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In questo lavoro si presenta un modulo didattico dal titolo La Lingua Matematica, rivolto a studenti del primo anno di scuola secondaria di secondo grado.Il modulo è stato presentato in classi di Liceo Matematico, ma può essere proposto anche in altri contesti. Esso ha un duplice obiettivo: introdurre questioni relative a matematica e linguaggio da un lato e introdurre i concetti di teorema e dimostrazione dall’altro. Il percorso ha permesso di introdurre nozioni di aritmetica modulare e ha condotto gli studenti a confrontarsi, elaborare ipotesi, produrre argomentazioni e generare nuovi teoremi.

The COVID-19-induced distance education was perceived as highly challenging by teachers and students. A cross-national comparison of five European countries identified several challenges occurred during the distance learning period. On this basis, the article aims to develop a theoretical framework and design requirements for distance and online learning tools. As one example for online learning in mathematics education, the ASYMPTOTE system is introduced. It will be freely available by May 2022. ASYMPTOTE is aimed at the adaptive and synchronous delivery of online education by taking a mobile learning approach. Its core is the so-called digital classroom, which not only allows students to interact with each other or with the teacher but also enables teachers to monitor their students’ work progress in real time. With respect to the theoretical framework, this article analyses to what extent the ASYMPTOTE system meets the requirements of online learning. Overall, the digital classroom can be seen as a promising tool for teachers to carry out appropriate formative assessment and—partly—to maintain personal and content-related interaction at a distance. Moreover, we highlight the availability of this tool. Due to its mobile learning approach, almost all students will be able to participate in lessons conducted with ASYMPTOTE.

The objective of this work is to show an educational path for combinatorics and graph theory that has the aim, on one hand, of helping students understand some discrete mathematics properties, and on the other, of developing modelling skills through a robust understanding. In particular, for the path proposed to middle-school students, we used a connection betweenk-permutations and colourings of graphs: we indicated a way to solve problems related to counting all the possible arrangements of given objects in ak-tuple under given constraints. We solve this kind of problem by associating a graph with the constraints related to thek-tuple and by using graphs’ colourings, in which every colour is associated with one of the objects. The number of arrangements is given by finding the number of colourings through an algorithm called the Connection-Contraction Algorithm. The educational path is set within the Teaching for Robust Understanding framework and the goal, from the mathematical skills perspective, is to enhance modelling, passing from real situations (the fish problem in our experiment) to mathematical problems (the graph’s colouring in our experiment) and vice versa through the use of technology (the Connection-Contraction Algorithm with yEd editor, in our experiment), by using an extended modelling cycle. The meetings with students were videotaped and some results of the experimentation are given.

Il 20 febbraio 2018, a Catania, è mancato improvvisamente Biagio Micale, Professore ordinario di Matematiche complementari all’Università di Catania. Avrebbe compiuto 72 anni il 16 novembre 2018. È stato membro della Commissione Italiana per l’Insegnamento della Matematica e presidente della sezione Mathesis di Catania. Ha coordinato il Nucleo di Ricerca Didattica che opera presso il Dipartimento di Matematica e Informatica dell’Università di Catania. L’attività scientifica ha riguardato vari aspetti della matematica. In particolare nel periodo che va dagli studi post-laurea al 1986 ha indirizzato la ricerca principalmente verso questioni di fondamenti della matematica riguardanti soprattutto le algebre universali, in particolare gli \( \Omega \)-gruppi di Higgins, e i sistemi algebrici. A partire dal 1987 ha arontato temi di ricerca riguardati la Combinatorica, in modo particolare i \( t \)-design (sistemi di quadruple di Steiner, sistemi di terne e di quaterne orientate). Dal 1978 ha sviluppato un’ampia attività di ricerca riguardante tematiche di fondamenti della matematica e di didattica della matematica, con particolare riguardo per la geometria. Relativamente alla didattica della matematica, sviluppa una organica ricerca prevalentemente indirizzata su problemi riguardanti la didattica della geometria nelle scuole secondarie in relazione alle innovazioni contenute nei programmi di insegnamento in vigore o in via di sperimentazione. In modo particolare dà vita ad una serie di ricerche su problematiche didattiche legate allo sviluppo del tema sulle trasformazioni geometriche, avendo come obiettivo unitario quello di innestare e amalgamare tale tema con la tradizione “euclidea” dell’insegnamento della geometria nella nostra scuola.

In this paper, we examine the impact of a massive open online course (MOOC) in the context of outdoor mathematics on the participating teachers’ professional development. We firstly introduce the theoretical background on outdoor mathematics, focusing on math trails with the digital tool MathCityMap and professional development to be accomplished using MOOCs. By taking into account the MOOC “Task Design for Math Trails”, with 93 finalists, we analyze the learning progress of 19 selected case studies from different nations and learning levels by taking into account their answers in a pre- and post-questionnaire and their posts on a specific communication message board, with a special focus on the MOOC’s topics’ task design for outdoor mathematics and the digital tool MathCityMap. The analysis is performed using different quantitative and qualitative approaches. The results show that the teachers studied have benefited from professional development, which is evident in the expansion/evolution of their knowledge from a content, pedagogical, and technological perspective. Finally, we formulate consequences for professional development in STEM education, and conclude the paper with limitations to be drawn and a perspective for further research.

The COVID-19 pandemic not only impacted people’s lives globally, but also pushed faculty to quickly adapt to an online teaching environment and continue it until the end of the school or academic y...

In 1905, at the meeting of the Deutschen Mathematiker-Vereinigung in Meran, the “Meraner Lehrplan,” a mathematics syllabus, was proposed. This document, which contains many of Felix Klein’s ideas on teaching geometry in school, proposed approaching geometry via intuitive geometry, which is the ability to see in space, in order to provide elements for both interpreting the real world and developing logical skills (see Treutlein in Der geometrische anschauungsunterricht als unterstufe eines zweistufigen geometrischen unterrichtes an unseren hoheren schulen. Teubner, Leipzig/Berlin 1911). Klein’s ideas still hold today: An intuitive approach to geometry can be facilitated using information technology. Some activities related to a space geometry approach based on the analogy among figures and on the use of a dynamic geometry system will be presented in this chapter.

The focus of this research is how Sicilian state university mathematics professors faced the challenge of teaching via distance education during the first wave of the COVID-19 pandemic. Since the pandemic entered our lives suddenly, the professors found themselves having to lecture using an e-learning platform that they had never used before, and for which they could not receive training due to the health emergency. In addition to the emotional aspects related to the particular situation of the pandemic, there are two aspects to consider when teaching mathematics at a distance. The first is related to the fact that at university level, lecturers generally teach mathematics in a formal way, using many symbols and formulas that they are used to writing. The second aspect is that the way mathematics is taught is also related to the students to whom the teaching is addressed. In fact, not only online, but also in face-to-face modality, the teaching of mathematics to students on the mathematics degree course involves a different approach to lessons (as well as to the choice of topics to explain) than teaching mathematics in another degree course. In order to investigate how the Sicilian State university mathematics professors taught mathematics at distance, a questionnaire was prepared and administered one month after the beginning of the lockdown in Italy. Both quantitative and qualitative analyses were made, which allowed us to observe the way that university professors have adapted to the new teaching modality: they started to appropriate new artifacts (writing tablets, mathematical software, e-learning platform) to replicate their face-to-face teaching modality, mostly maintaining their blackboard teacher status. Their answers also reveal their beliefs related to teaching mathematics at university level, noting what has been an advantageous or disadvantageous for them in distance teaching.

In this paper, we use geometric transformations to find some interesting properties related with geometric loci. In particular, given a triangle or a cyclic quadrilateral, the locus generated by the centroid or by the orthocentre (for triangles) or by the anticentre (for cyclic quadrilaterals) when one vertex moves on the circumcircle of the figure are considered. These loci are studied in paragraphs 3 and 4. By means of the homothetic transformations some properties of triangles and quadrilaterals are found. The study of these properties can be used, with profit, in a classroom activity supported by Dynamic Geometry System.

In this paper, we show some properties of centroids of geometric figures, such as triangles, quadrilaterals and tetrahedra. In particular, we will prove the properties by means of geometric transformations and by introducing extensions of triangles and quadrilaterals, i.e. by adding one, two or three new vertices to the figure. The study of these properties can be used, with profit, in a classroom activity supported by a dynamic geometry system.

In this paper we present an experimental teaching activity conduced in some primary, middle and high schools in Sicily. The activity concerned several topics of graph theory. Here we highlight, in particular, the approach to teaching Eulerian graphs. The aim of the whole project was to present a fun, easy approach to mathematics in order to promote a good attitude towards mathematics in primary school children and to improve it in middle school kids and in high school young people. This goal is pursued also by showing some connections of mathematics with real life, making mathematics less abstract than the topics too often taught in school. Through this activity we also reach mathematical knowledge and practical abilities (related to graph theory), and above all mathematical competencies related to reasoning and mathematization, in particular by the use of graphs in mathematical models to solve problems. The teaching experiments were different, according to the different school level, but unified by the method, based on laboratorial activities, by presenting a problem to be solved together with classmates, by manipulating objects and guided by the teacher. These activities were realized by the use of artefacts: in the sense of Vygotskijan semiotic mediation, we used signs, symbols, maps, language and, in many cases, new technology’s artefacts, to mediate mathematical concepts. Lessons involved also the body as a mean to learning, especially with children, according to embodied cognition theory.

In this chapter we present an experimental activity conducted with eight teachers in several high-schools in the South of Italy. The activity deals with the study of some geometric loci and is based on the use of a Dynamic Geometry System (DGS) on the one hand and on properties of geometric transformations on the other hand. Here we explain the motivations at the base of our activity and show the contents, the modalities of building up the activity and the results of the teaching experiment.

In this paper, we present the results of an experimental classroom activity, history-based with a phylogenetic approach, to achieve algebra properties through geometry. In particular, we used Euclidean propositions, processed them by a dynamic geometry system and translate them into algebraic special products.

We present a sequence of classroom activities on Euclidean geometry, both plane and space geometry, used to make three dimensional geometry more catchy and simple. The activity consists of a guided research activity that leads the students to discover unexpected properties of two apparently distant geometrical entities, quadrilaterals and tetrahedra. The activity has been realised by means of an efficacious conceptual tool, the analogy, and an operative one, a dynamic geometry system.

Let Q be a convex quadrilateral ABCD. We denote by TA, TB, Tc, TD, the four triangles BCD, CDA, DAB, ABC, respectivelyThe barycentres (or centroids), orthocentres, incentres and circumcentres of such triangles determine other quadrilaterals in their turn that we call the quadrilateral of the barycentres, of the orthocentres, of the incentres and of the circumcentres, respectively. We denote these quadrilaterals by Qb, Q0, Qi, Qc, respectively.

Kažemo da je lik F poseban za trokut T ukoliko F može biti dobiven iz T korištenjem ravnala i šestara neovisno o redoslijedu vrhova od T. U ovom radu razmatra se postojanje trokuta s danim posebnim likom koji je točka, pravac, kružnica ili trokut., A figure F is said to be special for a triangle T if F can be determined from T by a ruler and compass construction, that does not depend on the order of vertices of T. In this paper, we deal with the problem of the existence of triangles with a given special figure that is a point, or a line, or a circle, or a triangle.

The station location problem consists of placing new stations along the railway tracks of an existing network in order to increase the number of users. In this paper we consider the problem for a railway network consisting of two intersecting lines forming an angle α. An approach for solving the problem in polynomial time for sufficiently large angles α is presented.

In this paper, we use a property of the centroids to study two general types of geometric loci: one in the plane and one in the space. Then from this result, we examine in detail the case of quadrilaterals and tetrahedra and, in particular, we determine some properties of the nine-point circle of a triangle. The study of those geometric loci is carried out by means of the homothetic transformations that facilitate and make more immediate the proofs of the properties.

In this note we show the existence of a surprising analogy between quadrilaterals and tetrahedra, determining some properties that hold for both families of figures. This analogy suggests some new properties of quadrilaterals.