daily use in classrooms. In this article we present two educational experiences developed in a second and fifth grades Italian classroom on the cultural transposition of the pictorial equations approach of the Chinese curriculum inserted in the typical structure of Chinese word problems, called problems with variation (Bartolini Bussi et al., 2013). According to Cai and Knuth (2011) in China this symbolic representation is used since the early school years as a possible support for “constructing” a bridge to the “informal algebra”. In the Russian curriculum it is possible to find an analogous approach called intermediate strategies of graphic representation (Davydov, 1982) referred to similar pictorial equations. Our experience of cultural transposition in the Italian context have shown the potentialities of the use of problems with variation as well as the crucial role of the figural equation.

The research on the intercultural phenomenon in Italy increased with the growing phenomenon of migration, highlighting the problem of cultural diversity and social policies. The school has, in this process, a key role. The goal of a good educational project is to understand that the intercultural requires continual reference to the concrete experiences of the people. Theoretical assumptions of our project are the concepts of “interaction”, “empathy”, “decentralization” and “cognitive transitivity" (Nanni 1998). In this theoretical direction, we present an experimental multidisciplinary mathematical laboratory where students and teachers can understand that a discipline that seems static and rigid is the result of an evolution involving different peoples, with very different, often conflicting, traditions and religions. Multi-ethnicity, which increasingly characterizes our society, is surely an important opportunity of growth and enrichment for people.

Let us start with an important statement: Mathematics and Science teachers do a good, and often an outstanding, job in teaching young people the basic knowledge of their respective fields! It is not the intent of this book to criticize what they do or how they do it. Keeping that in mind, and noting the fact that the teaching content of these fields intersects and overlaps, we observed – and this took us by surprise – that there is hardly any collaboration or consultancy between mathematics and science teachers (or textbook authors). Mathematics teachers often use science contexts in tasks, and science teachers often use mathematics, however they are usually working independently. Science context is often arbitrarily chosen, mathematics used with little regard towards learning. Looking through existing teaching and learning materials, we quickly discovered that these materials, too, were mostly designed by either science, or mathematics educators, and that they do not offer active support or lots of opportunities for collaboration. Being an international team of mathematics and science teacher educators, we set out to improve the situation. We developed materials that are useful for both mathematics and science teachers, materials that are designed to increase the competences in both subjects at the same time, allowing for interdisciplinary learning and for collaboration between science and mathematics teachers, ranging from common lesson planning to team teaching. These materials have been piloted and tested by students, teachers and teacher educators in several countries, as well as reviewed by two education specialists. Based on this feedback, the materials were then revised and brought into their final form. The materials in this book, containing lesson descriptions, work sheets etc., can be used as they are by mathematics and science teachers. They however also can be taken apart and set together in a new and different way, or bits and pieces of them can be used in teaching, as the teachers see fit. They also can be used in teacher training, making science and mathematics teacher trainees aware that working together – now and in their later careers – can improve their experience and the learning of their future students. We hope that with this book we encourage teachers to actively seek collaboration, so regardless whether you are a science or a mathematics teacher, go ahead and join forces with a colleague from the other field!

In this contribution we discuss the application of a quantitative, non-hierarchical clustering method to make sense of the answers that 120 engineering undergraduates students at the University of Palermo, Italy, gave to four open-ended questions on the meaning of the modeling processes in Science. We will show that the use of non-hierarchical analysis allows us to easily separate students into groups that can be recognized and characterized by common traits in students’ answers without any prior knowledge on the part of the researcher of what form those groups would take (unbiased classification).

Il processo che porta alla formazione della conoscenza musicale è tutt’altro che semplice e consta di diversi elementi, apparentemente lontani tra loro. In primo luogo è opportuno evidenziare l’esistenza di una componente primitiva nell’ascolto della musica che lega sensazioni e reazioni emotive a specifiche gamme timbriche e tonali

Our main research goal lies in a proposal to discuss the lack of, and improve, activities in the Italian school curriculum about discrete mathematics, computer algorithms and cryptography, especially for 3rd to 8th grade students. Activities of this kind are missing almost entirely, both in the school programs and in textbooks, despite many agree that they can be really useful to improve both general skills, such as reasoning and modeling, and skills particular to discrete mathematics, such as algorithmic and recursive thinking. A survey among various grades teachers confirmed this. The activity we are going to describe fits into a wider research project. Design research, chosen as the methodology to use, feels quite appropriate, as we are facing a brand new experience in an environment that we need to analyze carefully, i.e. on a local scale, considering all the different elements in the learning environment. The activity exposed in our paper is part of some lessons about sorting algorithms for 5th and 6th graders. From simple ordering to more efficient algorithms (quicksort as an example), we get to sorting networks and other variants. We focus on one single activity which is presented to students as a group task, giving them the rules, but with the goal that they find themselves the algorithm for the solution. The methodology used follows the principles of the Guided Reinvention of Mathematics and of RME (Freudenthal, 1973) and allowed us to highlight one particular and very important aspect discussed with the student: the relation between algorithms called greedy and others which are more group-oriented. A qualitative analysis of the results, through some videos recorded in the classroom, shows, according to Vygotsky’s perspective on the zone of proximal development (Vygotsky, 1981), how children, playing together, realize that some greedy algorithms might never work, if we want to achieve the group success. Some dynamics in which the game cannot finish if they seek to optimize their own result over the group result are shown.

Il presente contributo, attraverso delle riflessioni teorico/sperimentali, prova a mettere in luce atteggiamenti, soddisfazioni, aspettative e paure di bambini di Scuola Primaria e dell'Infanzia relative al loro mondo di vedere la Scuola. Attraverso le loro verbalizzazioni spontanee gli autori hanno cercato di definire un possibile framework teorico capace di teorizzare, seppur in una prima analisi, lo scollamento vissuto in alcuni casi dai bambini intervistati tra le attività di gioco della Scuola dell'infanzia e quelle più formalizzate della Scuola Primaria.

The paper, examining the increasing presence of different culture students in Italy, discusses the didactical problematic of multiculturalism at School. In particular a possible “comparison” between East (China, Korea and Japan) and West mathematical epistemology is presented through a historical and linguistic cultural approaches. Some evidences of analogies and differences between the cognitive styles of Chinese and Italian learners in mathematics are presented as examples to show the complexity of teaching/learning mathematics in multicultural classroom

his paper describes a quantitative method to analyse an open- ended questionnaire. Student responses to a specially designed written questionnaire are quantitatively analysed by not hierarchical clustering called k-means method. Through this we can characterise behaviour students with respect their expertise to formulate explanations for phenomena or processes and/or use a given model in the different context. The physics topic is about the Boltzmann Factor, which allows the students to have a unifying view of different phenomena in different contexts